text:
\section{Introduction}
In this article we study the structure of the invariant measures for typical continuous maps of a $C^1$ compact, connected manifold $M$ of finite dimension $m \geq 1$, with or without boundary.    This generalizes the work in our previous
article \cite{CT} where we studied the case when $M$ is an interval.   The study of the invariant measures for typical maps was
initiated recently by Abdenur and Andersson \cite{AA}. Previously studies of the dynamics of typical maps have concentrated on the  topological properties; see \cite{AHK,AP,H1,H2,Oprocha&als,O,OU,PP,Y} and the references therein.

Let $\mathcal E_f$ denote the set of ergodic, $f$-invariant Borel probability measures,
$\mbox{Per}_f$ the set of invariant measures supported on a single  periodic orbit, and $\mathcal O_f$ denote the set of pseudo-physical measures for $f$, (see subsection \ref{sec-pseudophysical}
for the definition).
We always have $\mbox{Per}_{f} \subset {\mathcal E}_f$.
Let $\mathcal C(M)$ denote the set of continuous maps of $M$ to itself and $\mathcal H(M)$ denote the set of homeomorphisms of $M$ to itself.
 We endow these spaces with the $C^0$ topology and
 say that a family of maps in $\mathcal C(M)$ (resp.\ in $\mathcal H(M)$) is typical if it contains a countable intersection of open and dense family.
Our main result  is the following theorem:

\begin{theorem}\label{Theorem-main}
If $f$ is   typical   in $\mathcal C(M)$
(resp.\ $\mathcal H(M)$), then
$$ \overline{ \mathcal E_f }  =\overline{ \mbox{Per}_f} =  \mathcal O_f. $$

\end{theorem}

The following questions arise from  Theorem \ref{Theorem-main}.   Are all the invariant measures  in the closure of the ergodic measures? Are  all the invariant measures   pseudo-physical? We prove that the answer is negative for $C^0$-typical maps:

\begin{theorem}\label{Theorem-main2}
If $f$ is  typical  in $\mathcal C(M)$  or if $f$ is   typical    in $\mathcal H(M)$, then the set
$  \mathcal O_f $  has empty interior in the set of all $f$-invariant measures.\end{theorem}

In particular an open and dense set of $f$-invariant measures is not in the closure of the ergodic measures.
Theorem \ref{Theorem-main2} implies that the typical behaviour of homeomorphisms on manifolds widely differ from the typical behaviour of $C^1$ diffeomorphisms. In fact,
Gelfert and Kwietniak proved that for $C^1$-typical diffeomorphisms the set of ergodic measures is dense in the space of invariant measures \cite[Theorem 8.1]{Gelfert-K}.

\vspace{.3cm}

The proof of Theorem \ref{Theorem-main} is split  into several pieces; its main components are  Theorems \ref{TheoremPeriodicSRB-l},
 \ref{theoremBigcapClosShr_q&SRBl-l}, \ref{TheoremSRB-l=ClosureErgodic} and
Corollary \ref{corollaryOprocha&als}.
The main components of the proof of Theorem \ref{Theorem-main2} are Theorems \ref{theoremLebesgue-a-e-isInShrinkingInterval} and \ref{TheoremAddedNonUniqErgodic}, Proposition \ref{propositionAA_1}, and Lemma \ref{Lemma-ConvexCombination}.

{The article is organized as follows.  In Section \ref{sec2} we give some background material and prove a technical lemma on the approximation of measures. In Section \ref{sec3} we define the notion of shrinking sets, and
we prove an extension of  a result of Abdenur and Andersson, as well as some consequences of this result.
We prove Theorem \ref{Theorem-main2} in Section \ref{sec4}. Finally,
Sections \ref{sec5} and \ref{sec6} are dedicated to the proof of
 Theorem \ref{Theorem-main}.}

\section{The set-up}\label{sec2}
Let $M$ be a compact, connected, $C^1$ manifold of finite dimension $m \ge 1$, with or without boundary.
  Once  a Riemannian structure is chosen on $M$, it defines a volume measure  which we will refer to as the Lebesgue measure. It also defines a distance,  $dist(\cdot,\cdot)$, between points. We denote  the $C^0$ distance on $C(M)$ by
  $$\rho(f,g) :=\max_{x\in M} dist (f(x), g(x)),$$
 {This metric makes the set $C(M)$ a complete metric space and generates the $C^0$ topology on $C(M)$.
 Similarly, $\mathcal H(M)$ is a complete  metric space with the $C^0$ topology generated by  the distance  $$\rho_{H} :=\max\{\rho(f,g),\rho(f^{-1},g^{-1}) \}.$$
}

The results of Theorems  \ref{Theorem-main} and \ref{Theorem-main2} hold independently of the choice of the Remannian structure on $M$. In fact,
the topological and Borel-measurable properties of the system   depend  on the  metrizable topology of the manifold $M$, but not on the particular choice
of its metric $dist$.  The properties related to  the pseudo-physical measures   depend only on the set of   pseudo-physical measures inside
the space of all the Borel invariant measures. But  this set  is preserved if  we substitute  the previously chosen Lebesgue measure by any finite Borel measure
equivalent to it (see Definition \ref{definitionpseudo-physical}). Hence, the results remain unchanged if we change the choice of  the volume form.

\subsection{Pseudo-physical measures}\label{sec-pseudophysical}

For any point $x \in M$ and $f \in C(M)$,  let  $p\omega(x)$  be the set of the Borel probability measures on $M$
that are the limits in the weak$^*$ topology of the convergent
subsequences of the sequence
$$\left \{  \frac1n \sum_{j=0}^{n-1} \delta_{f^jx} \right\}_{n \in \mathbb{N}}$$
where $\delta_y$ is the Dirac probability measure supported in $y \in M$.

 A measure $\mu$  is called \emph{physical} if the set of those $x \in M$ for which
$p\omega(x) = \mu$ has positive Lebesgue measure. Note that we do not require physical measures to be ergodic.

In this article we consider a generalization of the above definition, introduced in \cite{CE1} and studied in the $C^1$ case in \cite{CE2}, \cite{CCE1}, \cite{CCE2}.
We fix a distance, $d(\cdot,\cdot)$,  in the space of probability measures that endows  the weak$^*$ topology. It is easy to check that the following definition does not depend on the choice of this distance (see \cite{CE1}).

\begin{definition} \em \label{definitionpseudo-physical}
A probability measure $\mu$ is called \em pseudo-physical  \em if for all $\varepsilon > 0$ the set $A_\varepsilon(\mu) \mathrel{\mathop:}= \{x \in M: d(p\omega(x),\mu) <  \nolinebreak
\varepsilon\}$ has positive Lebesgue
measure.  We denote by $\O_f$   the set of  pseudo-physical measures for $f$.

Note that $\O_f$ is always closed and non-empty, and that any pseudo-physical measure is automatically $f$-invariant, and   we do not require a pseudo-physical measure to be ergodic. (In \cite{CE1,CE2,CCE1} pseudo-physical measures were called SRB-like.)
\end{definition}

\subsection{The simplexes }\label{sec-setup}

We consider an   atlas $\{(U_\alpha,\phi_\alpha)\}$ of the manifold $M$, where $U_\alpha$ are open sets whose union covers $M$, and
$\phi_\alpha : U_\alpha \mapsto \mathbb{R}^m$ are $C^1$ diffeomorphisms.

An Euclidean \emph{$k$-simplex} ($0 \le k \le m$) is the convex hull of a finite set $\{x_0, x_1,\dots,x_k\} \subset \mathbb{R}^m$ such that $\{x_j - x_0: 1 \le j \le k\}$ are
linearly independent.

A \em $k$-simplex \em of $M$ is a nonempty compact set $\bar T$ contained in a chart $(U_{\alpha},\phi_{\alpha})$ such that $\phi_\alpha(\bar T)$ is  an Eucliean $k$-simplex of $\mathbb{R}^m$.   Throughout the article $T$ will denote the nonempty interior of an $m$-simplex $\overline{T}$.

$M$ is \em triangulable \em if there exits a finite family $\{\bar T_1, \dots , \bar T_\ell\}$  of $m$-simplexes (called a {\em triangulation}) such that $T_i \cap T_j = \emptyset$ if $i \ne j$ and $\cup_i \bar T_i = M$.

It is well known that any compact $C^1$ manifold is triangulable \cite{W},\cite{M}.
For any $\varepsilon > 0$ there exists a triangulation of $M$ such that the diameters of all the simplexes $\bar T_i$ are smaller than $\varepsilon$. To prove this it suffices to notice that
an Euclidean $m$-simplex can be decomposed into a finite number of $m$-simplexes of arbitrarily small diameter.

Consider a triangulation $\mathcal T := \mathcal T_m := \{ \bar T^m_1,\dots, \bar T^m_{h_m}\}$ of $M$.
Define
$\partial \mathcal T$ to be  the union of the topological boundaries  $\partial \bar T$  of the simplexes $\bar T$ of  $\mathcal T$.

Consider an  $m$-simplex $\bar T$ of $M$ and its associated chart $(U_\alpha,\phi_\alpha)$.  Let $ c $ be a point in the interior of $ \bar T$, which we will call \lq\lq centroid\rq\rq. Let $\{x_0, x_1,\dots,x_m\} \subset \mathbb{R}^m$ be the vertices of  $\phi_{\alpha}(\bar T)$. Up to translation of the coordinates, it is not restrictive to assume that
   $\phi_{\alpha}(c)$ of $\phi_{\alpha}(\bar T)$ is located at the origin. Suppose $\lambda > 0$ is not too large. We denote $\lambda \bar T  \subset U_\alpha$ the new simplex such that $\phi_\alpha (\lambda \bar T)$ is the Euclidean $m$-simplex with
vertices $\{\lambda x_0, \lambda x_1,\dots,  \lambda x_m\} \subset \phi_\alpha(\bar U_\alpha)$. Clearly, there exists a $\lambda_{0} > 1$ (depending on $\bar T$  and the chosen centroid $c$) such that $\lambda \bar T$
is well defined for all $\lambda \in (0,\lambda_{0})$.

\begin{figure}[h]
\begin{tikzpicture}
\centering
\draw[thin] (-1,-1) -- (1,-1) -- (0,0.73) -- (-1,-1);
\draw[densely dotted] (-0.45,-0.5) -- (0.55,-0.5) -- (0.05,0.36) -- (-0.45,-0.5) ;
\draw[black,fill=black] (0.1,0) circle (.2ex);
\end{tikzpicture}
\caption{The simplex $\bar T$ is drawn with a solid line, while the simplex $\lambda \bar T$ for a $\lambda =1/2$ is drawn with a dotted line. The solid dot   is
the chosen centroid of the triangle located at the point $c \in T = \mbox{int}(\bar T) \subset U_\alpha$.}\label{fig-centroid}
\end{figure}

\subsection{A technical lemma}\label{s2}

In the metrizable space of probability measures endowed with the weak$^*$ topology, consider the following distance between probabilities $\mu, \nu$:
$$d(\mu, \nu) := \sum_{i=1}^{\infty} \frac{1}{2^i}\left |    \int \Psi_i \, d\mu - \int \Psi_i \, d\nu \right |, $$
where $\{\Psi_i\}_{i=1}^{\infty}$ is a countable dense family in the space $\mathcal{C}^0(M,[0,1])$.

We will use the following lemma  on the approximation of measures.

\begin{lemma}\label{lemma-dist}
For any   $\varepsilon >0 $, there exists  $q \geq 1$ such that,
if $\mu$ and $\nu$ are  probability measures of $M$ satisfying
$supp (\nu) \cup supp (\mu) \subset \cup_{j=1}^m I_j$ for some pairwise disjoint connected closed sets,  if $\mbox{diam}(I_j) \leq 1/q$, and if
$|\nu(I_j) - \mu(I_j) | \le 1/qm$ for each $j$,
then $d(\mu, \nu) < \varepsilon$.
\end{lemma}

\begin{proof}
 Fix $n \geq 1$ such that $\sum_{i= n+1} ^{+ \infty} 2^{-i} < {\varepsilon}/{2}.$ Then, for any pair of probability measures $\mu, \nu$ we obtain
 \begin{equation}
 \label{eqn01}
 d (\mu, \nu) < \frac{\varepsilon}{2} +  \sum_{i= 1}^n   \frac{1}{2^i} \left |    \int_M \Psi_i \, d\mu - \int_M \Psi_i \, d\nu \right |.
 \end{equation}
The uniform continuity of the finite family of functions $\{\Psi_i\}_{1 \leq i \leq n}$, implies that there exists $\delta > 0$ such that, if  $dist(x_1,x_2) < \delta$, then $dist(\Psi_i(x_1) , \Psi_i(x_2))< \varepsilon/4$ for all $1 \leq i \leq n$.
Fix $q  \in \mathbb{N}^+$  such that
  $ {1}/{q} <  \min (\delta, \varepsilon/4) $.

The mean value theorem for integrals, yields for all $1 \leq i \leq n$
\begin{equation*}
  \begin{split}
   \int_{\bigcup_{j=1}^m I_j}\Psi_i \, d\mu &= \sum _{j= 1} ^m \int_{I_j} \Psi_i  \, d \mu = \sum _{j= 1} ^m\Psi_i(x_j) \mu (I_j) \mbox{ for some } x_j \in I_j,\\
    \int_{\bigcup_{j=1}^m I_j}\Psi_i \, d\nu & = \sum _{j= 1} ^m \int_{I_j} \Psi_i  \, d \nu = \sum _{j= 1} ^m\Psi_i(x'_j) \nu (I_j) \mbox{ for some } x'_j \in I_j.
  \end{split}
  \end{equation*}
\nopagebreak[4]
From this we deduce
$$
 \int_{\bigcup_{j=1}^m I_j}\Psi_i \, d\mu - \int_{\bigcup_{j=1}^m I_j} \Psi_i \, d\nu     = \sum _{j= 1} ^m \left (\Psi_i(x_j) \mu (I_j) -\Psi_i(x'_j) \nu (I_j) \right ) $$ $$
     =\sum _{j= 1} ^m  \left (\Psi_i(x_j) (\mu (I_j) -  \nu (I_j) ) +  (\Psi_i(x_j)   -\Psi_i(x'_j) ) \nu (I_j) \right ).
$$
Since  $|\nu(I_j) - \mu(I_j) | \le 1/qm$ for each $j$, we conclude
$$ \left |\int_M\Psi_i \, d\mu - \int_M \Psi_i \, d\nu \right | \leq \sum _{j= 1} ^m  \frac{1}{qm}    +  \frac{\varepsilon}{4} \sum _{j= 1} ^m   \nu (I_j) < \varepsilon/2 \ \ \ \forall \ 1 \leq i \leq n.$$
Substituting this inequality  in (\ref{eqn01}), finishes the proof of Lemma \ref{lemma-dist}.
\end{proof}

\section{Shrinking sets}\label{sec3}
 A  \em periodic  shrinking set of period $p \geq 1$,  \em is a nonempty open set $I$ whose closure $\bar I$ is an $m$-simplex, such that  $\{f^j( \bar I)\}_{0 \leq j \leq p-1}$ is a family of pairwise disjoint sets,  $f^p(\bar I) \subset I$, and   $\mbox{diam} (f^j(I) ) <  \mbox{diam}(I)$ for all $1 \leq j \leq p-1$.

