Thanks to our good friends at Norstar Genetics we bring you a new and exciting cross. The Nepali OG is very unusual and combined with the grape flavor of Querkle with undertones of Space queen this is one great smelling plant. Rock hard buds are produced even outdoors and in light dep. The buds are hard enough to shoot in a slingshot which adds to the total weight. The overall smell is of Hash mixed with fruits berries and cherries.
This amazing new cross is unlike any other "Subcools The Dank" strain that has been created to date IMO. The combination of the rare Nepali OG with a hint of Querkle that adds a grape flavoring that makes you lick your lips. It also has an amazing Hash flavor from the Nepali. What I tasted was grape hash and a hint of Berries. It has a hashy background taste that is very intriguing. The high is deep and thoughtful while also being very relaxing. While the grape flavor is in the forefront there are so many undertones, Hashy, Cherry, Earthy and Fruity with a hint of Vanilla, a true full experience of the taste spectrum.
Rock hard buds are produced even outdoors and in light dep. The buds are hard enough to shoot in a slingshot which adds to the total weight. The overall smell is of Hash mixed with fruits berries and cherries. The high is deep and thoughtful while also being very relaxing. While the grape flavor is in the forefront there are so many undertones, Hashy, Cherry, Earthy and Fruity with a hint of Vanilla, a true full experience of the taste spectrum.
Mostly Nepali Dominant with a hint or grape taste and purple color from the querkle. Incredible colors can be achieved with proper growing methods.
Super hard dense buds lend to the weight of this strain but the hash returns are staggering!
Calming but very potent the flavor keeps you taking hit after hit until your nicely toasted. I was surprised how potent the hash was I took only two small hits and was smiling like a fool.


text:
\section{Introduction}

In the present paper we contribute to the theory of multifractal analysis for asymptotically additive sequences of sequentially saturated maps(see Definition \ref{def sequentially saturated}), that include maps that satisfy the specification property(as a large class of uniformly hyperbolic maps). The main result is that the irregular set of maps sequentially saturated for a asymptotically additive potential sequence $\Phi=(\phi_n)_n$ of a topological dynamical system $(X,f)$ carries the full asymptotically additive topological pressure. This generalizes results of \cite{ercai,thompson2012,Thomp2009,Zhao2011} and \cite{Tian2017}. For this purpose, we give a version of the variational principle(in the context ``asymptotically additive") for certain non compacts as in \cite{Pfister2007,Pei10,Thomp2009}. The details are given below.

Let $(X,d,f)$ be a topological dynamical system(TDS), where  $f:X\rightarrow X$ is a continuous map and $X$ is a compact metric space. Let $C^0(f,X)$ denote the space of continuous functions on $X$. A sequence of continuous functions $\Phi:=(\phi_n)_n$ is called \textit{asymptotically additive} for $f$ if for each $\delta>0$, there exists a continuous function $\psi_{\delta}$ such that

\[\limsup_{n\rightarrow \infty}\frac{1}{n}\|\phi_n-S_n\psi_{\delta}\|<\delta,\]
where $\|\cdot\|$ is the supremum norm and $S_n\psi_{\delta}(x)=\sum_{i=0}^{n-1}\psi_{\delta}(f^i(x))$. We denote by $\mathfrak{C}^0(f,X)$   the space of asymptotically additive sequences endowed with the product topology.

Let $\mathcal{M}(X)$ be the space of Borel probability measures on $X$,  $\mathcal{M}_f(X)\subset \mathcal{M}(X)$ denote the space of $f$-invariant Borel probability measures and $\mathcal{M}_f^e(X)\subseteq \mathcal{M}_f(X)$ the subset of the ergodic measures.

In \cite{Feng2010}, Feng et al defined the Lyapunov exponent of a asymptotically additive sequence $\Phi$ at $x$. For a asymptotically additive sequence $\Phi=(\phi_n)_n$  on $X$ and $x\in X$, the Lyapunov exponent of $\Phi$ at $x$ is the limit(whenever it exists)

\begin{eqnarray}\label{limit lyapunov}
\lambda_{\Phi}(x)=\lim_{n\rightarrow \infty}\frac{\phi_n(x)}{n}.
\end{eqnarray}
By Kingman's sub-additive ergodic theorem, for any $\mu \in \mathcal{M}_f^e(X)$,

\[\lambda_{\Phi}(x)=\Phi^{*}(\mu)\;\;\;\;\;\mbox{for} \;\mu-\mbox{a.e.}\, x\in X,\]
where $\Phi^{*}(\mu):=\lim_{n\rightarrow \infty}\int \frac{\phi_n(x)}{n}d\mu(x)$, that always exists, see Proposition A.1 of \cite{Feng2010}. The authors consider the distribution of the Lyapunov exponents of $\Phi$ on $\alpha$-\textit{level set}  of $\lambda_{\Phi}$. More precisely, for any $\alpha \in \mathbb{R}$ we define the set

\[E({\Phi},f,\alpha)=\{x\in X:\lambda_{\Phi}=\alpha\}.\]
They study the topological entropy $h_{f}(E(\Phi,f,\alpha))$ in the sense of Bowen\cite{Bow73}. If we consider the multifractal decomposition

\[X=R(\Phi,f)\cup I(\Phi,f), \]
where $R(\Phi,f)=\bigcup_{\alpha \in \mathbb{R}}E(\Phi,f,\alpha)$ is the $\Phi$-regular set($x\in R(\Phi,f)$ is called $\Phi$-regular  point)   and $I(\Phi,f):=\{x \in X: \lim_{n\rightarrow \infty}\frac{\phi_n(x)}{n}\, \mbox{does not exist}\}$ is the $\Phi$-irregular set($x\in I(\Phi,f)$ is called $\Phi$-irregular point), an interest question is about the entropy of the set $I(\Phi,f)$. Here, we address this question. More precisely, we studied the set $I(f):=\bigcup_{\Phi \in \mathfrak{C}^0(X)}I(\Phi,f)$, called of irregular set. As the product $\Pi_{n=1}^{\infty}C^0(f,X)$ is a separable set, then $\mathfrak{C}^0(f,X)\subset \Pi_{n=1}^{\infty}C^0(f,X)$ it is also a separable set, i.e., there exists a enumerable and dense set $\{(\Phi_n)_n;  n \in \mathbb{N}\}$ in $\mathfrak{C}^0(f,X)$ where $ \Phi_n=(\phi^n_k)_{k\in \mathbb{N}}$ such that, by Kingman's Theorem,   $\mu\left(\bigcap_{n=1}^{\infty}R(\Phi_n,f)\right)=1$. Then, for any $\Phi=(\phi_k)_{k\in \mathbb{N}} \in \mathfrak{C}^0(f,X)$ and $\varepsilon>0$, we can choose a $l\in \mathbb{N}$ such that
$$
\|\Phi-\Phi_l\|\leq \varepsilon.
$$
Then, for any $x\in R(f):= \bigcap_{n=1}^{\infty}R(\Phi_n,f)$,
$$
\limsup_{n\rightarrow \infty}\frac{\phi_n(x)}{n}\geq \lim_{n\rightarrow \infty}\frac{\phi_n^l(x)}{n}-\varepsilon
$$
$$
\liminf_{n\rightarrow \infty}\frac{\phi_n(x)}{n}\leq \lim_{n\rightarrow \infty}\frac{\phi_n^l(x)}{n}+\varepsilon
$$
then, $\limsup_{n\rightarrow \infty}\frac{\phi_n(x)}{n}=\liminf_{n\rightarrow \infty}\frac{\phi_n(x)}{n}$. Therefore, $I(f)=X\backslash R(f)$ is a mensurable set.

Various  authors have studied the irregular set and proved, in many cases, this  has full entropy and topological pressure. Pesin and Pitskel \cite{Pesin1984}, showed that it carried the full entropy in the case of the Bernoulli shift on two symbols, Barreira and Schmeling(\cite{Barreira2000}) for case of
generic Hölder continuous function on a conformal
repeller. See more in \cite{li2013,li2014}, \cite{Thomp2010}, \cite{thompson2012}, \cite{BarLiVal} \cite{bomfim2015} and \cite{bomfim2017}, for others authors.

The studies cited above were made for irregular sets under one observable. In \cite{Tian2017}, Tian proposes a study for irregular sets under a collection of observable functions. We make a contribution to the study of irregular sets under a collection finite or infinite of asymptotically additive potentials. More precisely, let $\mathfrak{\hat{C}}^0(f,X):= \{\Phi \in \mathfrak{C}^0(f,X):I(\Phi,f)\neq \emptyset\}$. We prove that the set $I(\Phi,f)$ carries full additive asymptotically topological pressure. In fact, we consider the additive asymptotically topological pressure
$P_f(\bigcap_{\Psi \in D}I(\Psi,f),\Phi)$, where $D$ is an subset of $\mathfrak{\hat{C}}^0(f,X)$(and even uncountable), see Theorem B below.

Let $Z\subset X$  $f$-invariant Borel set(i.e., $f^{-1}(Z)=Z$), denote by $\mathcal{E}(Z,f)=\{\mu \in \mathcal{M}_f^e(X):\mu(Z)=1\} $ and $\mathcal{M}_xf$ the space of limit measures of the sequence of measures, in weak$^*$ topology,
\[\mathcal{E}_n(x):=\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^j(x)}.\]

In \cite{Pesin1984}, Theorem A2.1, Pesin and Pitskel proved that for a $f$-invariant Borel set $Z\subset X$, if we consider the Borel $f$-invariant set  $\mathcal{Z}=\{x\in Z:\mathcal{E}(Z,f)\cap \mathcal{M}_xf \neq \emptyset\}$, we have for any continuous function $\varphi \in C(X)$

\[P_f(\mathcal{Z},\varphi)=\sup\{h_{\mu}(f)+\int_{Z}\varphi d\mu: \mu \in \mathcal{E}(Z,f)\}.\]

\begin{definition}\label{def sequentially saturated}
	A TDS $f:X\rightarrow X$ is \textit{sequentially saturated}, if for any $\Phi \in \mathfrak{C}^0(f,X) $ and any compact connected nonempty set $K\subseteq \mathcal{M}_f(X)$ we have

	\[P_f(G_K,\Phi)=\inf\{h_{\mu}(f)+\Phi^{*}(\mu): \mu \in K\},\]
	where $G_K=\{x \in X:\mathcal{M}_xf=K\}$.
\end{definition}
We proved that
\begin{teoB}\label{teo B}
	Let $(X,f)$ be a dynamical system sequentially saturated and assume that $\mathfrak{\hat{C}}^0(f,X)\neq \emptyset$. Then, for any  subset $D\subseteq \mathfrak{\hat{C}}^0(f,X)$ and for each $\Phi \in \mathfrak{C}^0(f,X) $ we have

	\[P_f(\bigcap_{\Psi \in D}I(\Psi,f),\Phi)=P_f(\Phi).\]

\end{teoB}

A subset $D\subset X$ is saturated if $x\in D$ and the sequences $\mathcal{E}_n(x)$ and $\mathcal{E}_n(y)$ have the same limit-point set, then $y\in D$. Of particular interest are the generic points of $\mu$, i.e. points that satisfy $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^j(x))=\int \varphi d\mu$. We denote the saturated set of generic points of $\mu$ by $G_{\mu}$. In \cite{Bow73} Bowen proved that if $\mu$ is ergodic, then
\begin{eqnarray}\label{variational bowen}
h_{f}(G_{\mu})=h_{\mu}(f).
\end{eqnarray}
We know that if $\mu\in \mathcal{M}_f(X)$, then $\mu$ is ergodic if and only if $\mu(G_{\mu})=1$. For non-ergodic measures, the Equation (\ref{variational bowen}) cannot hold. It is not difficult give examples such that $G_{\mu}=\emptyset$ and $h_{\mu}(f)>0$.

In \cite{Pfister2007} Pfister and Sullivan treat of the case of non-ergodic measures, introducing two conditions on the dynamics: the
$g$\textit{-almost product property} and the \textit{uniform separation property}. They proved that

\begin{theorem}\cite{Pfister2007}\label{theo pfister}
	If the $g$-almost product property and the uniform separation property hold, then for any compact connected non-empty set $K\subset \mathcal{M}_f(X)$
	\[\inf\{h_{\mu}(f):\mu \in K\}=h_{f}(G_K).\]
\end{theorem}
In \cite{Pei10}, Pei and Chen generalized Theorem \ref{theo pfister} for the case of the topological pressure.

\begin{theorem}\label{theo pei}\cite{Pei10}
	If the $g$-almost product property and the uniform separation property hold, then for any compact connected non-empty set $K\subset \mathcal{M}_f(X)$
	\[\inf\{h_{\mu}(f)+\int \varphi d\mu:\mu \in K\}=P_f(\varphi,G_K).\]

\end{theorem}

 We prove a variational principle for the  asymptotically additive topological pressure on certain non-compact sets. We prove that

\begin{teoA}\label{teoA}
	If $f$ satisfies the $g$-almost product property and the uniform separation property, then $f:X\rightarrow X$ is sequentially saturated.

\end{teoA}

\begin{remark}
The proof of Theorem A is given in the subsection  \ref{subsection A} following the lines of \cite{Pfister2007}. We note here that the proof of Theorem A can also be given in a simpler way. We give the details below:

As $\Phi=\{\phi_n\}_n \in \mathfrak{C}^0(f,X)$ is a limit of additive sequences, for every each $k\in \mathbb{N}$, there exist $\psi_k$ such that
$$
\limsup_{n\rightarrow \infty}\frac{1}{n}\|\phi_n-S_n\psi_k\|<\frac{1}{k}.
$$
By Definition \ref{def pressure assinto.} of  asymptotically additive topological pressure  and the definition of topological pressure for continuous maps, it's not hard to see that for $Z\subset X$ we have
$$
\|P(Z,\Phi)-P(Z,\psi_k)\|<\frac{1}{k}\Rightarrow P(Z,\Phi)=\lim_{k\rightarrow \infty}P(Z,\psi_k)
$$
and that for each $\mu \in \mathcal{M}_f(X)$
$$
|\lim_{n\rightarrow \infty}\int \frac{\phi_n}{n}d\mu-\int \psi_k d\mu|<\frac{1}{k} \Rightarrow \Phi^*(\mu)=\lim_{k\rightarrow \infty}\int \psi_k d\mu
$$
By Theorem \ref{theo pei}, for each $k$
$$
P(G_K,\psi_k)=\inf\{h_{\mu}(f)+\int \psi_k d\mu:\mu \in K\}
$$
then
$$
P(G_K,\Phi)=\inf\{h_{\mu}(f)+ \Phi^*(\mu):\mu \in K\}
$$
\end{remark}

In the Section \ref{aplication} we will give some applications and examples of the main results on Cocycles under shift of finite type,  Nonconformal repeller and others.

\section{Preliminaries}
In this section, we remember some concepts and we given some notations for the proof of the Theorem A and Theorem B.

For $a,b \in \mathbb{N}$, $a\leq b$, we denote $[a,b]:=\{c\in \mathbb{N}:a\leq c\leq b\}$ and $\Lambda_n:=[0,n-1]$. The cardinality of a set $\Lambda$ is denoted by $\lvert\Lambda\rvert$ or $\#\Lambda$ .