An nonempty open set $J$, whose closure $\bar J$ is an $m$-simplex,  is \em eventually periodic  shrinking,  \em if there exists a periodic  shrinking set  $I$ and a {\em transience time}
$n \in \mathbb{N}^+ $ such that  $ f^n(J) \subset I$, and $\mbox{diam}(f^j(J)) <  \mbox{diam}(J)$ for all $1 \leq j \leq n-1$.

Note that the same periodic or eventually periodic  shrinking set   for some $f \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$)  is also a periodic or eventually periodic  shrinking set  for all $g$ in a small neighborhood of $f$ in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$).

If $I$ is a periodic or eventually periodic shrinking set  with period $p$, then,
by the Brouwer fixed point theorem, the map $f^p$ has a fixed point $x_0 \in I$; hence, the point  $x_0$  is  periodic for $f$  with period $p$.

\begin{theorem} \label{theoremLebesgue-a-e-isInShrinkingInterval}
For a typical map  $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M))$,  Lebesgue a.e.\ point $x \in M$ belongs to a sequence  $\{I_q\}_{q \in {\mathbb N}^+}$ of  eventually periodic  shrinking sets $I_q$ such that   $\mbox{diam}(I_q) < 1/q$.

\end{theorem}

\begin{proof}
For  given natural numbers $q,k \in \mathbb{N}^+$, denote by ${\mathcal S}_{q,k}$  the set of maps in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$), for which there exists a  finite family of nonempty open sets,
which we denote by $\{  I_ {1}, \ldots,   I_l\}$,     such that:

\noindent (i)  $diam(I_i) < 1/q$ for   $i= 1, 2, \ldots, l$,

\noindent (ii) $I_i$ is a periodic or eventually periodic shrinking set,

\noindent (iii)  $Leb (M \setminus \bigcup_{i= 1}^l I_i) < 1/k$.

The set ${\mathcal S}_{q,k}$ is open   in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$) since, for each $f \in {\mathcal S}_{q,k}$, the same family of such shrinking sets of $f$, is also a family of shrinking sets satisfying conditions (i), (ii), (iii) for any other map $g$ in a sufficiently small neighborhood of $f$.

Now, let us prove that  ${\mathcal S}_{q,k}$ is dense in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$). This will complete the proof
of Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}, since the set
$${\mathcal S} \mathrel{\mathop:}= \bigcap_{q \ge 1}\bigcap_{k  \ge 1} {\mathcal S}_{q,k}$$
is a dense $G_\delta$-set, and by  construction of ${\mathcal S}_{q,k}$, any map in ${\mathcal S}$ satisfies the conclusion of the  Theorem.

To prove the denseness of ${\mathcal S}_{q,k}$
fix any map $f \in \mathcal C(M)$ (resp.\ $\mathcal H(M))$. For any   $\varepsilon >0$, we will construct  a perturbation $g \in {\mathcal S}_{q,k}$ such that $\rho(f,g) <\varepsilon$  (resp.\  $\rho_{H} < \varepsilon$).

Choose  $0<\delta  < \min\{\varepsilon, 1/q\}$  such that $$dist(x,y) < \delta \ \ \Rightarrow \ \ dist(f(x) , f(y))< \frac{\varepsilon}{3},$$ and if besides $f \in \mathcal H(M)$, then also $ dist(f^{-1}(x), f^{-1}(y)) < \frac{\varepsilon}{3}$.
Consider a triangulation $\mathcal T := \{\overline{T}_1,\dots,\overline{T}_l \}$ of $M$ such that the diameters of all the simplexes $\overline T_i$ are smaller than $\delta$.
For each $i$ we would like to choose a point  $x_i \in T_i = \mbox{int}(\bar T_i)$ such that  $f(x_i) \in T_{j(i)} = \mbox{int}( \bar T_{j(i)})$ for some $j$ which depends on $x_i$.
 If $f$ is an homeomorphism the set of points in $T_i$ whose image is not in $\partial \mathcal T$ is
a non-empty open set, and so we can choose such points $x_i$.  If $f$ is merely continuous it is possible that such points do not exist. Then, we
perturb $f$ to a continuous map $f_1 \in \mathcal C(M)$ satisfying this property such that $\rho(f,f_1) < \varepsilon/{12}$ and such that
$dist(x ,y) < \delta \ \Rightarrow dist(f_1(x) , f_1(y)) < \varepsilon/2$.  For the sake of consistency of notation,
if $f \in \mathcal H(M)$  or if $f \in \mathcal C(M)$ already satisfied this requirement we set $f_1 := f$.

Now we define ``spherical coordinates'' on each $\bar T_i$. To do this consider a chart $(U_\alpha,\phi_\alpha)$ such that $\bar T_i \subset U_\alpha$.
Consider $S_i := \phi_\alpha(\bar T_i)$. Up to  a translation of the coordinates, it is not restrictive to assume that $0= \phi_\alpha(x_i)$.
Recall that $x \in T_i = \mbox{int}(\overline T_i)$; hence
      $0 \in \mbox{int}(S_i)$. Therefore, there exists
a small solid $m$-dimensional ball $B(0,r_0)$  lying  inside $S_i$, and the spherical coordinate
system $(r,\theta): r\in [0,r_0], \theta \in \mathbb S^m$ in it.  Using the convexity of $S_i$ we can extend this coordinate system to all of $S_i$ by defining
$R(\theta)$ to be the supremum of all $r$ such that the ray  in the direction $\theta$ lies inside $S_i$ up to the distance $r$.
For simplicity of notation, we normalize the $r$ coordinate, by setting $s := r/R(\theta)$, thus $s \in [0,1]$ for all $\theta$.
Finally we  pull back these
coordinates to $\bar T_i$ via the map $\phi^{-1}_\alpha$.
For any $i$ and any $s_0 \in (0,1]$ we will denote
$$B^i_S :=  \{(s,\theta) \in \bar T_i: s \in [0,s_0]  \}.$$

We consider an integer number $n \geq 1$ to be specified later.  We define an homeomorphism $h$ of $M$ as follows.
$h: \bar T_i \mapsto \bar T_i$ is given by
  by setting
  $$h(s,\theta) \mathrel{\mathop:}=  ( s^{n}  ,\theta  ).$$
Note that the map $h$ is uniquely defined on $\partial \mathcal T$, because for all $i$ the definition of $h$ on $\partial \bar T _i$ yields the identity map.
Furthermore, note that since $h$ and $h^{-1}$ map each simplex $\bar T_i$ to itself, and the diameters of the $\bar T_i$ are at most $\delta$ we have
$$dist(x,h(x)) \le  \delta \quad \mbox{ and } \quad dist(x,h^{-1}(x)) \le \delta.$$
The   map $g$ is then defined by
$$g = f_1 \circ h.$$
We claim that, for an adequate choice of $n$, the map $g$ has the desired properties, i.e., $g \in {\mathcal S}_{q,k}$ and $\rho(g,f) < \varepsilon$ (and $\rho(g^{-1},f^{-1}) < \varepsilon$ in the homeomorphism case).

Consider $y \in M$. Since $dist(h(y),y) < \delta$ we have
$$dist(g(y),f_1(y)) = dist(f_1(h(y)),f_1(y)) < \frac{\varepsilon}{2}$$
and thus
$$\rho(f,g) < \rho(f,f_1) + \rho(f_1,g) < \varepsilon.$$

In the case that $f = f_1 \in \mathcal H(M)$, we have  $g^{-1} = h^{-1} \circ f_1^{-1}$.
Consider $y \in M$, let  $z = f^{-1}(y) = f_1^{-1}(y)$ , then
$$dist(g^{-1}(y),f ^{-1}(y))  =dist(g^{-1}(y),f _1^{-1}(y)) =  dist(h^{-1}(z),z) \le \delta < \varepsilon$$
and thus
$$\rho(g^{-1} , f^{-1}) < \varepsilon.$$

Now, to end the proof of Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}, it is enough to choose $n$ such that the map $g$ above constructed belongs to ${\mathcal S}_{q,k}$.

Choose $x_i$ as the centroid of $T_i$,   choose $\lambda_1 \in (0,1)$ close to 1,  and $\lambda_2 \in (0, \lambda_1)$ close to 0, and construct the simplexes $\lambda_2 \bar T_i \subset \lambda_1 \bar T_i \subset T_i = \mbox{int}(\bar T_i)$  whose interiors contain $x_i$.  If $\lambda_1$ is close enough  1,  we have:
\begin{equation}\label{iii}
Leb\big(M \setminus (\cup_i \lambda_1 \bar T_i)\big) < 1/k.
\end{equation}
Besides, if $\lambda_1 \in (0,1)$ is close enough 1, then $f_1(x_i) \in \mbox{int}(\lambda_1 \bar T_{j(i)})$. So, if $\lambda_2 \in (0,\lambda_1)$ is small enough, we have  $f(x_i) \in f_1(\lambda_2 \bar T_i) \subset \mbox{int}(\lambda_1 \bar T_{j(i)}) $ with $\mbox{diam}  (f_1(\lambda_2 \bar T_i)) <\min_j{\mbox{diam}(\lambda_1 \bar T_j)} $ for all $i$.

We claim that there exists $n$ such that   the sets $\lambda_1 T_i$ are eventually periodic  shrinking sets for $g$. In fact, choose a real number $s_1 > 0$ such that $\lambda_1 \bar T_i \subset B^i_{s_1}$ and
 choose other real number $s_2 \in (0,s_1)$ such that
 $B^i_{s_2}  \subset \lambda_2 \bar T_i .$
Finally choose $n$ such that $s_1^{n} < s_2$. With these choices
we have $h(B^i_{s_1}) \subset B^i_{s_2}$
and thus
$$g(\lambda_1 \bar T_i) = f \circ h (\lambda_1 \bar T_i) \subset
f \circ h(B^i_{s_1}) \subset f(B^i_{s_2}) \subset f(\lambda_2 \bar T_i) \subset  {\lambda_1}  T_{j(i)}.$$
Since there exists only a finite number $l$ of simplexes in the triangulation, this assertion implies that $\lambda_1T_i$ is an  eventually periodic shrinking set with the sum of the transience time plus the period bounded by $l$. By construction, Properties (i), (ii) and (iii) are satisfied.   \end{proof}

When  $M$ is a manifold without boundary, the following result was proven by Abdenur and Andersson in \cite{AA} for typical $f \in \mathcal C(M)$ in any finite dimension and for typical $f \in \mathcal H(M)$ in dimension at least two.
Our proof works also when $M$ is a manifold with boundary, and also for $f \in \mathcal{H} (M)$ in dimension one.

\begin{corollary}
   \label{theoremAAa)}

\noindent Let $f $ be a typical map in $C(M)$ (resp.\ $\mathcal H(M)$).  Then,
for Lebesgue almost every point $x \in M$, the  sequence $$\Big \{\frac{1} {n}  \sum_{j=0}^{n-1} \delta_{f^j(x)}\Big\} _{n \in \mathbb{N}^+}$$ of empirical probabilities is convergent,
\end{corollary}

\begin{proof}
Consider the families ${\mathcal S}_{q,k}$ for $q,k \in \mathbb{N}^+$, defined at the beginning of the proof of Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}. We proved that each ${\mathcal S}_{q,k}$ is open and dense, so typical $f$ belong to $\mathcal S:= \bigcap_{q,k \geq1} {\mathcal S}_{q,k}$. Thus, it is enough that the assertion of this corollary holds for all $f \in {\mathcal S}$.
 Consider a continuous function $\phi: M \to \mathbb{R}$. Since $M$ is compact, for every $\varepsilon > 0$ we can find a $\delta > 0$ such that if $dist(x,y) < \delta$ then
 $|\phi(x) - \phi(y)| < \varepsilon$.
Suppose $f  \in {\mathcal S}_{q,k}$ with  $q$ such that $1/q < \delta$.  Due to Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}, Lebesgue a.e.\ belongs to an eventually periodic shrinking set of diameter smaller than $1/q$.
It is enough to prove the convergence of the corollary for any iterate of $x$ instead of $x$; thus we can assume that $x \in I_i$ where $I_i$ is a periodic   shrinking set.
Denote the period of $I_i$ by $p$. Then for $n > p$ we write $n = \ell p + r$ with $0 \le r < p$. Since $I_i$ is a  periodic shrinking set with diameter smaller than $1/q$ we have
$dist(f^{\ell p + r}x, f^rx) \leq \mbox{diam}f^r(I_i) \leq \mbox{diam}(I_i) < 1/q <\delta$. Thus

\begin{equation}\label{eqcont}\left  |\sum_{j=0}^{\ell p - 1} \phi(f^j x) - \ell \sum_{j=0}^{ p - 1} \phi(f^j x) \right  | < \ell \varepsilon.\end{equation}
Then
\begin{equation}\label{eqconv}
\begin{split}
\Big | \frac1n \sum_{j=0}^{n-1} \phi(f^j x) & - \frac{1}{p} \sum_{j=0}^{p-1} \phi(f^j x) \Big | \le \\
& \Big | \frac1n \sum_{j=0}^{n-1} \phi(f^j x)  - \frac{1}{\ell p} \sum_{j=0}^{\ell p-1} \phi(f^j x) \Big | +\\
&\Big | \frac{1}{ \ell p} \sum_{j=0}^{\ell p-1} \phi(f^j x) - \frac1p \sum_{j=0}^{\ell p - 1} \phi(f^j x) \Big |.
\end{split}
\end{equation}

Using  \eqref{eqcont} we see that second term is bounded by $\varepsilon/p < \varepsilon$.
Using the triangle inequality once more we can bound the first term by
$$ \Big | \frac1n \sum_{j=\ell p}^{n-1} \phi(f^j x)  \Big |+ \Big |  \frac{r}{n \ell p} \sum_{j=0}^{\ell p-1} \phi(f^j x) \Big |. $$
Since $r$ is bounded, for $n$ sufficiently large this is bounded by $\varepsilon$, and  the difference in \eqref{eqconv} is bounded by $2\varepsilon$.

We deduce that
$$\limsup \frac1n \sum_{j=0}^{n-1} \phi(f^j x)-  \liminf \frac1n \sum_{j=0}^{n-1} \phi(f^j x) < 4\varepsilon.$$
Since $\varepsilon > 0$ is arbitrary, the result follows.
\end{proof}

A map $f \in \mathcal C(M)$ is \em Lebesgue-a.e.~strongly non positively expansive, \em  if for any real number $\alpha >0$,   and for Lebesgue a.e.~$x \in M$,
$$Leb\Big (\{ y \in M : dist(f^n(x), f^n(y))< \alpha \ \  \forall \ n \in  \mathbb{N}  \}   \Big) >0 .$$
So, according to \cite[Definition 2.1]{AM}, the Lebesgue measure is non positively expansive. Moreover, using the notation of  \cite[Section 2]{AM}: $$Leb(\Phi_{\alpha}(x)) \neq 0 \   Leb\mbox{-a.e.  } x \in M, \mbox{ where}$$
$$\Phi_{\alpha}(x):=\{y \in M: \mbox{dist}(f^n(y), f^n(x)) < \alpha.$$ Namely, the positive expansivity of the Lebesgue measure fails not only in some points, but   Lebesgue a.e..

\begin{corollary} \label{corollaryStronglyNonExpansive}
Typical  maps in $\mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$)  are Lebesgue-a.e.\ strongly non positively expansive.
\end{corollary}

\begin{proof}

Take any periodic or eventually periodic set $I$  with diameter smaller than $\alpha$. Then,  $\mbox{diam}(f^j(\overline I)) <\alpha$ for all $j \geq 0$. Any two points $x, y \in I$  satisfy $dist(f^j(x), f^j(y)) < \alpha $ for all $j \geq 0$. Thus for any point $x \in I$ we have
$$ Leb\Big (\{ y \in M:  dist(f^n(x), f^n(y))< \alpha \ \  \forall \ n \in  \mathbb{N}  \}   \Big) \geq Leb(I) > 0.$$
Thus the result follows directly from  Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}.
\end{proof}

An $f \in \mathcal C(M)$ is called \em positively expansive  \em  if there exists a   constant $\alpha >0$,  called the  \em expansivity constant, \em such that, for any two points $x, y \in M$, if  $dist(f^n(x), f^n(y) ) \leq \alpha $ for all $  n \in \mathbb{N},$ then $  x=y.$ Clearly a map which is  Lebesgue a.e.\ strongly non positively expansive is not positively expansive.  For compact connected manifolds with boundary it is known that every map is not positively expansive (Theorem 2.2.19, \cite{AH}). On the other hand, if $M$ has no boundary, then we have examples of positively expansive maps for example hyperbolic expanding toral diffeomorphisms.

\subsection{The sets $AA$ and $AA_1$}

For any  $x \in M$ we define
\begin{equation}
\label{eqn-mu_x}
\mu_x \mathrel{\mathop:}= \lim_{n \to \infty} \frac{1} {n}  \sum_{j=0}^{n-1} \delta_{f^j(x)}\end{equation} when this limit exists; namely, when $p\omega(x) = \{\mu_x\}$.
Let
\begin{equation}
\label{eqnEquationAA}
AA \mathrel{\mathop:}= \{  x \in M: p\omega(x) = \{\mu_x\}\} \mbox{ and }
AA_1 \mathrel{\mathop:}= \{  x \in AA: \mu_x \in \O_f\}.\end{equation}
For typical continuous maps or homeomorphisms,  Corollary \ref{theoremAAa)}    states
\begin{equation}\label{eqAA}
Leb(AA) = Leb(M),
\end{equation}
while   \cite[Proposition 3.1 and Definition 3.2]{CE2} implies that
\begin{equation}\label{eqAA1}
Leb(AA_1) = Leb(M).
\end{equation}
\begin{proposition} \label{propositionAA_1}
If $f $ is typical  in $\mathcal C(M)$ (resp.\  $\mathcal H(M)$) then
$$\O_f = \overline{\{ \mu_x: x \in AA_1 \}}.$$
The set $\{x \in AA_1: d(\mu_x, \mu_{x_0}) < \varepsilon\}$ has positive Lebesgue measure for every $\varepsilon >0$ and for every $x_0 \in AA_1$.
\end{proposition}

\begin{proof}  By definition of the set $AA_1$ we have
$\{\mu_x: x \in AA_1\}  \subset \O_f$. Besides, since $\O_f$ is closed in the weak$^*$ topology (see \cite[Theorem 1.3]{CE1}), we have
 $\overline{\{\mu_x: x \in AA_1\}}  \subset \O_f$.

 We turn to the other inclusion.  Suppose $\mu \in \O_f$, i.e.,
 $$Leb\{ x \in M: d(p\omega(x),\mu) < \varepsilon)\} > 0 \text{ for all } \varepsilon > 0.$$
Since $Leb(AA_1) =1$ and $p\omega(x) = \mu_x$ for $x \in AA_1$,
this is equivalent to
  $$Leb\{ x \in AA_1: d(\mu_x,\mu) < \varepsilon)\} > 0  \text{ for all } \varepsilon > 0.$$
Thus $\mu \in \overline{\{ \mu_x: x \in AA_1 \}}$. Since $\mu \in \O_f$ is arbitrary we conclude
 $\O_f \subset \overline{\{ \mu_x: x \in AA_1 \}}.$ The first assertion of Proposition \ref{propositionAA_1} is proven.

 To prove the second assertion recall  the definition of pseudo-physical measure,  $\mu \in {\mathcal O}_f$ if the set $\{x \in M \colon d(p\omega(x), \mu) < \varepsilon\}$ has positive Lebesgue measure. Combining this with Equality (\ref{eqAA1}), we deduce that $\{x \in AA_1 \colon d(\mu_x, \mu) < \varepsilon\}$ has positive Lebesgue measure, for any $\mu \in {\mathcal O}_f$, in particular for $\mu_{x_0}$ for any $x_0 \in AA_1$.
\end{proof}

\subsection{Approximation of pseudo-physical measures by periodic measures.}

In this subsection we will prove Theorem \ref{TheoremPeriodicSRB-l}:  each pseudo-physical measure can be arbitrarily approximated by atomic measures supported on periodic point. We will use the following lemma.
\begin{lemma}
\label{lemmaInvMeasOnShrinkingIntervals} Let $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$) and $\mu$ be an $f$-invariant measure. Assume that $I \subset M$ is a periodic shrinking set of period $p$, and denote  $K \mathrel{\mathop:}=  \bigcup_{j= 0}^{p-1} f^j(\overline I).$ Then \begin{equation}
\label{eqnno.}
\mu (f^j(\overline I)) = \mu(\overline I) = \frac{\mu(K)}{p}   \ \ \ \forall \ j  \geq 0. \end{equation}
If $\mu$ is additionally ergodic, then $\mu(K)$ is either zero or one.
\end{lemma}

\begin{proof}
 For any $j \geq 0$ let $j= kp - i$  with $k \in \mathbb{N}^+$ and $1 \leq i \leq p$. Using that $\mu$  is $f$-invariant, that $f^p(\overline I) \subset I$ and that $f^{-j} (f^j (\overline I)) \supset \overline I$ for all $j \geq 0$, we obtain:
 $$\mu (f^j(\overline I)) =  \mu (f^{-j}(f^j(\overline I)) \geq \mu (\overline I) \geq \mu(f^{kp}(\overline I)) = \mu (f^{-i} (f^{kp}(\overline I)) \geq \mu(f^j(\overline I)).$$
Hence, all the inequalities above are equalities; and thus $\mu(f^j(\overline I)) = \mu(\overline I)$ for all $j \geq 0$.
Assertion (\ref{eqnno.}) follows, since $\mu(K) = \sum_{j= 0}^{p-1} \mu ( f^j(\overline I) )$.

Finally, since $f(K) \subset K$, we have $K \subset f^{-1}(f(K)) \subset f^{-1}(K)$.  Since $\mu$ is ergodic, the set $K$  must have $\mu$-measure equal to zero or one.
\end{proof}

 \begin{theorem}\label{TheoremPeriodicSRB-l}
 Typical maps  $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M))$,  satisfy
 ${\mathcal O}_ f \subset \overline{\mbox{Per}_{f}}.$
 \end{theorem}

 \begin{proof}
Choose  $q_n \geq 1$ such that  any two  measures  satisfying the $q_n$-approach conditions of  Lemma \ref{lemma-dist}, are mutually at distance smaller than $1/n$.
Let $\mu \in \O_f$.
By the definition of pseudo-physical measure, the set $A_n \subset M$ of points $y$ such that $$
d(p \omega (y), \mu)< 1/n, $$ has positive Lebesgue measure. If $f$ is typical in $\mathcal C(M)$ or $\mathcal H(M)$, then Lebesgue a.e.~$y_n \in A_n$ satisfy $p\omega(y_n) = \{\mu_{y_n}\}$ (Corollary \ref{theoremAAa)}) and is contained in an arbitrarily small eventually periodic shrinking set (Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}).
Therefore,  \begin{equation}
\label{eqn09}
d(\mu_{y_n}, \mu)< 1/n, \end{equation} and $\mu_{y_n}\ $ is supported on $K_n \mathrel{\mathop:}= \bigcup_{j= 1}^{p_n} f^j(\overline I_n)$,  where $I_n$ is a periodic shrinking set of period $p_n$,  such that $\mbox{diam}(I_n) \leq 1/q_n$. Applying Lemma \ref{lemmaInvMeasOnShrinkingIntervals} and the definition of shrinking set yields $\mu_{y_n}(f^j(\overline I_n)) = 1/p_n$.

Since the set $I_n$ is shrinking periodic with period $p_n$, there exists at least one periodic orbit of period $p_n$ in $K_n$. Consider the periodic invariant measure $\nu_n $ supported on this periodic orbit. It also satisfies $\nu_n(f^j(\overline I)) = 1/p_n$. So, applying Lemma \ref{lemma-dist} we deduce that
\begin{equation}
\label{eqn10}   \  \exists \   \nu_n \in \mbox{Per}_{f}\colon \
 d(\mu_{y_n}, \nu_n) \leq 1/n. \end{equation}

Combining Inequalities (\ref{eqn09}) and (\ref{eqn10}), we deduce that there exists    sequence of periodic atomic measures $\nu_n$ such that
 $d(\nu_n, \mu) < 2/n$, finishing the proof of Theorem \ref{TheoremPeriodicSRB-l}.
\end{proof}

\section{Most invariant measures are not pseudo-physical}\label{sec4}
In this section we will prove Theorem \ref{Theorem-main2}.
We start with an extension of  Theorem of Abdenur and Andersson in \cite[Theorem 3.6]{AA}:

\begin{theorem}
 {\bf (Extension  of Abdenur-Andersson Theorem) } \label{theoremAA}
\noindent For typical maps in $C(M)$  there does not exist physical measures.
\end{theorem}

\begin{proof}
In \cite[Theorem 3.6]{AA}, Abdenur and Andersson   prove this  assertion  in the case that $M$ is a compact connected
$C^1$ manifold without boundary.
Now, let us prove that it also holds   when $M$ has boundary.   Given a manifold
with boundary $M$, take two disjoint copies $M_1, M_2$ and identify each boundary point in $M_1$ with the same point in $M_2$. We obtain the double cover
$D(M)$ of $M$ which is a $C^1$ manifold  without boundary  with the same dimension as $M$. See for example 9.32 in \cite{L} for details of this consruction.

Suppose that $M$ has boundary, and denote by ${\mathcal V}$ the family of all the continuous map  $f \in \mathcal C(M)$ such that $f(M) \subset M \setminus \partial M$.
 By construction ${\mathcal V}$  is   open and dense in $\mathcal C(M)$.  For each $f \in {\mathcal V}$, construct the nonempty  set ${\mathcal S} _{f} \subset C(D(M))$ composed by all the continuous maps $g: D(M) \mapsto D(M)$ such that
$g|_{M_1}= f$.

We claim that for any open set ${\mathcal U} \subset {\mathcal V}$ the set
\begin{equation} \label{eqnU'} {\mathcal U}' := \bigcup_{f \in {\mathcal U}} {\mathcal S} _f = \Big \{g \in C(D(M)) \colon  g|_{M_1}=: f \in {\mathcal U} \Big\}\nonumber\end{equation}
 is open in $C(D(M))$;  in particular ${\mathcal V}' $ is open in $C(D(M))$. This claim follows from that fact that
the restriction that sends each map $g \in C(D(M)) $ to the map $g|_{M_1} : M_1 \mapsto g(M_1) \subset M$ is a continuous operator. Therefore, the preimage ${\mathcal U}'$ by that restriction, of the open family ${\mathcal U} \subset {\mathcal V} \subset  C(M) \subset C(M_1, D(M)) $, is open.