We set $\langle \phi,\mu \rangle:=\int \phi d\mu$. We define a metric on $\mathcal{M}(X)$ by

$$d(\mu,\nu)=\|\mu-\nu\|:=\sum_{k\geq 1}2^{-k}\lvert\langle \psi_k,\mu-\nu\rangle\rvert.$$
where $\{\psi_k\}_{k\in \mathbb{N}}$ is a countable and dense set of continuous functions on these taking values on $[0,1]$. We use the metric on $X$ given by $d(x,y):=d(\delta_x,\delta_y)$.

We recall the definition of \textit{$g$-almost product property} introduced in \cite{Pfister2007}.
\begin{definition}
	Let $g:\mathbb{N}\rightarrow \mathbb{R}$ be a given nondecreasing unbounded map with the properties
	\[g(n)<n\;\;\;\; \mbox{and}\;\;\; \lim_{n\rightarrow \infty}\frac{g(n)}{n}=0.\]
	The function $g$ is called blowup function. Let $x\in X$ and $\varepsilon>0$. The $g$-blowup of $B_n(x,\varepsilon)$ is the closed set
	\[B_n(g;x,\varepsilon):=\left\{y\in X: \exists \Lambda\subset \Lambda_n, \lvert\Lambda_n\backslash \Lambda\rvert\leq g(n)\;\mbox{and}\; \max\{d(f^j(x),f^j(y)):j\in \Lambda\leq \varepsilon\right\}.\]
\end{definition}

\begin{definition}
	A TDS $f$ has the $g$-almost product property with blowup function $g$, if there exists a nonincreasing function $m:\mathbb{R}^+\rightarrow \mathbb{N}$, such that for any $k\in \mathbb{N}$, any $x_1\in X,...,x_k\in X$, any positive $\varepsilon_1,...,\varepsilon_k$, and any integers $n_1\geq m(\varepsilon_1),...,n_k\geq m(\varepsilon_k)$, the intersection

	\[\bigcap_{j=1}^kf^{-M_{j-1}}B_{n_j}(g;x_j,\varepsilon_j)\neq \emptyset\] where $M_0=0, M_i=n_1+...+n_i$, $i=1,...,k-1$.
\end{definition}
For $\delta>0$ and $\varepsilon>0$, two points $x$ and $y$ are $(\delta,n,\varepsilon)$-separated if

\[\#\{j:d(f^j(x),f^j(y))>\varepsilon,0\leq j\leq n-1\}\geq \delta n.\]
\begin{remark}
	Note that $g$-almost product property is weaker than Bowen's specification property because it requires only partial shadowing of the specified orbit
	segments. All $\beta$-shifts satisfy the $g$-almost product property, see \cite{Pfister2007}.
\end{remark}

A subset $E$ is $(\delta,n,\varepsilon)$-separated if any pair of different points of $E$ are $(\delta,n,\varepsilon)$-separated.
Let $F\subseteq \mathcal{M}(X)$ be a neighborhood of $\nu\in \mathcal{M}_f(X)$, we set $X_{n,F}=\{x\in X:\mathcal{E}_n(x)\in F\}$.

\begin{proposition}\cite{Pfister2005}\label{prop separation}
	Let $\nu \in \mathcal{M}_f(X)$ be ergodic and $h^*<h_{\nu}(f)$. Then there exist $\delta^*>0$ and $\varepsilon^*>0$ so that for each neighborhood $F$ of $\nu$ in $\mathcal{M}(X)$, there exists $n_{F,\nu}^*\in \mathbb{N}$ such that for any $n\geq n_{F,\nu}^*\in \mathbb{N}$, there exists a $(\delta^*,n,\varepsilon^*)$-separated set $\Gamma_n$, such that

	\[\Gamma_n\subset X_{n,F} \;\;\mbox{and}\;\; \lvert\Gamma_n\rvert\geq e^{nh^*}.\]
\end{proposition}

Let $N(F,\delta,n,\varepsilon)$ be the maximal cardinality of an $(\delta,n,\varepsilon)$-separated set of $X_{n,F}$ and
$N(F,n,\varepsilon)$  the maximal cardinality of an $(n,\varepsilon)$-separated set of $X_{n,F}$.

\begin{definition}
	A TDS $f$ has uniform separation property if the following holds. For any $\eta>0$, there exists $\delta^*>0$ and $\varepsilon^*>0$ so that for $\mu$ ergodic and any neighborhood $F\subset \mathcal{M}_f(X)$ of $\mu$, there exists $n_{F,\mu,\eta}^*$ such that, for $n\geq n_{F,\mu,\eta}^*$, we have

	\[N(F,\delta^*,n,\varepsilon^*)\geq 2^{n(h_{\mu}(f)-\eta)}.\]
\end{definition}

The previous definition says that the conclusion of Proposition \ref{prop separation} holds uniformly(i.e. $\delta^*, \varepsilon^*$ does not depend of the measure $\nu$) for a TDS with the uniform separation time.

\begin{remark}
	It is easy to see  that uniform separation implies that $h_{top}(f)$ is finite.

\end{remark}
\begin{remark}
	In \cite{Pfister2007}, the authors proved that expansive and asymptotically $h$-expansive maps have the uniform separation property.

\end{remark}
We define
\[\underline{s}(\nu,\varepsilon)=\inf_{\nu \in F}\liminf_{n\rightarrow \infty}\frac{1}{n}\log N(F,n,\varepsilon)\;\;\;\;\mbox{and}\;\;\;\;\overline{s}(\nu,\varepsilon)=\inf_{\nu \in F}\limsup_{n\rightarrow \infty}\frac{1}{n}\log N(F,n,\varepsilon)\]
where the infimum is take over any base of neighborhood of $\nu$.
Let $\overline{s}(\mu)=\lim_{\varepsilon \rightarrow 0}\overline{s}(\nu,\varepsilon)$ and $\underline{s}(\mu)=\lim_{\varepsilon \rightarrow 0}\underline{s}(\nu,\varepsilon)$.

\begin{proposition}[\cite{Pfister2007}]\label{proposition limit entropy}
	If $\mu \in \mathcal{M}_f(X)$, then $\overline{s}(\mu)\leq h_{\mu}(f)$.
\end{proposition}

\begin{definition}
	The ergodic measures are entropy-dense if for any $\nu \in \mathcal{M}_f(X)$, each neighborhood $F$ of $\nu$, and $h^*<h_{\nu}(f)$, there exists ergodic measure $\rho \in F$ such that $h^*<h_{\rho}(f)$.
\end{definition}

\begin{proposition}[\cite{Pfister2007}]
	Assume that $f$ has the uniform separation property and the ergodic measures are entropy-dense. For any $\eta>0$, there exist $\delta^*>0$ and $\varepsilon^*>0$ so that for $\mu \in \mathcal{M}_f(X)$ and any neighborhood $F\subset \mathcal{M}_f(X)$ of $\mu$, there exists $\eta_{F,\mu,\eta}^*$ such that
	\[N(F;\delta^*,n,\varepsilon^*)\geq e^{n(h_{\mu}(f)-\eta)}\;\;\mbox{if}\;\; n\geq \eta_{F,\mu,\eta}^*. \]
\end{proposition}
\begin{proposition}[\cite{Pfister2007}]
	If the uniform separation property is true and the ergodic measures are entropy-dense, then $s(\mu):=\overline{s}(\mu)=\underline{s}(\mu)$ is well-defined and $s(\mu)=h_{\mu}(f)$ for all $\mu \in \mathcal{M}_f(X)$.
\end{proposition}