 Next we claim that a converse-like statement is true: for any open set ${\mathcal U}' \subset {\mathcal V}' \subset C(D(M))$ the set
  \begin{equation} \label{eqnU} {\mathcal U} :=    \Big \{f \in {\mathcal V} \colon   \ \exists \ g \in {\mathcal U'} \subset C(D(M)) \mbox{ such that }  g|_{M_1}= f \Big\}\end{equation}
  is open in $C(M)$.  In fact, since ${\mathcal U}' \subset {\mathcal V}'$,  the restriction $g|_{M_1} $ for each $g \in {\mathcal U}'$, belongs to $ {\mathcal V} $. Besides, the restriction that sends each map $g \in {\mathcal V'} $ to the map $f := g|_{M_1} \in {\mathcal V} $ is an open operator. Therefore, the  image ${\mathcal U}$ by that restriction, of the open family ${\mathcal U}' $, is open.

    Since   Theorem  \ref{theoremAA} holds for typical maps $g \in C(D(M))$, and ${\mathcal V'}$ is nonempty and open in $C(D(M)) $, it also holds for typical maps $g \in {\mathcal V}' $. Namely, there exists a sequence  $\{{\mathcal U}'_n\}_{n \geq 1}$ of open and dense subsets of ${\mathcal V}'$ such that assertions a) and  b)  hold for all $g \in \bigcap_ {n \geq 1} {\mathcal U}'_n \subset {\mathcal V}' $. Consider  the preimages ${\mathcal U}_n \subset {\mathcal V}$ defined in (\ref{eqnU}), of the sets ${\mathcal U}_n$.  Since ${\mathcal U}_n$ is open and dense in ${\mathcal V'}$, the above assertions imply  that ${\mathcal U}_n$ is open and dense in ${\mathcal V}$; hence also in $\mathcal C(M)$.

If Theorem \ref{theoremAA}  holds for a map $g \in {\mathcal V}' \subset C(D(M))$, then  it also holds for the restriction $g|_{M_1} \in {\mathcal V} \subset \mathcal C(M)$. We conclude that it holds for the countable intersection $\bigcap_{n \geq 1} {\mathcal U}_n$ of open and dense sets in $\mathcal C(M)$. In other words, Theorem \ref{theoremAA} holds for typical maps in $\mathcal C(M)$.
\end{proof}

 {The following theorem follows easily  from results of Hurley and his coauthors on the existence of
many periodic points for homeomorphism of manifolds of dimension at least two without boundary \cite{AHK,H1}.
We give a different proof which also works for manifolds with boundary and also in dimension one.}

\begin{theorem}
\label{TheoremAddedNonUniqErgodic}
Typical maps in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$) are not uniquely ergodic.
\end{theorem}

\begin{proof}
First, let us argue in $\mathcal C(M)$. Recall that Lebesgue a.e.\ $x \in M$ belongs to the set $AA_1$ defined by Equality \eqref{eqnEquationAA}. We claim that
if $f \in \mathcal C(M) $ is typical,   then
 for any $\varepsilon > 0$ and any  $x_0 \in AA_1$ the \nolinebreak set \linebreak $\{x \in AA_1: \mu_x \neq \mu_{x_0} \mbox{ and } d(\mu_x, \mu_{x_0}) < \varepsilon\}$ has positive Lebesgue measure.

To prove the claim we argue by  contradiction, assume that there exists $x_0 \in AA_1$ and $\varepsilon >0$ such that \begin{equation}
\label{eqn08}
\mbox{Leb}\{x \in AA_1\colon \mu_x \neq \mu_{x_0} \mbox{ and } d(\mu_{x_0}, \mu_x) < \varepsilon\} = 0.\end{equation}
Since $\mu_{x_0}$ is pseudo-physical,  by definition we have
$$\mbox{Leb}(\{x \in M\colon d(p\omega (x), \mu_{x_0}) < \varepsilon\}) >0.$$
Since $Leb(AA_1) = 1$ we deduce that
$$\mbox{Leb}(\{x \in AA_1\colon d(\mu_x, \mu_{x_0}) < \varepsilon\}) >0.$$
Combining this last assertion with Inequality (\ref{eqn08}), we obtain that
$$\mbox{Leb}(\{x \in AA_1\colon  \mu_x = \mu_{x_0}) \}) >0.$$
So, $\mu_{x_0}$ is a  physical measure, contradicting  Theorem \ref{theoremAA}, the claim is proven.

Since the measures $\mu_x $ are $f$-invariant for all $x \in AA_1$, we deduce  from the claim that $f$ is not uniquely ergodic.

Now let us prove Theorem \ref{TheoremAddedNonUniqErgodic} for typical $f \in \mathcal H(M)$.
From Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}, there exists a periodic shrinking set $I$ {with arbitrarily small diameter}. Denote by $p$ the period of $I$, and by $x_0 \in I$ a periodic point of period $p$.
 Recall that $\mbox{diam}(f^j(I)) < \mbox{diam}(I)$.
Besides the sets $f^i(\bar I)$ are pairwise disjoint for $j \in \{0,\dots,p-1\}$, $f^p (\bar I) \subset I$,
and this inclusion is strict since $\mbox{diam}(f^p(I)) < \mbox{diam}(I)$. Therefore,    if $\mbox{diam}(I)$ is small enough, then
\begin{equation}\nonumber
\label{eqnKneqM} K:= \bigcup_{j=1}^{p} f^j(\overline I) \neq M.
\end{equation}

Denote by $\nu$ the atomic invariant measure supported on the periodic orbit of $x_0$.  We clain that
there exists $\delta>0$  such that for any Borel probability measure $\mu$ (not necessarily invariant), if $\mbox{dist}(\mu, \nu)< \delta$, then $\mu(I) >0$.

To prove the claim, we argue by contradiction. Assume that there exists $\mu_n \rightarrow \nu$ such that $\mu_n(I)=0$. Consider $\psi:M \mapsto [0,1]$ defined by
\begin{equation}
\label{eqn-psi}
\psi(x):=\frac{\mbox{dist}(x, M \setminus I)}{\mbox{dist}(x , f^p(\overline I)) + \mbox{dist}(x, M \setminus I) }.\end{equation}
This function is continuous and satisfies $\psi|_{f^p(\overline I)} =1, 0 < \psi(x) <1$ if $x \in I \setminus f^p(\overline I))$ and $\psi(x)= 0$ if $x \not \in I$. Since $\mu_n(I)=0$, we obtain $\int \psi \, d \mu_n =0$, and taking the weak$^*$-limit of $\mu_n$, we deduce $\int \psi \, d \nu=0$; hence $\psi=0$ $\nu$-a.e., which contradicts that $\nu(I) = 1/p >0$ and $\psi>0$ for all $x \in I$, finishing the proof of the claim.

Now, let us prove that $f$ is not uniquely ergodic. Arguing by contradiction, if $\nu$ were the unique invariant measure, then  $(1/n) \sum_{j=0}^{n-1} \delta_{f^j(x)}$ would converge uniformly to $\nu$ for all $x \in M$ as $n \rightarrow + \infty$. Therefore, there would exist $n_0 \geq 1$ such that
$$\mbox{dist} \Big(\nu, (1/n_0) \sum_{j=0}^{n_0-1} \delta_{f^j(x)}        \Big)< \delta \ \ \forall \ x \in M.$$
Applying  the claim for all $x \in M$, there would exist $0 \leq j = j(x) \leq n_0-1$ such that $f^j(x) \in I$. Since $I$ is shrinking periodic, we would deduce that $f^{n_0}(x) \in K$  for all $x \in M$. Therefore, applying assertion (\ref{eqnKneqM}), the image of $M$ by the homeomorphism $f^{n_0}: M \mapsto M$ would not be $M$, which is a contradiction.
\end{proof}

Now, we state the main lemma to be used to prove Theorem \ref{Theorem-main2}.

\begin{lemma} \label{Lemma-ConvexCombination}
Suppose that $f$ is a typical map in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$),
 $\mu_1$ is   $f$-invariant and supported on  $K = \bigcup_{j= 0}^{p-1} f^j(\bar I)  $, where $  I$ is a periodic shrinking set of period $p $, and that  $\mu_2$ is   $f$-invariant such that $\mu_2(K)=0$. Then, no convex combination $ \nu= \lambda \mu_1 + (1 - \lambda) \mu_2$  with $0 < \lambda < 1$  is pseudo-physical.
\end{lemma}

\begin{proof} Since $I$ is a shrinking periodic set with period $p$,  it is is open, $\overline{I}$ is an $m$-simplex, and $f^p(\overline{I}) \subset I$. Consider the continuous real function $\psi:M \rightarrow [0,1]$ defined by Equality (\ref{eqn-psi}).
It satisfies
 $\psi|_{f^p(\overline I)}= 1$, $0 < \psi(x) <1$ if $x  \in I \setminus f^p(\overline I) $, and $\psi(x) = 0$ if $x \not \in I$.

Since $\mu_1 $ is supported on $K =   \bigcup_{j= 0}^{p-1} f^j(\overline I) $,  we can apply Lemma \ref{lemmaInvMeasOnShrinkingIntervals} to deduce that $$\mu_1(\overline I) = \mu_1 (f^p(\overline I)) = \frac{1}{p}; \ \ \mbox{ hence }  \mu_1 (I \setminus f^p(\overline I)) = 0.  $$ We obtain $
  \int \psi \, d \mu_1 = \mu_1 (f^p(\overline I)) =  {1}/{p} $ and $ \int \psi \, d \mu_2 = \mu_2 (I) = 0.$
  Therefore
 $ 0 <\int \psi \, d \nu =  \lambda /p  <  {1}/{p}.  $

Choose $\varepsilon >0$  such that for any measure $\mu$, if $d(\nu, \mu) < \varepsilon$, then $\int \psi \, d \mu > 0$ and $\int \psi \, d \mu <   1 /p. $

Consider the set $$A'_{\varepsilon}(\nu) \mathrel{\mathop:}=    \{ x \in AA: \mbox{ dist} (\mu_x, \nu) < \varepsilon\},$$
where the set $AA$ is defined in Equality (\ref{eqnEquationAA}).
We claim that  the set $A'_{\varepsilon} (\nu)$ is empty. Arguing by contradiction, assume that there exists $x \in A'_{\varepsilon}(\nu)$.
Then,  from the choice of $\varepsilon$, we have   \begin{equation}
   \label{eqn07}0 <\int \psi \, d \mu_x < 1/p, \end{equation} and  from (\ref{eqn-mu_x}) we deduce that there exists $n_0 \geq 1$ such that $$ \frac{1}{n} \sum_{j= 0} ^{n-1} \psi (f^j(x)) = \int \psi \, \Big (\frac{1}{n} \sum_{j= 0}^{n-1} \delta_{f^j(x)}
   \Big )  >0 \ \forall \ n \geq n_0.$$
Thus,  there exists $n_1 \geq 1$ such that $\psi(f^{n_1}(x))    >0$; hence $f^{n_1}(x) \in I$. This implies that  the future orbit of $f^{n_1}{x}$ is contained in $K$. Hence,  $\mu_x$ is supported on $K$. Since $\mu_x$ is $f$-invariant, we apply Lemma \ref{lemmaInvMeasOnShrinkingIntervals} to deduce that $\mu_x (f^p(\overline I)) = 1/p$ and $\mu_x(I \setminus f^p(\overline I)) = 0$. Therefore,
$$\int \psi \, d \mu_x = \mu_x(f^p(\overline I)) = \frac{1}{p},$$
contradicting  \eqref{eqn07}. We have proved that $A'_{\varepsilon} (\nu) $ is   empty.

The definition of  pseudo-physical measure $\mu$   asserts that  $$\mbox{Leb}(A_{\varepsilon}(\mu)) >0 \ \ \forall \ \varepsilon >0. $$
But \eqref{eqAA} implies that $Leb(A'_\varepsilon(\nu)) = Leb(A_\varepsilon(\nu))$, which  equals zero because
 $A'_{\varepsilon}(\nu) = \emptyset$.  We conclude that $\nu$ is not  pseudo-physical.
\end{proof}

\subsection{End of the proof of Theorem \ref{Theorem-main2}}

{
\begin{proof}
From \cite[Theorem 1.3]{CE1} the set $\O_f$ is closed. Let us prove that  its interior  in ${\mathcal M}_f$ is empty.
Fix $\mu \in \O_f$.  Since a typical map is not uniquely ergodic (recall Theorem \ref{TheoremAddedNonUniqErgodic}), there exists an ergodic measure $\nu \neq \mu$. Denote by $\varepsilon:= d(\mu_x, \nu)>0$.
Consider an arbitrary $\delta \in (0,\varepsilon/2)$.

From Proposition \ref{propositionAA_1}, we can find $x_0 \in AA_1$ such that $d (\mu_{x_0},\mu) < \delta$. So, to prove that $\mu$  does not belong to the interior of ${\mathcal O}_f$ in ${\mathcal M}_f$, it is enough to prove that the $\delta$-neighborhood of $\mu_{x_0}$ in ${\mathcal M}_f$ is not contained in ${\mathcal O}_f$. To do that, it is enough to find a sequence $\{\mu_n\}_n \subset {\mathcal M}_f$ converging to $\mu_{x_0}$ and such that $\mu_n \not \in {\mathcal O}_f$.

Applying   Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval} and Equality (\ref{eqAA1}), and taking into account the second part of Proposition \ref{propositionAA_1},   the point $x_0$ could  be chosen to belong  to arbitrarily small eventually periodic shrinking sets. Therefore, for any fixed $q \in \mathbb{N}^+$ (that will be chosen later),  the probability measure $\mu_{x_0}$ is supported on the orbit of a periodic shrinking set  $I_q$ of  diameter smaller than $1/q$. Denote by $p$ the period of $I_q$, and denote
$$K = \bigcup_{j=0}^{p-1} f^j(\overline I_q).$$
We have $\mu_{x_0}(K)=1$. So,
applying Lemma \ref{lemmaInvMeasOnShrinkingIntervals} we deduce $$\mu_{x_0} (f^j(\overline I))= {1}/{p} \ \ \ 0 \leq j \leq p-1.$$

Applying Lemma \ref{lemma-dist}, construct $q \geq 1$ such that if $\mu$ is a probability measure satisfying $\mu(f^j(\overline I)) = \mu_x(f^j(\overline I)) = 1/p$ with  $\mbox{diam}(f^j(\overline I)) <1/q$ for all $0 \leq j \leq p-1$, then $d(\mu, \mu_x)< \delta$. Since $d(\mu, \nu)= \varepsilon > \delta$, we deduce that $\nu(f^j(\overline I)) \neq 1/p$ for some $j$. But applying Lemma \ref{lemmaInvMeasOnShrinkingIntervals} $\nu(f^j(\overline I)) = \nu(K)/p$ for all $j$, with $\nu(K) \in \{0,1\}$. Since $\nu$ is ergodic, we conclude that $\nu(K)=0$. Therefore, applying Lemma \ref{Lemma-ConvexCombination}, $\mu_n:=\lambda_n \mu_x + (1- \lambda_n) \nu \in {\mathcal M}_f \setminus {\mathcal O}_f$ for all $0 <\lambda_n<1$.
Taking $\lambda_n \rightarrow 1^-$, we obtain $\mu_n \rightarrow \mu_x$, with $\mu_n \in {\mathcal M}_f \setminus {\mathcal O}_f$, as wanted.
\end{proof}
}

\section{Pseudo-physical,  ergodic and periodic measures.}\label{sec5}
A \em  $\delta$-pseudo-orbit of $f$ \em is a sequence $\{y_n\}_{n \in  \mathbb{N}} \subset M$ such that $$dist(f(y_n), y_{n+1} ) < \delta \ \ \forall \ n \in \mathbb{N}.$$
A \em $\delta$-pseudo-orbit  $\{y_n\}_{n \in  \mathbb{N}} $ is  periodic \em with period $p \geq 1$, if $$y_{n+p}= y_n \ \ \ \forall \ n \in \mathbb{N}.$$
A  map $f \in \mathcal C(M)$ has the   \emph{periodic shadowing property} if for all $\varepsilon > 0$, there exists $\delta>0$ such that,  if $\{y_n\}_{n \in \mathbb{N}}$ is any periodic $\delta$-pseudo-orbit, then, at least one  periodic orbit $\{f^n(x)\}_{n \in \mathbb{N}}$ satisfies
$$dist(f^n(x), y_n) < \varepsilon \ \ \ \forall \ n \in \mathbb{N}.$$

We will also use the following result, which is the accumulation of many authors' work.
\begin{theorem}
\label{TheoremOprocha&als}
Typical maps $f \in \mathcal C(M)$ (resp.\  $\mathcal H(M)$) have the periodic shadowing property.
\end{theorem}
\begin{proof} The statement follows from \cite[Theorem 1.2]{Oprocha&als} and \cite[Theorem 1]{PP}.  Earlier special cases where treated in \cite{Y} and \cite{O}.
\end{proof}

Recall that ${\mathcal E}_f$ denotes the set of ergodic measures, and $\mbox{Per}_{f}$ denotes the set of invariant measures supported on periodic orbits of $f$.
As a consequence of Theorem \ref{TheoremOprocha&als}:
\begin{corollary}
\label{corollaryOprocha&als}
For a typical map $f \in \mathcal C(M)$ (resp.\  $f \in \mathcal H(M)$)  we have $\mathcal E_f \subset \overline{\mbox{Per}_{f}}. $
\end{corollary}

\noindent{\bf Remark. }
 The closing lemma and the closeability properties (see for instance \cite[Definition 2.1]{Coudene-S} and \cite[Definitions 4.1 and 4.5]{Gelfert-K}), state  that for all $\epsilon>0$ there exists $\delta>0$ such that, for a finite piece  of orbit $\{f^j(y\}_{0 \leq j \leq p}$ verifying $dist(f^p(y), y) < \delta$,  there exists a periodic point $x$, whose period  is $p$  or at least   $(\epsilon \cdot p)$- near  $p$, satisfying $dist(f^j(y), f^j(x)) < \epsilon$ for all $0 \leq j \leq p$.  It is  known that the closing lemma and the closeability properties imply   ${\mathcal E}_f \subset \overline{\mbox{Per}_f}$ (see for instance  \cite[Lemma 2.2]{Coudene-S} and  \cite[Theorems 4.10]{Gelfert-K}).
 Nevertheless,  we do not know if typical maps in $C^0(M)$  satisfy the closing lemma or the closeability properties.

\begin{proof}[Proof of Corollary \ref{corollaryOprocha&als}]
From the definition of distance in the weak$^*$ topology of the space of probability measures it is standard to check that   for all $\varepsilon_0 >0$, there exists $\varepsilon>0$, such that, for any two points $x_1, x_2 \in M$, $$dist(x_1, x_2) < \varepsilon \ \ \Rightarrow \ \ d(\delta_{x_1}, \delta_{x_2})< \varepsilon_0.  $$

Fix any $\mu  \in {\mathcal E}_f$. Since $\mu$ is  ergodic, we have
$p \omega (x) = \{\mu\}  \ \ \mbox{ for } \mu\mbox{-a.e. } x \in M.$ Fix such a point $x$;  then there exists $n_0 \geq 1$ such that
\begin{equation}
\label{eqn11}
d \left (\frac{1}{n} \sum_{j= 0}^{n-1} \delta_{f^j(x)}, \ \mu \right ) < \varepsilon_0 \ \ \ \forall \ n \geq n_0.\end{equation}
Given $\varepsilon$, choose $\delta >0$ given by Theorem \ref{TheoremOprocha&als}. By Poincar\'{e} Recurrence Lemma, the point $x$ can be chosen to be recurrent. Thus
$$ d(f^{p - 1} (x), x) < \delta \ \ \mbox{ for some } p \geq n_0.$$
Construct the periodic $\delta$-pseudo-orbit  $\{y_n\}_{n \in \mathbb{N}}$ of period $p$ defined by   $y_n= f^n(x)$ for all $0 \leq n < p$,
$y_{n + p} = y_n$ for all $n \geq 0$.  Applying Theorem \ref{TheoremOprocha&als}, there exists a periodic orbit $\{f^n(z)\}_{n \geq 0}$, such that
$dist (f^n(z), y_n) < \varepsilon \ \ \forall \ n \geq 0$.
By construction,  if  $ip \le  n < (i+1)p$    and   $i \ge 0$ then
$dist(f^n(z), f^{n-ip}(x)) < \varepsilon $.
Thus, from the choice of $\varepsilon$, we obtain
$d(\delta_{f^n(z)}, \delta_{f^{n-ip}(x)}) < \varepsilon_0$
Denote by $q$ the period of $z$.
Taking into account that  balls are convex in the
 weak$^*$-distance in the space of probabilities, we deduce
$$d\left ( \frac{1}{qp} \sum_{j= 0 }^{qp-1} \delta_{f^j(z)},  \ \ \frac{1}{qp}  q \cdot \sum_{j= 0 }^{p-1} \delta_{f^j(x)}   \right) < \varepsilon_0. $$
For the atomic invariant measure $\nu$ supported on the periodic orbit of $z$, we have
$$\nu = \frac{1}{q} \sum_{j= 0}^{q-1} \delta_{f^j(z)} =  \frac{1}{qp} \sum_{j= 0 }^{qp-1} \delta_{f^j(z)}.$$
Thus,
$$d \left ( \nu ,  \ \ \frac{1}{p} \sum_{j= 0 }^{p-1} \delta_{f^j(x)}   \right) < \varepsilon_0.$$
Together with  (\ref{eqn11}), this implies that for any $\varepsilon_0$
the given ergodic measure $\mu$ is $2  \varepsilon_0$-approximated by  some measure $\nu \in \mbox{Per}_{f}$,
finishing the proof of   Corollary \ref{corollaryOprocha&als}.
\end{proof}

 An invariant measure $\mu$ is called \em infinitely shrinked \em if there exists a sequence $\{I_q\}_{q \geq 0}$ of periodic shrinking intervals $I_q$, of periods $p_q$, such that $\mbox{diam}(I_q) < 1/q$ and $\mu(\bigcup_{i= 1}^{p_q}f^j(\bar I_q) )= 1$ for all $q \geq 1$.
We denote by $\mbox{Shr}_f \subset {\mathcal M}_f$ the set of  infinitely shrinked  invariant measures.
Define
$$AA_2  \mathrel{\mathop:}= \{ x \in AA_1: \ \  \mu_x \in \mbox{Shr}_f\}. $$
\begin{theorem}
\label{theoremSRB-l=ClosureSrh}
For a typical map $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$),  $$ \mbox{ Leb}(AA_2)  = 1 \ \ \mbox{ and  } \ \  \O_f  =\overline {\{\mu_x \colon  x \in AA_2\}} =  \overline {\mbox{\em Shr}_f}.$$
\end{theorem}

\begin{proof}
From Theorem \ref{theoremLebesgue-a-e-isInShrinkingInterval}, Lebesgue-a.e.~$x \in M$ belongs to a sequence of eventually periodic or periodic shrinking sets $J_q$ with $\mbox{diam}(J_q) < 1/q$. Every eventually periodic shrinking set $J_q$  wanders under  $f$ until it drops into a periodic shrinking set $I_q$ with
$ \mbox{diam}(I_q)< \mbox{diam}(J_q)< 1/q$.  By the definition of periodic shrinking set, every point of $I_q$ has all the measures of $p\omega (x) $ supported on the compact set $$K_{x,q} \mathrel{\mathop:}= \bigcup_{j= 0}^{p_q-1} f^j(\overline I_q).$$ In particular for Lebesgue almost all $x \in AA_1$, the limit measure $\mu_x$  defined by   (\ref{eqn-mu_x}), is supported on $K_{x,q}$. Thus, for a.e.\ $x \in AA_1$ we have $\mu_x \in \mbox{Shr}_f$. Taking into account (\ref{eqAA1}), the above assertion implies  that the set $AA_2$ has full Lebesgue measure.

By construction,  $AA_2 \subset AA_1$. So, applying Proposition \ref{propositionAA_1}, we obtain:
$$ \overline {\{\mu_x \colon  x \in AA_2\}} \subset \overline {\{\mu_x \colon  x \in AA_1\}} = \O_f. $$
To obtain the opposite inclusion, we apply \cite[Theorem 1.5]{CE1}):   $\O_f$ is the minimal weak$^*$-compact set of probability measures, that contains $p \omega (x)$ for Lebesgue a.e.~$x$. Since $ \overline {\{\mu_x \colon  x \in AA_2\}}$ is weak$^*$-compact and contains $p\omega_x = \{\mu_x\}$ for Lebesgue almost all $x$ (because $\mbox{Leb}(AA_2) = Leb (M)$), we conclude that
    $$   \overline {\{\mu_x \colon  x \in AA_2\}} \supset \O_f.$$

    The inclusion   $\overline {\{\mu_x \colon  x \in AA_2\}} \subset \overline{\mbox{Shr}_f}$ follows trivially from the definition of the set $AA_2$. Now, let us prove the opposite inclusion. We will prove that every shrinking measure is pseudo-physical.  Let $\mu \in \mbox{Shr}_f$. For any  $\varepsilon>0$, choose  $q \geq 1$ as in Lemma \ref{lemma-dist}.  By the definition of shrinking measure, there exists a periodic shrinking set $I'_{q'}$ such that $\mu$ is supported on
    $$K_\mu  = \bigcup_{j= 0}^{p_{q'}-1} f^j(\overline{ I'_{q'}}) $$
 for the periodic shrinking set $I'_{q'}$ of period $p_{q'}$, such that  $\mbox{diam}(\overline {I'_{q'}}) <1/{q'}$; hence  $\mbox{diam}(f^j(\overline I'_{q'})) <1/{q'} \ \ \forall \ 1 \leq j \leq p_{q'}$,
$\mu(K_{\mu}) = 1$.

    Besides, for any point $x \in I'_{q'}$, any  measure in $p\omega_x$ is also supported on $K_{\mu}$. If additionally $x \in AA_1$, then $p\omega_x = \{\mu_x\}$, so  $\mu_x (K_{\mu}) = 1$.
    Finally, applying Lemmas \ref{lemma-dist} and  \ref{lemmaInvMeasOnShrinkingIntervals}, we deduce that
      the measures $\mu \in \mbox{Shr}_f$ and $\mu_x$ given above satisfy $$d(\mu, \mu_x) < \varepsilon \ \mbox{ for any } x \in I_q \cap AA_1. $$ Since $Leb (I'_{q'} \cap AA_1) = Leb (I'_{q'}) >0$, the basin $A_\varepsilon (\mu)$ has positive Lebesgue measure; namely $\mu $ is pseudo-physical.

    We have shown that every shrinking measure is pseudo-physical. Since the set $\O_f$ of pseudo-physical measures is closed, we conclude $$\overline {\mbox{Shr}_f} \subset \mathcal O_f, $$
finishing the proof of Theorem \ref{theoremSRB-l=ClosureSrh}.
\end{proof}

\begin{theorem} \label{theoremShrIncludedErgodic-SRB-l-includedClosureErgodic}
For any map $f \in \mathcal C(M)$ (resp.\  $f \in \mathcal H(M)$), if $\mu \in \mbox{Shr}_f$, then it is ergodic.
\end{theorem}
Before proving Theorem \ref{theoremShrIncludedErgodic-SRB-l-includedClosureErgodic} let us deduce its main consequence:
\begin{corollary} \label{corollaryShrink}
For a typical map $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$):
$$\O_f  = \overline{\mbox{Shr}_f}\subset \overline{{\mathcal E}_f} =  \overline{\mbox{Per}}_f.$$
\end{corollary}

The corollary immediately follows by combining Corollary \ref{corollaryOprocha&als} with
Theorems  \ref{theoremSRB-l=ClosureSrh} and \ref{theoremShrIncludedErgodic-SRB-l-includedClosureErgodic}.
At the end of the next section we will prove that  for typical maps these sets are all equal.

\begin{proof}[Proof of Theorem \ref{theoremShrIncludedErgodic-SRB-l-includedClosureErgodic}]
Fix $f \in \mathcal C(M)$. Suppose $\mu \in \mbox{Shr}_f$, and  $\mu_1, \mu_2 \in \mathcal{M}_f$ such that
\begin{equation}
\label{eqn13}
\mu = \lambda \mu_1 + (1 -\lambda)\mu_2, \ \ \ \mbox{with} \ \  0 < \lambda <1,\end{equation}
We shall prove that $\mu_1 = \mu_2 = \mu$; namely  $\mu$ is extremal in the convex compact set of invariant measures; hence ergodic.