In \cite{Zhao2011}, the authors gives a definition of asymptotically additive  topological pressure for an set $Z\subseteq X$ as in \cite{Bar06} and \cite{Mum2006}, motivated by work of \cite{Thomp2010} and \cite{Zhao2008}. Fix $\varepsilon>0$. Let $Z\subset X$, $\Gamma=\{B_{n}(x,\varepsilon)\}_n$ a cover of $Z$ and $\Phi=(\phi_n)_n$  an asymptotically additive sequence. For $\alpha \in \mathbb{R}$  we define the following quantities:

\[Q(Z,\alpha, \Gamma, \Phi):= \sum_{B_{n}(x,\varepsilon)\in \Gamma}e^{-\alpha n+\sup_{x\in B_{n}(x,\varepsilon)}\phi_{n}(x)}\,\;\;\mbox{and}\;\;\]

\[M(Z,\alpha,\varepsilon,N,\Phi)=\inf_{\Gamma}Q(Z,\alpha, \Gamma, \Phi).\]
where the infimum is taken over all covers finite or countable of the form $\Gamma=\{B_{n}(x,\varepsilon)\}_n$ of $Z$ with $n(\Gamma):= \min_n\{n\}\geq N$. We define
\[m(Z,\alpha,\varepsilon,\Phi)=\lim_{N\rightarrow \infty} M(Z,\alpha,\varepsilon,N,\Phi).\]
As the function $M(Z,\alpha,\varepsilon,N,\Phi)$ is non-decreasing on $N$, then the limit always exist. We can show that
\[
P_f(Z,\Phi,\varepsilon):=\inf\{\alpha:m(Z,\alpha,\varepsilon,\Phi)=0\}=\sup\{\alpha:m(Z,\alpha,\varepsilon,\Phi)=\infty\}.\]

\begin{definition}\label{def pressure assinto.}
	The topological pressure of $\Phi$ on $Z$ is given by
$$
P_f(Z,\Phi)=\lim_{\varepsilon \rightarrow 0}P_f(Z,\Phi, \varepsilon).$$
\end{definition}
A sequence $\Phi$ is an almost-additive sequence if
\[\phi_n(x)+\phi_m(f^n(x))-C_{\Phi}\leq \phi_{n+m}(x)\leq \phi_n(x)+\phi_m(f^n(x))+C_{\Phi},\]
for all $x\in X$, $n,m \in \mathbb{N}$ and some constant $C_{\Phi}$.
\begin{remark}

	Feng and Huang proved, in \cite{Feng2010}, that an almost-additive sequence is indeed asymptotically additive. Then, by Proposition 4.7 in \cite{Cao08} we have  variation principle for topological pressure of asymptotically additive sequence:
\begin{eqnarray}\label{eq variational principle}
P_f(\Phi):=P_f(X,\Phi)=\sup\{h_{\mu}(f)+\Phi^{*}(\mu):\mu \in \mathcal{M}_f(X)\}.\end{eqnarray}

	\begin{remark}
		A asymptotically additive sequence may not be  almost-additive sequence. The next examples illustrate this.
	\end{remark}

	\begin{example}\label{Ban-Cao-Hu2010}\cite{Ban-Cao-Hu2010}
		Let a $C^1$ map $f:M\rightarrow M$ defined on $C^{\infty}$ $m$-dimensional Riemannian manifold. Let $\Lambda\subset M$ is a compact $f$-invariant subset. The set $\Lambda$ is called an average conformal repeller if for any $f$-invariant ergodic measure $\mu$ the Lyapunov exponents of $\mu$ $\lambda_i(\mu)$, $i=1,...,m$ are equal and positive. The authors show that the limit

		\[\lim_{n\rightarrow \infty}\log\frac{\|Df^n(x)\|}{\|Df^n(x)^{-1}\|^{-1}}=0\]
		converges uniformly on the average conformal repeller $\Lambda$. It is not hard to check hat the sequences  $\Phi_1=(\log \|Df^n(x)^{-1}\|^{-1})_n$ or $\Phi_2=(\log \|Df^n(x)\|)_n$ are asymptotically additive but, these may not be almost additive.
	\end{example}

	\begin{example}\label{Barreira1996}\cite{Bar06}
		Let $(X,f,d)$ be a TDS. Assume that $f$ is expanding on $X$, in the sense that there exists constants $a\geq b\geq 1$ and $\varepsilon_0>0$ such that

		\[B(f(x),b\varepsilon)\subset f(B(x,\varepsilon))\subset B(f(x),a\varepsilon)\;\;\;\; \mbox{for all}\; x\in X \,\, \mbox{and}\,\, 0<\varepsilon<\varepsilon_0.\]
		Then, $X$ has a Markov partition $R_1,..,R_m$ of arbitrarily small diameters, see \cite{Bar06}. We define a $m \times m$ matrix $A=(a_{ij})$ with $a_{ij}=1$ if $R_i\cap f^{-1}(R_j)\neq \emptyset$ and $a_{ij}=0$ otherwise. For each $\omega=(i_1,i_2,...)\in \Sigma_A^+$, $n\geq 1$ and $k\geq 0$, put

		\[\underline{\lambda}_k(\omega,n)=\min \inf\left\{\frac{d(x,y)}{d(f^n(x),f^n(y)}:x,y \in R_{j_1...j_{n+k}}\,\,\mbox{and}\,\, x\neq y\right\}\,\,\mbox{and}\]
		\[\overline{\lambda}_k(\omega,n)=\max \sup\left\{\frac{d(x,y)}{d(f^n(x),f^n(y)}:x,y \in R_{j_1...j_{n+k}}\,\,\mbox{and}\,\, x\neq y\right\},\]
		where $R_{j_1...j_n}=\{x\in X: f^{i-1}(x)\in R_{j_i}\,\,\mbox{for}\,\,i=1,...,n\}$ is the cylinder of length $n$, and the maximum and minimum are taken over the $j_1...j_{n+k}\in \Sigma_A^+$ such that $j_1...j_{n}=(i_1...i_n)$. We say that $f$ is asymptotically conformal if there exists $k\geq 0$ such that the limit
		\[\lim_{n\rightarrow\infty} \frac{1}{n}\log\frac{\overline{\lambda}_k(\omega,n)}{\underline{\lambda}_k(\omega,n)}=0,\]
		converges uniformly on $\Sigma_A^+$. As $\Phi_1=(\log\overline{\lambda}_k(\omega,n))_n$ is sub-additive and $\Phi_2=(\log\underline{\lambda}_k(\omega,n))_n$ is a sup-additive its easy to see that $\Phi_1=(\log\overline{\lambda}_k(\omega,n))_n$ or $\Phi_2=(\log\underline{\lambda}_k(\omega,n))_n$ are asymptotically additive, but may not are almost additive.
	\end{example}

\end{remark}

\section{Proofs}

\subsection{Proof of Theorem A}\label{subsection A}
In this subsection we will prove Theorem A. We will give some results that help us in the proof.
\begin{proposition}[Corollary 3.1 of \cite{Pfister2007}]\label{prop. separation uniform}
	Assume that $(X,d,T)$ has the uniform separation property, and that the ergodic measures are entropy dense. For any $\eta>0$, there exist $\delta^*>0$ and $\varepsilon^*>0$ so that for $\mu \in \mathcal{M}_f(X)$ and any neighborhood $F\subset \mathcal{M}(X)$ of $\mu$, there exists $n_{F,\mu,\eta}$ such that

	\[N(F;\delta^*,n,\varepsilon^*)\geq e^{n(h_{\mu}(f)-\eta)}\;\;\;\mbox{if}\;\;\; n\geq n_{F,\mu,\eta}. \]
	For any $\mu \in \mathcal{M}_f(X)$,
	\[h_{\mu}(f)\leq \lim_{\varepsilon \rightarrow 0}\lim_{\delta \rightarrow 0}\inf_{\mu \in F}\liminf_{n\rightarrow \infty}\frac{1}{n}\log N(F;\delta,n,\varepsilon).\]
\end{proposition}
We define $^K G:=\{x \in X; \{\mathcal{E}_n(x)\}_n \;\mbox{has a limit-point on $K$}\}$.

\begin{theorem}\label{thm specification/saturated}
	Let $(X,d,f)$ be  a TDS and $K\subset \mathcal{M}_f(X)$ be compact subset. Then,

	\[P_f(^K G,\Phi)\leq \sup\{h_{\mu}(f)+\Phi^{*}(\mu):\mu \in K\}.\]
	Consequently, $P_f(G_K,\Phi)\leq \inf\{h_{\mu}(f)+\Phi^{*}(\mu):\mu \in K\}$.