Take arbitrary $\varepsilon >0$ and fix $q \geq 1$ as in Lemma \ref{lemma-dist}. By the definition of infinitely shrinking measures, there exists a periodic shrinking set $I_q$, with $\mbox{diam}(I_q) < 1/q$, and period $p_q$, whose $f$-orbit $K_q$ supports $\mu$.  The definition of periodic shrinking set and Lemma \ref{lemmaInvMeasOnShrinkingIntervals} tell us:
$$\mu(f^j(\overline I_q))  = \frac{1}{p_q},  \ \ \ \mbox{diam} (f^j(\overline I_q)) < 1/q \ \ \ \ \forall \ 1 \leq j \leq p_q.$$

Since $\mu(K_q) = 1$,  from  (\ref{eqn13}) we deduce $\mu_1(K_q) = \mu_2(K_q) = 1 $. Applying  Lemma \ref{lemmaInvMeasOnShrinkingIntervals} we obtain
 $$\mu_1(f^j(\overline I_q))  = \mu_2(f^j(\overline I_q))  = \frac{1}{p_q}\ \ \ \ \forall \ 1 \leq j \leq p_q. $$
So,   Lemma \ref{lemma-dist} implies
$d(\mu_1, \mu) < \varepsilon,$ and $d(\mu_2, \mu)  < \varepsilon$.
Since $\varepsilon> 0$ is arbitrary, we conclude that $\mu = \mu_1 = \mu_2$; hence $\mu $ is ergodic.
\end{proof}

\section{All ergodic measures are pseudo-physical.}\label{sec6}
\begin{definition} \em  \label{definition-qShrinking}
Let $q \geq 1$ and $x_0$ be a periodic point with period $r \geq 1$. We call the  (invariant) measure  $$\nu= \frac{1}{r} \sum_{j=0} ^{r-1}\delta_{f^j (x_0)} \in Per_f$$   a \em  $q$-shrinked periodic measure,  \em  if there exists some periodic shrinking set  $I $, with diameter smaller than $1/q$, with period $p \geq 1$ such that $\nu$ is supported on   $K \mathrel{\mathop:}= \bigcup_{j= 1}^p f(\bar I)$. From the definition of periodic shrinking set,    the period $p$ must divide $r$.
We denote by $\mbox{Shr}_q\mbox{Per}_{f}$ the set of $q$-shrinked periodic measures.

We say that an invariant measure $\mu$,  \em is $\varepsilon$-approached by $q$-shrinked  periodic measures \em if there exists $\nu \in \mbox{Shr}_q\mbox{Per}_{f}$ such that $d(\mu, \nu) < \varepsilon $. We denote by $\mbox{AShr}_{\varepsilon, q}\mbox{Per}_{f}$ the set of measures that are  $\varepsilon$-approached by $q$-shrinked periodic measures.

\end{definition}

\begin{theorem} \label{theoremBigcapClosShr_q&SRBl-l}
For any map $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$)
$$  \bigcap_{\varepsilon >0}\bigcap_{q \geq1}\overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}}_f  \subset \O_f.$$

\end{theorem}
\begin{proof}
Fix  $\varepsilon > 0$, and choose $q \geq 1$ as in Lemma \ref{lemma-dist},  such that $1/q < \varepsilon$.
For any $\mu_q  \in \mbox{AShr}_{1/q, q}\mbox{Per}_f $, denote by $\nu_q $ a measure in $ {\mbox{Shr}_q\mbox{Per}}_f$  such that $$d(\mu_q, \nu_q) < 1/q< \varepsilon.$$
Consider  the periodic shrinking set $I = I(\nu_q)$ and the compact set $K$ for $\nu_q$ from   Definition \ref{definition-qShrinking}. From the definition of periodic shrinking set,  any point $x \in I$ satisfies $f^n (x) \subset K= \bigcup_{j= 0}^{p-1} f^j(\bar I)$ for all $n \geq 0$. So, any  measure $\mu_x \in p\omega (x)$ is supported on $K$. Also $\nu_q$ is supported on $K$. Thus, applying Lemma \ref{lemmaInvMeasOnShrinkingIntervals}, we deduce that
$$\mu_x( f^j(\bar I)) = \nu_q(f^j(\bar I)) = \frac{1}{p} \ \ \forall \ 1 \leq j \leq p; \ \ \ \mbox{diam}(f^j(\bar I)) < \frac{1}{q}.$$
Now, from Lemma \ref{lemma-dist}, we obtain  $d(\nu_q, \mu_x) < \varepsilon$ for all $\mu_x \in p\omega(x)$, for all $x \in I$. Thus, for any $x \in I$
\begin{equation}
\label{eqn14}
d(p \omega(x), \nu_q)< \varepsilon,  \mbox{ hence } d(p \omega(x), \mu_q)< 2 \varepsilon.\end{equation}

Note that when we vary the value of $\varepsilon >0$, the value of $q$, and thus also the measures $\nu_q$ and $\mu_q$ and the set $I$, may change. So, from the above inequality we \em can not deduce \em that each $\mu_q$ is pseudo-physical. Nevertheless, we have proved that for any fixed value of  $\varepsilon> 0$ there exists $q \geq 1$ such that Inequality (\ref{eqn14}) holds  for all $\mu_q  \in \mbox{AShr}_{1/q, q}\mbox{Per}_f $.

Now, consider any measure $\mu'_q \in  \overline{\mbox{AShr}_{1/q, q}\mbox{Per}_f}$. Thus, there exists $\mu_q \in \mbox{AShr}_{1/q, q}\mbox{Per}_f$ such that
$$d(\mu'_q, \mu_q) < \varepsilon.$$
Combining this with (\ref{eqn14}) we deduce that,  for  all $\varepsilon >0$   there exists   $q \geq 1$ such that, for any measure $\mu'_q  \in \overline{\mbox{AShr}_{1/q, q}\mbox{Per}_f}$ there exists an open set $I$ (the periodic shrinking set $I(\nu_q)$ for the measure $\nu_q$ associated to $\mu'_q$)  such that
\begin{equation}
\label{eqn15}
d(p \omega(x), \mu'_q) < 3\varepsilon \ \ \ \forall \ x \in I.\end{equation}

So, if $\mu \in \bigcap_{\varepsilon >0}  \bigcap_{q \geq 1}\overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f}$, then,  for all $\varepsilon >0$ there exists an open set $I$ satisfying assertion (\ref{eqn15}). Thus  $\mbox{Leb}(A_{3\varepsilon}(\mu)) \ge \mbox{Leb}(I) >0$ for all $\varepsilon >0$; hence $\mu \in \O_f$, as wanted.
\end{proof}

The last ingredient of the proof of Theorem \ref{Theorem-main} is the following theorem.

\begin{theorem}
\label{TheoremSRB-l=ClosureErgodic}
For a typical map $f \in \mathcal C(M)$ (resp.\ $f \in \mathcal H(M)$),
\begin{equation}
\label{eqn16}
\mbox{Per}_f \subset \bigcap_{q \geq1}\overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f} \ \ \forall \ \varepsilon>0.\end{equation}
\end{theorem}

Before proving Theorem \ref{TheoremSRB-l=ClosureErgodic}, let us introduce the following definition:

\begin{definition}
\label{definitionP_q,r} \em

Fix  $q,r \in \mathbb{N}^+$.
  A \em good $q, r$-covering  \em $\mathcal U_{q,r}$ for $f \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$), is a finite family of \em open \em simplexes (i.e., the interiors of simplexes)  such that

(1)  $\mathcal U_{q,r}$ covers the compact set  $Per(f,r) \mathrel{\mathop:}= \{x \in M: f^rx = x\}$.

(2)  $\mbox{diam}(U_i)< 1/q$ for any $U_i \in \mathcal U_{q,r}$.

(3) For any $U_i \in \mathcal U_{q,r}$, there exists a periodic shrinking set $I_i$, with period $p_i \leq r$, with $p_i$ that divides $r$, such that $\bar I_i \subset U_i$.

\vspace{.2cm}

We call a map $f \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$)  \em a good $q,r$-covered map,  \em  if  there exists a  good $q, r$-covering  $\mathcal U_{q,r}$ for $f$. We denote by ${\mathcal P}_{q,r} \subset \mathcal C(M)$ (resp.\ $\mathcal H(M)$) the set of all good $q,r$-covered maps.
\end{definition}

\begin{proof}[Proof of Theorem \ref{TheoremSRB-l=ClosureErgodic}] We claim that, for fixed $q,r \geq 1$, the set ${\mathcal P}_{q,r}$  is open in $\mathcal C(M)$.
Fix $f \in {\mathcal P}_{q,r} $, and denote its good $q,r$-covering by  $\mathcal U_{q,r} = \{U_1, U_2, \ldots, U_h\}$. The compact set $K= M \setminus \bigcup_{i= 1}^h U_i$ does not intersect the compact set $\{f^r(x) =x\}$. Let us prove that for all $g \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$) close enough to $f$, the same compact set $K$ (defined for the \em same \em covering ${\mathcal U}_{q,r}$) does not intersect   $\{g^r(x) = x\}$.  In fact,    the real function   $\phi_f(\cdot) : = \mbox {dist} (f^r(\cdot), \cdot) $  depends continuously on $f$. Since $\min _{x \in K} \phi_f(x) >0$, we deduce $\min _{x \in K} \phi_g(x) >0$ for all $g \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$) close enough to $f$.
In other words, ${\mathcal U}_{q,r}$ also covers the fixed points of $g^r$.  Thus, the good $q,r$-covering of $f$, is also a covering satisfying conditions (1) and (2) of Definition \ref{definitionP_q,r},  for any $g \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$) close enough to  $f$.
Now, let  us prove that Condition (3) for $g$  is satisfied   by the same   covering ${\mathcal U}_{q,r}$, provided that $g $ is close  enough to
 $f$.
Consider a $f$-shrinking periodic set $I_i \subset U_i \in  {\mathcal U}_{q,r}$, of period $p_i$.
Now  $I_i$ is a periodic shrinking set with the same period $p_i$  for all $g$ sufficient close  to $f$. Since the family $\{I_i\}_{1 \leq i \leq h}$ of shrinking periodic sets to be preserved is finite, we conclude that
(3) is also satisfied for any $g$ sufficiently close to $f$
and thus ${\mathcal P}_{q,r}$ is open in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$).

In Lemma \ref{lemmaP_qrIsDense}, we will  prove that  ${\mathcal P}_{q,r}$ is  dense in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$). Let us conclude the proof of Theorem \ref{TheoremSRB-l=ClosureErgodic} assuming that Lemma \ref{lemmaP_qrIsDense} is proven.
Observe that, for fixed $q,r \geq 1$, any $f \in {\mathcal P}_{q,r}$ has the following property:
any   point $x_0$ fixed by $f^r$ (in particular any periodic point $x_0$ of period $r$) is $(1/q)$-near all the points of a periodic shrinking set $I_0$ with diameter smaller than $1/q$, and with period $p_0  \leq r$, $p_0 $ dividing $r$.

Besides, any periodic shrinked set of period $p_0$ has at least one periodic point $y_0$, fixed by $f^{p_0}$. We deduce that $I_0 $, whose diameter is smaller than $1/q$, contains  a periodic point $y_0$. Using the  definition of the set of measures $\mbox{Shr}_q \mbox{Per}_f$, we summerize this assertion as follows:
 \begin{equation}
\label{eqn18} \mbox{for all }  \ x_0 \mbox{ with period } r, \ \mbox{ there exists }  \end{equation} $$
\nu_0 \mathrel{\mathop:}=  \frac{1}{r} \sum_{j= 0}^{r-1} \delta_{f^j(y_0)} \in \mbox{Shr}_q \mbox{Per}_f,
\mbox{ with } \ \  dist(y_0, x_0) < 1/q.$$

For   $r \geq 1$ fixed, consider $f \in \bigcap_{q \geq 1} \mathcal P_{q,r}$.  Let $\mu_0 \mathrel{\mathop:}=  \frac{1}{r} \sum_{j= 0}^{r-1} \delta_{f^j(x_0)}$.
Fix $\varepsilon > 0$ and choose $q' \ge 1$ as in Lemma \ref{lemma-dist}.
Since $f^j$ is continuous for each $j$, we can find a $q > q'  $ so that if $dist(x,y) < 1/q$, then $dist(\delta_{f^j x}, \delta_{ f^j y}) < 1/q'$ for all $j \in  \{0,1,\dots, r\}$.
Then Lemma \ref{lemma-dist}  implies that $d(\mu_0,\nu_0) < \varepsilon$.

We have shown that for any given periodic orbit $\{f^j(x_0)\}_{ 0 \leq j \leq r-1}$ of period $r$, the distance between the periodic measure  supported on it, and some measure $\nu_q \in \mbox{Shr}_q \mbox{Per}_f$, for all $q$ large enough, is smaller than $\varepsilon$.
  In other words, any periodic measure supported on a periodic orbit of period $r$, belongs to $\bigcap_{q \geq 1} \overline {\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f}$ for all $\varepsilon>0$.