\end{theorem}

\begin{proof}

	Let $K\subset \mathcal{M}_f(X)$ be compact subset and $\mu \in \mathcal{M}_f(X)$. Put $s:=\sup\{h_{\theta}(f)+\Phi^{*}(\theta),\,\theta \in K\}$. Assume, without loss of generality, that $s<\infty$. Let $s'=s+2\delta$, with $\delta>0$.

For a neighborhood $F$ of $\mu$, denoted by $F_{\mu}$, the function $N(F_{\mu},n,\cdot)$ be non increasing, by Proposition \ref{proposition limit entropy} we have
	\[\inf_{F_{\mu}}\limsup\frac{1}{n}\log N(F_{\mu},n,\varepsilon)\leq h_{\mu}(f)\,\,\; \mbox{for all}\;\; \varepsilon>0.\]
	Then, for $\varepsilon>0$, there exist a neighborhood $B(\mu,\zeta_{\varepsilon})\subseteq F_{\mu}$ of $\mu$ and a number $M(B(\mu,\zeta_{\varepsilon}),\varepsilon)$ such that

	\[\frac{1}{n}\log N(B(\mu,\zeta_{\varepsilon}),n,\varepsilon)\leq h_{\mu}(f)+\delta,\]
	for each $n\geq M(B(\mu,\zeta_{\varepsilon}),\varepsilon)$.

	Let $E$ be a maximal $(n,\zeta_{\varepsilon})$-separated of $X_{n,B(\mu,\zeta_{\varepsilon})}$(which also is a $(n,\zeta_{\varepsilon})$-spanning of $X_{n,B(\mu,\zeta_{\varepsilon})}$) with cardinality $N(B(\mu,\zeta_{\varepsilon}),n,\varepsilon)$. Then,

	\begin{eqnarray*}
		M(X_{n,B(\mu,\zeta_{\varepsilon})},s',\Phi,n,\varepsilon)&\leq& \sum_{B_n(x,\varepsilon)\in \Gamma}e^{-s'n+\sup_{y\in B_n(x,\varepsilon)}\phi_n(y)},
	\end{eqnarray*}
	where $\Gamma:=\cup_{x\in E}B_n(x,\varepsilon/3)\supseteq X_{n,B(\mu,\zeta_{\varepsilon})}$.

	For $x\in E\subseteq X_{n,B(\mu,\zeta_{\varepsilon})}$ and since $\Phi$ is asymptotically additive, for $\zeta_{\varepsilon}>0$ there exists $\varphi_{\zeta_{\varepsilon}}:=\varphi_{\zeta}\in C^0(X)$ and $n_0$ such that for all $n>n_0$, we have
	\begin{eqnarray*}
		\lvert\frac{1}{n}\phi_n(x)-\Phi^*(\mu)\rvert&\leq& \lvert\frac{1}{n}\phi_n(x)-\frac{1}{n}S_n \varphi_{\zeta}(x)\rvert+\lvert\frac{1}{n}S_n \varphi_{\zeta}(x)-\int \varphi_{\zeta}d\mu\rvert\\
		&+&\lvert\int \varphi_{\zeta}d\mu-\Phi^*(\mu)\rvert
		=\lvert\frac{1}{n}\phi_n(x)-\frac{1}{n}S_n \varphi_{\zeta}(x)\rvert\\
		&+&\lvert\int \varphi_{\zeta}(x)-\int \varphi_{\zeta}d\mu\rvert
		+\lvert\int \varphi_{\zeta}d\mu-\Phi^*(\mu)\rvert
		\leq 3\zeta_{\varepsilon}
	\end{eqnarray*}
	and $\sup_{y\in B_n(x,\varepsilon)}\phi_n(y)\leq n(2\zeta_{\varepsilon}+\lvert\varphi_{\zeta}\rvert_{\varepsilon})+\phi_n(x)$, where $\lvert\varphi_{\zeta}\rvert_{\varepsilon}:=\sup\{\lvert\varphi_{\zeta}(x)-\varphi_{\zeta}(y)\rvert: d(x,y)<\varepsilon\}$.
	Then,

	\begin{eqnarray*}
		M(X_{n,B(\mu,\zeta_{\varepsilon})},s',\Phi,n,\varepsilon)&\leq& \sum_{B_n(x,\varepsilon)\in \Gamma}e^{n(-s'+\Phi^*(\mu)+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon})}\\
		&\leq& N(B(\mu,\zeta_{\varepsilon}),n,\varepsilon)e^{n(-s'+\Phi^*(\mu)+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon})}\\
		&\leq& e^{n(-\delta+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon})}.
	\end{eqnarray*}
	For a fixed $\delta$, there exists $\varepsilon_0$ such that $-\delta+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon}<0$, when $\varepsilon<\varepsilon_0$.

	As $K$ is compact set, given a fixed $\varepsilon>0$, we can choose a finite open cover $\{B(\mu_j,\zeta_{\varepsilon})\}_j$, with $j=1,...,m_{\varepsilon}$ of $K$. For sufficiently large $M>0$, $\bigcup_{n\geq M}\bigcup_{j=1}^{m_{\varepsilon}}X_{n,B(\mu_j,\zeta_{\varepsilon})}$ is a cover of $^KG$. Then, for $M\geq \max_{j}M(B(\mu_j),\varepsilon)$, we have

	\[M(^KG,s',\Phi,n,\varepsilon)\leq \sum_{n\geq M}\sum_{j=1}^{m_{\varepsilon}}e^{n(-\delta+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon})}\leq m_{\varepsilon}\sum_{n\geq M}e^{n(-\delta+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon})}.\]
	When $M\rightarrow \infty$, we have $m_{\varepsilon}\bigcup_{n\geq M}e^{n(-\delta+\lvert\varphi_{\zeta}\rvert_{\varepsilon}+5\zeta_{\varepsilon}))}\rightarrow 0$. Thus, $P(^KG, \Phi, \varepsilon)\leq s$ for all $\varepsilon<\varepsilon_0$. Therefore,

	\[P_f(^KG,\Phi)=\lim_{\varepsilon\rightarrow 0}P_f(^KG, \Phi, \varepsilon)\leq \sup_{\mu \in K}\{h_f(\mu)+\Phi^*(\mu)\}.\]

	For other part, note that $G_K\subset ^{\{\mu\}}G$ for all $\mu \in K$. Then,

	\[P_f(G_K,\Phi)\leq P_f(^{\{\mu\}}G,\Phi)\leq h_{\mu}(f)+\Phi^{*}(\mu)\;\;\mbox{for all}\;\; \mu \in K \]
	and $P_f(G_K,\Phi)\leq \inf\{h_{\mu}(f)+\Phi^{*}(\mu):\mu \in K\}$.
\end{proof}