  Finally, if $f \in {\mathcal P}\mathrel{\mathop:}= \bigcap_{r \geq 1} \bigcap_{q \geq 1}\mathcal P_{q,r}$, then all its periodic measures (supported on periodic orbits of any period $r$) will belong to $\overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f} $ for all $q \geq 1$ and for all $\varepsilon>0$. In brief, if $f \in   \mathcal P$, then
  $$\bigcap_{q \geq 1} \mbox{Per}_f \subset \overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f}  \ \ \ \forall \ \varepsilon > 0. $$

   As ${\mathcal P}_{q,r}$ is open and dense in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$), the maps $f \in \mathcal P$ are typical. This ends the proof of Theorem \ref{TheoremSRB-l=ClosureErgodic}, provided that Lemma \ref{lemmaP_qrIsDense} is proven.
\end{proof}
\begin{lemma}
\label{lemmaP_qrIsDense}

For each  $q,r \geq 1$, the set ${\mathcal P}_{q,r}$
is dense in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$).
\end{lemma}

\begin{proof}   We  will use the notation that we defined in Subsection \ref{sec-setup}. Fix $f \in \mathcal C(M)$ (resp.\ $\mathcal H(M)$) and $\varepsilon>0$.
Consider $0<\delta \mathrel{\mathop:}= \frac{1}{q'}< \min\Big \{\frac{1}{q}, \ \frac{\varepsilon}{2}  \Big \}$  such that $dist(x,y) < \delta \ $ implies $  dist(f(x) , f(y))< \varepsilon$.
We wish to $\varepsilon$-perturb $f$  into new map $g \in {\mathcal P}_{q,r}$.
Consider a triangulation $\mathcal T := \{\overline T_1,\dots,\overline T_h \}$ of $M$ such that the diameters of all the simplexes $\overline T_i$ are at most $\delta/3$.
By modifying $\mathcal T$ we can suppose that if  $\bar T_i \cap Per(f,r) \ne \emptyset$ then  $T_i \cap Per(f,r) \ne \emptyset$,  where $T_i = \mbox{int}(\overline T_i)$; and furthermore, for some
$x_i  \in T_i \cap Per(f,r)$   its orbit $\{x_i, f(x_i), \dots, f^{r-1}(x_i) \} \cap \partial \mathcal T = \emptyset$.
 Indeed, suppose some $\overline T_i$ does not satisfy these conditions.
 If no point of $\{x : f^r x = x  \}$ is in the interior of $\overline T_i$ but some is in $\partial T_i$, then we can modify the triangulation by first slightly moving one corner of $\overline T_i$ so that in the resulting triangulation  some point $x_i \in Per(f,r)$ belongs to the interior $T_i$, and all the other simplexes of $\mathcal T$ which satisfied this condition before,  still satisfy it.

If the orbit of $x_i$ does not lay in the interior of the simplexes,   let $1 \le  j \le r-1$ be the first time that $f^j(x_i) \in \partial \mathcal T$.  Again
we modify the triangulation by slightly moving one corner of the triangulation  so that in the modified triangulation $f^j(x_i) \not \in \partial \mathcal T$,
and all the other simplexes of $\mathcal T$ which satisfied this condition before,  still satisfy it.
  Repeat this procedure a finite number of times to
produce the desired triangulation. We can assume that the perturbations are so small that in  the resulting triangulation the  diameters of all the simplexes are at most $\delta/2$.

Let $\mathcal T^{(1)} \mathrel{\mathop:}=  \{\overline T_1^{(1)}, \dots, \overline T_{l}^{(1)}  \} \mathrel{\mathop:}= \{\overline T_i \in  \mathcal T:  {T_i }\cap Per(f,r) \ne \emptyset \} $  and for each $i$ let $x_i$ be a point in $T_i^{(1)} \cap Per(f,r)$. Using $x_i$ as a centroid, consider a real number $\lambda_{\overline T_i^{(1)}} >1$ near enough 1, such that $\lambda_{\overline T_i^{(1)}}{\overline T_i^{(1)}}$ is well defined.
Let $\lambda_{\mathcal T} \mathrel{\mathop:}= \min \{ \lambda_{T_i^{(1)}}: 1 \le i \le l\}$.
For $\lambda \in (1, \lambda_{\mathcal T})$ we have $x_i \in T_i \subset \bar T_i  \subset \lambda T_i$, and  $  \{\lambda T_1,\dots,\lambda T_l \}$ is an open cover of $Per(f,r)$.
We choose   such a value of  $\lambda_{max} \in (1,\lambda_{\mathcal T})$, so close to $1$ such that besides, each of the points $x_i$ does not lie inside $\lambda \bar T_j$ for any $j \ne i$, and furthermore such that the diameters of the $\lambda T_i$ are at most $\delta$.
For the rest of the proof we fix such a value of $\lambda \in (1,\lambda_{\mathcal T})$.

Denote    $$F : = \{x_{i}: 1 \leq i \leq l\}, \ \ \ \ \ F^r \mathrel{\mathop:}= \bigcup_{j= 0}^{r-1} f^j(F).$$
Hence $F$ is consisits of exactly  $l$ different points that are fixed by $f^r$, and $F^r$  consists of at most $r l$ different points, also fixed by $f^r$.

We will define an homeomorphism $h: M \mapsto M$.
Consider a chart $(U_\alpha,\phi_\alpha)$ such that $\lambda \bar T_i \subset U_\alpha$.
Choose $\eta > 0$ so small such that for each $x \in F^r$ the solid ball $B(\phi_\alpha(x),\eta) \subset \mathbb R^m$ does not intersect $\phi_\alpha(U_\alpha \cap  \partial (\lambda \mathcal T))$, such  that the diameter of  $\mathbf B(x,\eta) := \phi^{-1} (B(x,\eta))$ is at most $\delta$, and such that the finite family $\{\mathbf B(x,\eta): \ x \in F^r\}$ is composed of pairwise disjoint sets.
We define  $h(y) = y$ if $y \not \in \bigcup_{x \in F^r} \mathbf B(x,\eta))$.
On the complement we define $h$ as follows. Fix $x \in F^r$, and
 consider polar/spherical  coordinates $(s,\theta): s \in [0,1], \theta \in \mathbb S^m$ to describe  the ball $B(x,\eta)$.   Finally
  use $\phi_\alpha^{-1}$ to pull back these
coordinates to $\mathbf B(x,\eta)$.

Fix a certain $k \ge 1$ sufficiently large (that will be chosen later), and define
$h : \mathbf B(x,\eta) \mapsto \mathbf B(x,\eta)$  by setting  $h(s,\theta) \mathrel{\mathop:}= (s^{k},\theta)$. This construction for each point $x \in F^r$ completes the definition of the homeomorphism $h$.  Thus if $f$ is continuous then so is $$g: f \circ h,$$ and if $f$ is an homeomorphism, then $g$ is also an homeomorphism.

Note that for any $y \in M$ the map $h$ satisfies
$$dist(y,h(y)) < \delta \quad \mbox{ and } \quad dist(y,h^{-1}(y)) < \delta.$$
Since $g = f \circ h$ and $dist(h(y),y) <   \delta$ we have
$$\rho(f,g) < \varepsilon.$$
If besides $f$ is an homeomorphism, then $g^{-1} = h^{-1} \circ f^{-1}$ and
$$dist(g^{-1}(y),f^{-1}(y))  = dist(h^{-1}(z),z) < \delta < \varepsilon.$$
Thus
$$\rho(g^{-1} , f^{-1}) < \varepsilon.$$
We have proved that $g$ is an $\varepsilon$-perturbation of $f$ in $\mathcal C(M)$ (resp.\ $\mathcal H(M)$). Now, to end the proof of the lemma, it is enough to  choose $k$ such that the set $\mathbf B(x,\eta)$ contains a periodic shrinking set of period that divides $r$ for each point $x$ of the finite set $  F^r$.

From the above construction, we have $h(0, \theta)= 0$. Therefore, $g(x) = g(0,\theta) = f(0,\theta) = f(x)$ for all $x \in F^r$; thus  each point $x$ of $F^r$  will
be fixed by the map $g^r$.

We claim that if we choose $k$ large enough, then $g$ has a shrinking set around the point $x \in T_i \cap F^r$ contained in $\mathbf B(x,\eta)$.
We consider the $(s,\theta)$ coordinates in the set  $\mathbf B(x,\eta)$.
Fix $\eta_1 \in (0,\eta)$. For each  $j=0,1,\dots,r-1$ choose a simplex $\overline I_j$ containing $f^j(x)$ such that
 $\overline I_j \subset \mathbf B(f^j(x),\eta_1)$.
 Next choose  $\eta_2 \in (0, \eta_1)$ small enough such that $f(\mathbf B(f^jx,\eta_2)) \subset I_{j+1}$ for $j=0,1,\dots,r-1$.
 In these coordinates $\mathbf B(x,\eta_i)$ is given by $\{(s,\theta): s \in [0,\eta_i]\}$.
Choose $k > 1$ so that $\eta_1^k < \eta_2$. For each $j = 1,\dots,r$ we have
 $$g(\overline I_j) \subset g(\mathbf B(f^jx,\eta_1)) = f \circ h (\mathbf B(f^jx,\eta_1)) \subset f(B^jx,\eta_2) \subset I_{j+1}.$$
 Thus the simplexes $\overline I_j$ are the desired shrinking sets.
This finishes the proof of Lemma \ref{lemmaP_qrIsDense}.
\end{proof}

\subsection{End of the proof of Theorem \ref{Theorem-main}}

\begin{proof}

In Corollary \ref{corollaryShrink} we have proved that
 $\O_f   \subset \overline{{\mathcal E}_f} =  \overline{\mbox{Per}}_f.$
Combining Theorems   \ref{theoremBigcapClosShr_q&SRBl-l}  and \ref{TheoremSRB-l=ClosureErgodic}, we deduce that
\begin{equation}
\nonumber \mbox{Per}_f \subset \bigcap_{\varepsilon >0}\bigcap_{q \geq1}\overline{\mbox{AShr}_{\varepsilon, q}\mbox{Per}_f} \subset  \mathcal O_f. \end{equation}
Since $\mathcal O_f$ is closed,   we conclude that ${\mathcal O}_f = \overline{{\mathcal E}_f} =  \overline{\mbox{Per}}_f $, as wanted.
\end{proof}

{\footnotesize

redpajama_set_name: RedPajamaArXiv


text:
<?php

namespace Senegal\SecurityBundle\Security;

use GuzzleHttp\ClientInterface;
use GuzzleHttp\Exception\BadResponseException;
use Symfony\Component\DependencyInjection\ContainerInterface;
use Symfony\Component\Security\Core\Exception\AuthenticationException;
use Symfony\Component\Security\Core\Exception\UnsupportedUserException;
use Symfony\Component\Security\Core\User\UserInterface;

/**
 * Class ApiUserProvider.
 *
 * @codeCoverageIgnore
 */
class ApiUserProvider implements ApiUserProviderInterface
{
    private $client;
    private $container;

    public function __construct(ClientInterface $client, ContainerInterface $container)
    {
        $this->client = $client;
        $this->container = $container;
    }

    /**
     * @param $username
     * @param $password
     *
     * @return mixed
     *
     * @throws AuthenticationException
     */
    public function loadUserByUsernameAndPassword($username, $password)
    {
        try {
            $response = $this->client->post($this->container->getParameter('api_login_action'),
                [
                    'body' => [
                        'login' => $username,
                        'password' => $password,
                        'environment' => 'back',
                    ],
                ]
            );
        } catch (BadResponseException $e) {
            //var_dump($password, $e->getMessage());die();
            throw new AuthenticationException('form.login.invalid_username_or_password');
        }

        return new User($response->json());
    }

    /**
     * {@inheritdoc}
     */
    public function loadUserByUsername($username)
    {
        throw new \RuntimeException('loadUserByUsername not supported by the API');
    }

    /**
     * {@inheritdoc}
     */
    public function refreshUser(UserInterface $user)
    {
        if (!$user instanceof User) {
            throw new UnsupportedUserException(sprintf('Instances of "%s" are not supported.', get_class($user)));
        }

        return $user;
    }

    /**
     * {@inheritdoc}
     */
    public function supportsClass($class)
    {
        return $class === 'Senegal\SecurityBundle\Security\User';
    }
}

redpajama_set_name: RedPajamaGithub


Type of college/university: Public University, 23,000 students including undergraduate and graduate students.
"Learn all your student's names. Acknowledging their real presence in your classroom assists in building an environment of mutual respect and collaboration. Sometimes the simplest gestures can have the biggest impact." From a presentation by Kathleen Cumiskey, Chair of the Psychology Department at the College of Staten Island, CUNY.
Not a book or an article, but an event, Pedagogy Day (2015) at the CUNY Graduate Center was where I found a community of like-minded professors. Professor Aaron S. Richmond (Educational Psychology and Human Development at Metropolitan State University of Denver) presented an evidence-based guide to university teaching which has served as a foundation for my growth as an educator.
My favorite course to teach is Learning Theories 350. There are usually about 35 students and for most of them it's their last year and often their last class. I see this as an opportunity to make sure that students leave school knowing how learning has been studied, how to learn, and most importantly, loving to learn. My not-so-secret goal is to create lifelong learners.
Since most students are very interested in how they can learn better, we start the term with a study skills exercise including a reading, creating their own PowerPoint on the reading, and presenting it. After all that talk about deep encoding, the bounce back to Aristotle's "Laws of Association" makes sense and Pavlov's cortical mosaic concept is more accessible. From there-on-in the class follows the association theme from anticipatory association, to associations between behaviors and outcomes, right through to Hebbian synapses and all bright lights in the brain.
The syllabus progresses from lecture/discussion to an experiential assignment for each learning theory. Supplementing these basic elements are frequent 10 question quizzes and opportunities for extra credit. The quizzes serve to keep everyone's "head in the game". As experienced students, they know that if quiz grades get wobbly they need to study more. They can also take advantage of extra credit opportunities that may include three paragraph responses to "thought questions" like; "How has learning changed your behavior?"; or respond to a posted NY Times editorial on lecturing vs active learning.
Learning is a topic that has strong personal connections for my students and in many cases for their children. Many of the students are the first in their families to go to college, often their parents have worked very hard to give them this chance. A student of mine once commented that I teach like there is something at stake. I replied, "There is" and we both knew what I meant.
Early in the class I set a simple framework in place. I call roll for the first two weeks. There is so much power in connecting a name to a face and it's a sign of respect. That shared smile of recognition is so comforting.
The second structural element, is establishing teams of four to five students. These teams are not self-selected; everyone starts as strangers and through the term they become friends and study partners. Many students don't like teamwork, which I understand, but as someone who worked in the outside world, I know that learning how to collaborate is a skill that will be useful for the rest of their lives. I share this with the class and I'm met with a sea of "nobody's messing with my grade" stares. Class discussions about team dynamics and learning are helpful and by the end of the term, we have a classroom full of vibrant ideas and just the right amount of competition between the teams. Team projects are a significant percentage of their grade.
With this framework we build a community, where everyone has a role and a path to success. I teach, they learn; we all understand that neither role is passive.
Most of my class prep is done in my office at home. There is a window, a desk, my laptop, a view of the sky, and carefully managed piles of materials for each class. At my desk is an old-style metal office chair covered with this wonderful teal leatherette. Oh, and lots of books.
Learning is a Life Skill.
No disasters really, and embarrassment is an everyday fact of my teaching life. It is unfortunately true that I do not know everything about everything.
My students are pretty unflappable, but this might interest them-When I was 6 months old I moved to Shiraz, Iran for a year.
Blackboard and My iPhone make life so much easier. With mobile apps I can work wherever I am – a mixed blessing.
We often talk about how to get enough sleep.