A ball in $\mathcal{M}(X)$ is denoted by
$$
B(\nu, \xi);=\{\alpha \in \mathcal{M}(X): d(\alpha,\nu)<\xi\}
$$

\begin{proposition}[Lemma 2.1, \cite{Pfister2007}]\label{aprox. measures}
	Assume that $(X,d,f)$ has a $g$-almost product property. Let $x_1,...,x_k \in X$, $\varepsilon_1>0,...,\varepsilon_k>0$ and $n_1\geq m(\varepsilon_1)$,...,$n_k\geq m(\varepsilon_k)$ be given. Assume that
	\[\mathcal{E}_{n_j}(x_j)\in B(\nu_j,\xi_j),\;\;\mbox{for}\;\; j=1,...,k.\]
	Then for any $y\in \cap_{j=1}^k f^{-M_{j-1}}B_{n_j}(g;x_j,\varepsilon_j)$ and any Borel probability measure $\alpha$ on $X$ we have that
	\[d(\mathcal{E}_{M_k}(y),\alpha)\leq \sum_{j=1}^{k}\frac{n_j}{M_k}(\xi_j'+d(\nu_j,\alpha)),\]
	where $M_i=n_1+...+n_i$ and $\xi_i'=\xi_i+\varepsilon_i+\frac{g(n_i)}{n_i}$, $i=1,...,k$.
\end{proposition}

\begin{lemma}[\cite{Pfister2003}]
	Let $K\subset \mathcal{M}_f(X)$ be a compact connected non-empty set. Then there exists a sequence $\{\alpha_1,...,\alpha_k,...\}\subset K$ such that for each $n\in \mathbb{N}$

	\[\overline{\{\alpha_j:j\in \mathbb{N}, j>n\}}=K \;\;\mbox{and}\;\; \lim_{j\rightarrow \infty}d(\alpha_j,\alpha_{j+1})=0.\]
\end{lemma}
\begin{theorem}\label{theorem B2}
	Let $(X,d,f)$ be a TDS with the g-almost product property and uniform separation. Let $K$ be a connected non-empty compact subset of $\mathcal{M}_f(X)$. Then

	\[P_f(G_K,\Phi)\geq \inf\{h_{\mu}(f)+\Phi^*(\mu):\mu \in K\}.\]
\end{theorem}

The proof follow  the line in \cite{Pfister2007} and \cite{Pei10} with some changes. We shall repeat here part of the arguments.

Let $\eta>0$, and $h^*:= \inf\{h_{\mu}(f)+\Phi^*(\mu):\mu \in K\}-\eta$. For $s<h^*$, we put $h^*-s=2\delta$. By Proposition \ref{prop. separation uniform}, we can find $\delta^*>0$ and $\varepsilon^*>0$ such that for each neighborhood $F$ of $\mu$  there exists $n_{F,\mu,\eta}^*$ with

\begin{eqnarray}\label{equation cardinality}
N(F;\delta^*,n,\varepsilon^*)\geq e^{n(h_{\mu}(f)-\eta)}\;\; \mbox{for all}\;\; n\geq n_{F,\mu,\eta}^*.
\end{eqnarray}
Take $(\varepsilon_k)_k$ and $(\xi_k)_k$ two decreasing sequences converging for zero such that $\varepsilon<\varepsilon^*$ and  $\lvert\int \varphi_{\delta}d_{\mu}-\int\varphi_{\delta}d\alpha_k\rvert<\frac{\delta}{18}$, for all $\mu \in B(\alpha_k,\xi_k+2\varepsilon_k)$ where $\varphi_{\delta}$ is a continuous functions such that $\limsup_{n\rightarrow \infty}\frac{1}{n}\|\phi_n-S_n\varphi_{\delta}\|<\frac{\delta}{18}$. By Equation (\ref{equation cardinality}), there exists $n_k$ and a $(\delta^*,n_k,\varepsilon^*)$-separated sets $\Gamma_k\subset X_{n_k,B(\alpha_k,\xi_k)}$ with cardinality $\lvert\Gamma_k\rvert\geq e^{n_k (n_k(h_{\alpha}(f)-\eta))}$. We can assume that the sequence $n_k$ satisfies
\begin{equation}\label{eq.error function}
\delta^*n_k>2 g(n_k)+1 \;\; \mbox{and} \;\; \frac{g(n_k)}{n_k}\leq \varepsilon_k.
\end{equation}
By Proposition \ref{aprox. measures} and Equation (\ref{eq.error function}) we have that if $x\in \Gamma_k$, $y\in B_{n_k}(g;x,\varepsilon_k)$, then

\begin{equation}\label{eq. inclusion ball}
\mathcal{E}_{n_k}(y)\in B(\alpha_k,\xi_k+2\varepsilon_k).
\end{equation}
We choose a strictly increasing sequence of positive integers numbers $(N_k)_k$ satisfying

\[n_{k+1}\leq \xi_k\sum_{j=1}^{k}n_jN_j\;\;\mbox{and}\;\; \sum_{j=1}^{k-1}n_jN_j\leq \xi_k\sum_{j=1}^{k}n_jN_j,\]
and we define the sequences $(n_j'), (\alpha_j'), (\varepsilon_j')$ and $(\Gamma_j')$ by setting for  $j=N_1+...+N_{k-1}+q$, with $1\leq q\leq N_k$, $n_j':=n_k$, $\varepsilon_j':=\varepsilon_k$, $\xi_{j'}:=\xi_{k}$ and $\Gamma_j':=\Gamma_k$.

Let
\begin{equation*}
G_k:=\bigcap_{j=1}^k\left(\bigcup_{x_j.\in \Gamma_j'
}f^{-M_{j-1}}B_{n_j'}(g;x_j,\varepsilon_j')\right)\;\;\;\mbox{with}\;\;\; M_j:=\sum_{l=1}^{j}n_l'.
\end{equation*}
Note that $G_k$ is a non-empty closed set. Each element of $G$ can be indicated by $(x_1,...,x_k)$, where $x_j\in \Gamma_j'$. The proof of Theorem \ref{theorem B2} follow the Lemmas \ref{lemma pfister} and \ref{lemma key} below.

\begin{lemma}[\cite{Pfister2007}]\label{lemma pfister}
	Let $\varepsilon>0$ be such that $4\varepsilon=\varepsilon^*$ and $G:=\bigcap_{k\geq 1}G_k$.

	\begin{enumerate}
		\item Let $x_j,y_j \in \Gamma_j'$ with $x_j\neq y_j$. If $x\in B_{n_j'}(g;x_j,\varepsilon_j')$ and $y\in B_{n_j'}(g;y_j,\varepsilon_j')$, then
		\[\max\{d(f^m(x),f^m(y):0\leq m\leq n_j-1\}>2\varepsilon.\]

		\item $G$ is a closed set, which is the union of non-empty sets $G(x_1,x_2,...)$ where $x_j\in \Gamma_j'$.

		\item $G\subset G_K$.
	\end{enumerate}
\end{lemma}

\begin{lemma}\label{lemma key}
	$P_f(G,\Phi)\geq h^*$.
\end{lemma}
\begin{proof}
	We have $\frac{M_n}{M_{n+1}}\rightarrow 1$. Let $\Gamma_k$ be a $(\delta^*,n_k,\varepsilon^*)$-separated set of $X_{n_k, B(\alpha_k,\xi_k)}$ such that $\lvert\Gamma_k\rvert\geq e^{n_k(h_{\alpha_k}(f)-\eta)}$. We prove that $M(G,s,\Phi,n,\varepsilon)\geq 1$. Note that for each $^k x \in \Gamma_k$ we have $\mathcal{E}_{n_k}(^k x)\in B(\alpha_k,\xi_k)$. Then, for $n_k$ sufficiently large
	\begin{eqnarray*}
		\left\lvert\frac{\phi_{n_k}(^k x)}{n_k}-\Phi^*(\mu)\right\rvert&\leq& \left\lvert\frac{\phi_{n_k}(^k x)}{n_k}-\frac{S_{n_k}\varphi_{\delta}(^k x)}{n_k} \right\rvert+\left\lvert\frac{S_{n_k}\varphi_{\delta}(^k x)}{n_k}-\int \varphi_{\delta}d\alpha_k \right\rvert\\
		&+&\left\lvert\int \varphi_{\delta}d\alpha_k-\Phi^*(\mu)\right\rvert \leq \frac{\delta}{18}+\frac{\delta}{18}+\frac{\delta}{18}=\frac{\delta}{6}.
	\end{eqnarray*}
	Then,
	\begin{eqnarray*}
		\lvert\Gamma_k\rvert\geq e^{n_k\left(h_{\alpha_k}(f)+\Phi^*(\mu)-\eta\right)-\phi_{n_k}(^k x)-n_k\delta/6}\geq e^{n_kh^*-\phi_{n_k}(^k x)-n_k\delta/6}.
	\end{eqnarray*}
	As $G$ is compact set, we can choose finite covers $\mathcal{C}=\{B_m(x,\varepsilon)\}_{m}$ of $G$ such that $B_m(x,\varepsilon)\cap G\neq \emptyset$ for all $B_m(x,\varepsilon)\in \mathcal{C}$. For each $\mathcal{C}\in \mathcal{G}_n(G,\varepsilon)$, where $\mathcal{G}_n(G,\varepsilon)$ is the collection of all finite or countable covers of $G$ by
	sets of the form $B_m(x,\varepsilon)$ with $m \geq n$(for $n$ sufficiently large), we define the cover $\mathcal{C}'$ where we replace each ball
	$B_m(x,\varepsilon)$ by $B_{M_p}(x,\varepsilon)$ when $M_p\leq m < M_{p+1}$. Then

	\begin{eqnarray*}
		M(G,s, \Phi, n,\varepsilon)&=& \inf_{\mathcal{C}\in \mathcal{G}_n(G,\varepsilon)}\sum_{B_{m}(x,\varepsilon)\in \mathcal{C}}e^{-s m+\sup_{z \in B_{m}(x,\varepsilon)}\phi_{m}(z)} \\
		&\geq& \inf_{\mathcal{C}\in \mathcal{G}_n(G,\varepsilon)}\sum_{z\in B_{M_p}(x,\varepsilon)\cap G}e^{-s m+\phi_{m}(z)}.
	\end{eqnarray*}
	Consider $\mathcal{C}'$ and let $m$ be the largest such that there exist $B_{M_p}(x,\varepsilon)\in \mathcal{C}'$. We put	\[\mathcal{W}_k=\prod_{i=1}^{k}\Gamma'_i\;\;\mbox{and}\;\; \mathcal{\overline{W}}_m:=\bigcup_{k=1}^m\mathcal{W}_k.\]
	Each $z \in B_{M_p}(x,\varepsilon)\cap G$ corresponds to a point in $\mathcal{W}_p$ that is uniquely defined(Lemma \ref{lemma pfister}(i)). The word $v\in \mathcal{W}_j $ is a prefix of $w\in \mathcal{W}_k$ if the first $j$ entries of $w$ coincide with $v$. If $\mathcal{W}\subset \mathcal{\overline{W}}_m$ contains a prefix of each word of $\mathcal{W}_m$, then
	\[\sum_{k=1}^{m}\lvert\mathcal{W}\cap \mathcal{W}_k\rvert\lvert\mathcal{W}_k\rvert/\lvert\mathcal{W}_k\rvert\geq \lvert\mathcal{W}_k\rvert.\]
	Thus if $\mathcal{W}$ contains a prefix of each word of $\mathcal{W}_m$,
	\[\sum_{k=1}^{m}\lvert\mathcal{W}\cap \mathcal{W}_k\rvert\lvert\mathcal{W}_k\rvert\geq 1.\]
	It is easy to verify that

	\[\lvert\mathcal{W}_p\rvert\geq e^{M_p h^*-\sum_{i=1}^{p}\left(\phi_{n_i}(^ix)+n_i'\delta\right)}.\]
	Therefore, $\sum_{B_{M_p}(x,\varepsilon)\in \mathcal{C}'}e^{-M_p h^*+\sum_{i=1}^{p}\left(\phi_{n_i}(^ix)+n_i'\delta\right)}\geq 1$
	where $^i x \in \Gamma_i'$. We prove that
	\[M_ph^* -\sum_{i=1}^{p}(\phi_{n_i}(^{i'}x)+n_i'\delta/6)-sm+\phi_m(z)>0,\;\; \mbox{for}\;\; z\in G.\]
	By definition of asymptotically  additive we have, for $\delta>0$, that there exist $\varphi_{\delta}$ and $n_0$ such that for all $m,n_i'>n_0$ with $i=1,...,p$, we have

	\[-\phi_{n_i'}(^ix)+S_{n_i'}\varphi_{\delta}(^ix)>-n_i'\frac{\delta}{18} >-n_i'\frac{\delta}{6} \;\; \mbox{and}\;\; \phi_{m}(z)-S_{m}\varphi_{\delta}(z)> -m\frac{\delta}{6}. \]Then

	\begin{eqnarray*}
		&&\!\!\!\!\!\!\!\!\!\!\!m(h^*-s)-\sum_{i=1}^{p}(\phi_{n_i}(^{i}x)+n_i'\delta/6)+\phi_m(z)-(m-M_p)h^*\\
		&>&
		m(h^*-s-\frac{\delta}{6})-\sum_{i=1}^{p}\left(S_{n_i'}\varphi_{\delta}(^ix)+n'_i\frac{\delta}{3}\right)
		+ S_m\varphi_{\delta}(^ix)-(m-M_p)h^*\\
		&=&m(2\delta-\frac{\delta}{6})-M_{p}\frac{\delta}{3}+\sum_{i=1}^{p}\left(S_{n_i'}\varphi_{\delta}(f^{M_{i-1}}(z))-S_{n_{i}'}\varphi_{\delta}(^i x)\right)
		-(m-M_p)h^*\\
		&+& S_{m-M_p}\varphi_{\delta}(f^{M_p}(z)).
	\end{eqnarray*}
	And as
	$$G=\bigcap_{k\geq 1}\left(\bigcap_{j=1}^k\left(\bigcup_{x_j\in \Gamma_j'}f^{-M_{j-1}}B_{n_j'}(g;x_j,\varepsilon_j')\right)\right),$$
	we have that there exists $j$ such that $f^{M_{j-1}}(z) \in B_{n_j'}(g;x_j,\varepsilon_j')$. By Inclusion (\ref{eq. inclusion ball}) and $^ix \in \Gamma_i'$ we have

	\[\mathcal{E}_{n_i'}(f^{M_{i-1}}(z)) \in \mathcal{B}(\alpha,\xi_i'+2\varepsilon_i')\;\;\;\;\mbox{and}\;\;\; \mathcal{E}_{n_i'}(^ix)\in \mathcal{B}(\alpha,\xi_i').\] then

	\[\left\lvert\int \varphi_{\delta}d\mathcal{E}_{n_i'}(f^{M_{i-1}}(z))-\int \varphi_{\delta}d\mathcal{E}_{n_i'}(^ix)\right \rvert n_i'=\left\lvert S_{n_i'}\,\varphi_{\delta}(f^{M_{i-1}}(z))-S_{n_i'}\,\varphi_{\delta}(^ix)\right\rvert \leq n_i'\frac{\delta}{2}.\]
	Thus,

	\begin{eqnarray*}
		M_ph^*-\sum_{i=1}^{p}(\phi_{n_i}(^{i}x)+n_i'\delta/6)-sm+\phi_m(z)&\geq& m(2\delta-\frac{\delta}{6})-M_p(\frac{\delta}{2}+\frac{\delta}{3})\\
		&-&n_{p+1}'(\|g_k\|+h^*)\\
		&\geq&(2\delta-\frac{\delta}{6}-\frac{5\delta}{6})M_p -n_{p+1}'(\|g_k\|+h^*)\\