the eyes had it, until the talons did. tiny bits of lizard left on toes?
Juvenile Red-shouldered Hawk, checking out snack options.
Jangly, shimmery summer 90's music has been on the playlist (thank you dj husband) and doors are open in the studio. Yard fires, bird song, and that awesome musky blooming plant smell keep luring us outside. The tufty back pasture behind our place is a fabulous hunting ground for birds. Poo- and viscera-splattered fence posts are testimony to its attraction, in particular to Red-shouldered Hawks. Snakes, anoles, and little hairy unidentifiable things seem to be the primary prey.
happy confluence and source of post title, listening to Silver Jews / David Berman, and walking out to see this. Black Racer, and Dichondra carolinensis.
and the other side. hope this little one makes it.
Black Racers prefer the brushy perimeters. This one was cruising the edge of the yard with its head held up like a periscope. Usually these snakes scoot away when approached, but this individual had a cloudy eye and some scarring along its sides, so perhaps flight mode is compromised. Maybe a run-in with a weed-wacker or lawn mower? Black racers have excellent vision and hunt by sight. It watched the camera with its good and gorgeous eye, and rattled its tail when the lens got too close.
A star is born. A Starfish Stinkhorn, Aseroe rubra – removed from its egg-like structure under the leaves.
What a dumb play on words. But really, we had fun, we saw galls, and some were caused by fungus. It's true! Last week, an EF1 tornado blew through the neighborhood. It was a scary novelty exaggerated just a teeny bit by excitable local news media and booze-addled neighbors. Hail and huge raindrops pelted the ground before and after, so the ground was saturated – perfect conditions for some weird stuff to appear.
fleshy underbelly of stinky starfish.
and a closer view on concrete. Be thankful for no scratch and sniff option.
Starfish Stinkhorns, Aseroe rubra, have worldwide distribution, but are apparently not common in North Carolina, and may have been introduced. We've never seen them, so it was treated like an alien landing – we pointed, poked, prodded, and smelled. (Indeed, they are stinky.) Thank you Mushroom Expert for identification!
Exobasidium vaccinii, perhaps? This flower gall is at least twice the size of typical flowers, shadow visible upper right.
Broad-headed Skink, about 7.5 inches long.
During our wanderings, some scaly life forms also made an appearance in the damp maritime forest floor. A Broad-headed Skink scuttled up a tree to avoid getting squashed, making a racket worthy of a much larger scarier creature. And this little Yellow Rat Snake eyeballed us from the path edge, braving some close-up observation without flinching or bolting.


text:
US Immigration Reforms Revisited
By Almas Akhtar, TMO
President Barack Obama announced full support behind a comprehensive overhaul of U.S. immigration laws during his second term. On Feb 5, 2013 in Las Vegas, he said "now's the time" to replace a system which he regarded as "out of date and badly broken."
The president specified three pillars of immigration reform: better enforcement of immigration laws, dealing with citizenship for the more than 11 million undocumented immigrants already in the country, and revamping the legal immigration system.
A group of senators from both parties earlier unveiled the framework of a broad bipartisan reform to the nation's immigration laws, including a track towards citizenship for the 11 million undocumented aliens in the United States.
The detailed reform will carry the signatures of four Republicans and four Democrats. Senators Chuck Shumer, John McCain, Lindsey Graham, Dick Durbin and Marco Rubio are a few of the notable members of this group. This document will supposedly be drafted by March this year . It is also believed that it will produce tough but fair route to citizenship to the illegal immigrants living in the country.
Immigrants have played a very significant role in the financial growth of the country. Immigrants are indispensable to the U.S. economy, they are a significant part of the US workforce, research shows that one in every seven workers in the country is an immigrant. Immigrants are more likely to be entrepreneurs than native-born U.S. citizens. One of every four engineering and technology companies created in the United States between 1995 and 2005 had immigrant founders. Foreign-born scholars, scientists, and engineers have played imminent role in the prosperity of the country. In the last two decades over one third of Nobel Prizes in the U.S. were awarded to foreign-born scientists. Hispanics, Chinese and South Asian Americans are the three major immigrant groups residing in United States.
Palestine Awareness
United to Quell Civil Unrest in Kuwait
By Sumayyah Meehan, MMNS Ever since sectarian violence between Shi'a and Sunnis broke out in Iraq, as a result of…
Trump and Iran: What, Why and When did it happen?
October 19th , 2018 Vol 20 issue 13
Islamic Summit Opens With Calls For Syrian Dialogue Syrians Fight Close To Heart Of Damascus

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: flutter how to make card in leading of listview I want to make UI like my upper image but I cant make like that But it is not being made as you are seeing in the image below.
I want to make leading of ListTile part like this .

But I can't make it please help me.
this is my code and UI
import 'package:flutter/material.dart';

class MyHomePage extends StatefulWidget {
  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Container(
        padding: EdgeInsets.only(top: 30, left: 10, right: 10),
        child: Column(
          children: [
            Row(
              children: [
                Text(
                  'Your rooms',
                  style: TextStyle(fontSize: 25, fontWeight: FontWeight.bold),
                ),
                Spacer(),
                Card(
                  elevation: 1,
                  shape: RoundedRectangleBorder(
                    borderRadius: BorderRadius.circular(15.0),
                  ),
                  child: Padding(
                    padding: const EdgeInsets.all(5.0),
                    child: Row(
                      children: [
                        Text('add',
                            style: TextStyle(fontSize: 20, color: Colors.grey)),
                        Icon(
                          Icons.add,
                          size: 20,
                          color: Colors.grey,
                        )
                      ],
                    ),
                  ),
                )
              ],
            ),
            SizedBox(
              height: 20,
            ),
            Container(
              height: 110,
              width: 380,
              // color: Colors.red,
              decoration: new BoxDecoration(
                borderRadius: BorderRadius.circular(30),
                color: Color(0xffbbf0e3),
              ),
              child: Center(
                child: ListTile(
                  leading: Container(
                    // height: 10,
                    width: 100,
                    child: Card(
                      shape: RoundedRectangleBorder(
                          borderRadius: BorderRadius.circular(30.0)),
                      elevation: 1,
                      color: Colors.white,
                      child: Container(
                        decoration: BoxDecoration(
                          // borderRadius: BorderRadius.circular(25),
                          shape: BoxShape.circle,
                          // color: Colors.red,
                        ),
                        child: CircleAvatar(
                          radius: 25,
                          backgroundImage: NetworkImage(
                              'https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTkNWsvdJGKCRYW_Wb-K40Po2JQ8LpooWoilw&usqp=CAU'),
                        ),
                      ),
                    ),
                  ),
                  // leading: Card(
                  //     shape: RoundedRectangleBorder(
                  //         borderRadius: BorderRadius.circular(30.0)),
                  //     elevation: 1,
                  //     color: Colors.white,
                  //     child: Container(
                  //       height: 80,
                  //       width: 75,
                  //       decoration:  BoxDecoration(
                  //         // borderRadius: BorderRadius.circular(25),
                  //         shape: BoxShape.circle,
                  //         // color: Colors.red,
                  //       ),
                  //       child: CircleAvatar(
                  //         radius: 25,
                  //         backgroundImage: NetworkImage(
                  //             'https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTkNWsvdJGKCRYW_Wb-K40Po2JQ8LpooWoilw&usqp=CAU'),
                  //       ),
                  //     )),
                  title: Text(
                    "Living room",
                    style: TextStyle(fontSize: 20, fontWeight: FontWeight.w700),
                  ),
                  subtitle: Text(
                    '3 turned on',
                    style: TextStyle(
                        fontWeight: FontWeight.w600,
                        fontSize: 20,
                        color: Colors.green),
                  ),
                  trailing: Card(
                    elevation: 1,
                    shape: RoundedRectangleBorder(
                      borderRadius: BorderRadius.circular(15.0),
                    ),
                    child: Padding(
                      padding: const EdgeInsets.all(5.0),
                      child: Container(
                        width: 50,
                        child: Row(
                          mainAxisSize: MainAxisSize.min,
                          mainAxisAlignment: MainAxisAlignment.center,
                          children: [
                            Text('5',
                                style: TextStyle(
                                    fontSize: 20, color: Colors.grey)),
                            SizedBox(
                              width: 1,
                            ),
                            Icon(
                              Icons.electrical_services,
                              textDirection: TextDirection.rtl,
                              size: 20,
                              color: Colors.grey,
                            )
                          ],
                        ),
                      ),
                    ),
                  ),
                ),
              ),
            )
          ],
        ),
      ),
    );
  }
}

I am making like this .

A: I removed some unnecessary Lines of Code and voilà there you go:
import 'package:flutter/material.dart';

class MyHomePage extends StatefulWidget {
  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Container(
        padding: EdgeInsets.only(top: 30, left: 10, right: 10),
        child: Column(
          children: [
            Row(
              children: [
                Text(
                  'Your rooms',
                  style: TextStyle(fontSize: 25, fontWeight: FontWeight.bold),
                ),
                Spacer(),
                Card(
                  elevation: 1,
                  shape: RoundedRectangleBorder(
                    borderRadius: BorderRadius.circular(15.0),
                  ),
                  child: Padding(
                    padding: const EdgeInsets.all(5.0),
                    child: Row(
                      children: [
                        Text('add',
                            style: TextStyle(fontSize: 20, color: Colors.grey)),
                        Icon(
                          Icons.add,
                          size: 20,
                          color: Colors.grey,
                        )
                      ],
                    ),
                  ),
                )
              ],
            ),
            SizedBox(
              height: 20,
            ),
            Container(
              height: 110,
              width: 380,
              // color: Colors.red,
              decoration: new BoxDecoration(
                borderRadius: BorderRadius.circular(30),
                color: Color(0xffbbf0e3),
              ),
              child: Center(
                child: ListTile(
                  leading: CircleAvatar( // I ONLY CHANGED THIS PART
                          radius: 25,
                          backgroundImage: NetworkImage(
                              'https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTkNWsvdJGKCRYW_Wb-K40Po2JQ8LpooWoilw&usqp=CAU'),
                  ),
                  // leading: Card(
                  //     shape: RoundedRectangleBorder(
                  //         borderRadius: BorderRadius.circular(30.0)),
                  //     elevation: 1,
                  //     color: Colors.white,
                  //     child: Container(
                  //       height: 80,
                  //       width: 75,
                  //       decoration:  BoxDecoration(
                  //         // borderRadius: BorderRadius.circular(25),
                  //         shape: BoxShape.circle,
                  //         // color: Colors.red,
                  //       ),
                  //       child: CircleAvatar(
                  //         radius: 25,
                  //         backgroundImage: NetworkImage(
                  //             'https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTkNWsvdJGKCRYW_Wb-K40Po2JQ8LpooWoilw&usqp=CAU'),
                  //       ),
                  //     )),
                  title: Text(
                    "Living room",
                    style: TextStyle(fontSize: 20, fontWeight: FontWeight.w700),
                  ),
                  subtitle: Text(
                    '3 turned on',
                    style: TextStyle(
                        fontWeight: FontWeight.w600,
                        fontSize: 20,
                        color: Colors.green),
                  ),
                  trailing: Card(
                    elevation: 1,
                    shape: RoundedRectangleBorder(
                      borderRadius: BorderRadius.circular(15.0),
                    ),
                    child: Padding(
                      padding: const EdgeInsets.all(5.0),
                      child: Container(
                        width: 50,
                        child: Row(
                          mainAxisSize: MainAxisSize.min,
                          mainAxisAlignment: MainAxisAlignment.center,
                          children: [
                            Text('5',
                                style: TextStyle(
                                    fontSize: 20, color: Colors.grey)),
                            SizedBox(
                              width: 1,
                            ),
                            Icon(
                              Icons.electrical_services,
                              textDirection: TextDirection.rtl,
                              size: 20,
                              color: Colors.grey,
                            )
                          ],
                        ),
                      ),
                    ),
                  ),
                ),
              ),
            )
          ],
        ),
      ),
    );
  }
}

Output

redpajama_set_name: RedPajamaStackExchange


text:
Dear reader!
From Helsinki to Brazzaville, from Tangier to Izmir – the Transborder Summer Camp brought more than 500 activists from more than 90 cities together. An impressive composition, a series of exceptional discussions, plenaries and workshops – and many inspiring encounters and exchanges: in our evaluations we all agree, it was unforgettable and inspiring!
We once formulated the goal "to build and extend infrastructures for freedom of movement and equal rights"– and we think this is exactly what we managed to do in those six days at the ZAD near Nantes.
The initiative for the camp was started by Welcome to Europe, but many other networks got involved and mobilized and prepared for the camp. We decided, not to do an open call-out, but rather, to approach groups and initiatives, whom we knew from all our networks and through concrete projects, individually. As a result, friends, who are active on the ground, informed and invited more friends, who are also involved in their local struggles.
The atmosphere in the camp was characterized by mutual respect and an interest to learn from each other, by an eagerness to create more exchange and even better cooperation along all routes of flight and migration and in the transit and destination cities.
Of course, this brochure cannot reproduce and cover all exchanges and discussions in the camp. But it includes several texts and articles, which are exemplarily for the spectrum of topics and as well for the whole approach of this summer camp.
The message was very clear at the end of the camp: we go on, from the local to transnational level, and we will not give up fighting for our demands. Despite increasingly hard and repressive times and against the ongoing roll back of migration control, we will continue our daily struggles and structures against the border regime.
Within the last months we reflected again on the camp in several meetings and already we pre-decided: we want to come together latest again in a similar composition in summer 2022, after three years. This brochure might help to keep the idea vivid as well as the contacts. We hope, that more people, who like this kind of activism, who missed the last camp or who just started initiatives with a similar approach, can also find to us – for the daily struggles and for the next Transborder Summer Camp.