		&=&M_p\delta-n_{p+1}'(\|g_k\|+h^*).
	\end{eqnarray*}
	we can choose $p$ such that $M_p\delta-n_{p+1}'(\|g_k\|+h^*)>0$, because $\lim_{p\rightarrow \infty}\frac{n_{p+1}'}{M_p}=0$. Then,

	\begin{eqnarray*}
		M(G;s,\Phi,n,\varepsilon)\geq \inf_{\mathcal{C}\in \mathcal{G}_n(G,\varepsilon)}\sum_{B_{M_p}(x,\varepsilon)\in \mathcal{C}'}e^{M_ph^*-\sum_{i=1}^{p}(\phi_{n_i}(^{i}x)+n_i'\delta/6)}\geq 1
	\end{eqnarray*}
	this implies that $P(G,\Phi,\varepsilon)\geq s$. By fact that $s<h^*$ and the arbitrary choice of $\eta$ the Lemma \ref{lemma key} it's proved.

	As $G\subset G_K$ then $P_f(G,\Phi)\leq P_f(G_K,\Phi)$. Therefore, we have proved Theorem \ref{theorem B2}.

\end{proof}

\subsection{Proof of Theorem B}

In this subsection, we proved Theorem B. We will use the following Lemma:

\begin{lemma}\label{lemma.inf.sup}

	Let $(X,f)$ be a dynamical systems,  $\Phi:=(\phi_n)_n \in \mathfrak{C}^0(f,X)$ and $x \in X$. Then
	\[\Phi \in \hat{\mathfrak{C}}^0(f,X),\;\; x \in I(\Phi,f)\;\;\mbox{iff} \inf_{\mu \in \mathcal{M}_xf}\Phi^{*}(\mu)< \sup_{\mu \in \mathcal{M}_xf}\Phi^{*}(\mu).\]
\end{lemma}
\begin{proof}
	Assume that $\Phi \in \hat{\mathfrak{C}}^0(f,X) $ and $x \in I(\Phi,f)$. Then, $\frac{\phi_n(x)}{n}$ not converge pointwise for a constant, in particular, there exist a sequence $n_i$ converging for $+\infty$ and $\varepsilon_0>0$ such that
	\[\left\lvert\frac{1}{n_i}\phi_{n_i}(x)-\Phi^*(\mu_1)\right\rvert> \varepsilon_0\] where $\mu_1 \in \mathcal{M}_x(f)$;

	Take a subsequence(if necessary) of the sequence
	\[\nu_{i}:=\left(\frac{1}{n_i}\sum_{i=0}^{n_i-1}\delta_{f^j(x)}\right)_i\]
	such that $\nu_i \rightarrow \mu_2 $ in weak$^*$ topology. Obviously $\mu_2 \in\mathcal{M}_xf$. As $\Phi$ is asymptotically additive, for each $k$, there exists $\psi_{1/k}:=\psi_k \in C^{0}(X)$ such that
	$\limsup_{n\rightarrow \infty}\frac{1}{n}\|\phi_n - S_n\psi_k\|<\frac{1}{k} $.

	Therefore, for $i$ sufficiently large
	$$\frac{S_{n_i}\psi_k(x)}{n_i}-\frac{1}{k}\leq\frac{\phi_{n_i}(x)}{n_i} \leq \frac{S_{n_i}\psi_k(x)}{n_i}+\frac{1}{k}.$$

	Then, we have
	\[\lim_{i\rightarrow \infty}\int \psi_kd\nu_{i}-\frac{1}{k}\leq \liminf_{i\rightarrow \infty}\frac{\phi_{n_i}(x)}{n_i} \leq \limsup_{i\rightarrow \infty}\frac{\phi_{n_i}(x)}{n_i}\leq\lim_{i\rightarrow \infty} \int \psi_kd\nu_{i}+\frac{1}{k}. \]

	And as $\nu_{i}\rightarrow \mu_2$  we have

	\begin{eqnarray}\label{eq 1}
	\int \psi_kd\mu_2-\frac{1}{k}\leq \liminf_{i\rightarrow \infty}\frac{\phi_{n_i}(x)}{n_i} \leq \limsup_{i\rightarrow \infty}\frac{\phi_{n_i}(x)}{n_i}\leq\int \psi_kd\mu_2+\frac{1}{k}.
	\end{eqnarray}

	In the Equation (\ref{eq 1}), we have

	\[\int \psi_kd\mu_2-\frac{1}{k}\leq \limsup_{i\rightarrow \infty}\int\frac{\phi_{n_i}(x)}{n_i}d\mu_{2}=\Phi^*(\mu_2)\leq\int \psi_kd\mu_2+\frac{1}{k}.\]
	Therefore, as $k$ is arbitrary, we have
	$\limsup_{i\rightarrow \infty}\frac{\phi_{n_i}(x)}{n_i}=\Phi^{*}(\mu_2)\neq \Phi^*(\mu_1)$.

	Conversely, if for $\Phi \in\mathfrak{C}^0(f,X)$ and $x \in X$ we have
	\[\inf_{\mu \in \mathcal{M}_xf}\Phi^{*}(\mu)< \sup_{\mu \in \mathcal{M}_xf}\Phi^{*}(\mu),\]
	Then we can make two measures $\mu_1$ and $\mu_2$ in $\mathcal{M}_x(f)$ such that
	$\Phi^{*}(\mu_1)< \Phi^{*}(\mu_2)$ and two convergence subsequences

	\[\frac{1}{n_i}\sum_{i=0}^{n_i-1}\delta_{f^j(x)}\rightarrow \mu_1 \;\; \mbox{and}\;\; \frac{1}{m_i}\sum_{i=0}^{m_i-1}\delta_{f^j(x)}\rightarrow \mu_2.\] Then, as $\Phi$ is asymptotically additive, for each $k$, there exists $\varphi_{1/k}:=\varphi_k \in C^{0}(X)$ such that, for all $x\in X$ and large $n$ we have
	$\left\lvert\frac{1}{n}\phi_{n}(x)-\frac{1}{n}S_{n}\varphi_k(x)\right\rvert<\frac{1}{k}<\varepsilon$. Moreover, we can obtain that $\left\lvert\int\varphi_kd\mu_1-\Phi^{*}(\mu_1)\right\rvert<\frac{1}{k}<\varepsilon$. Then, for $i$ sufficiently large

	\begin{eqnarray*}
		\left\lvert\frac{1}{n_i}\phi_{n_i}-\Phi^{*}(\mu_1)\right\rvert&\leq& \left\lvert\frac{1}{n_i}\phi_{n_i}(x)-\frac{1}{n_i}S_{n_i}\varphi_k(x)\right\rvert  +\left\lvert\frac{1}{n_i}S_{n_i}\varphi_k(x)-\int \varphi_k d\mu_1\right\rvert\\
		&+&\left\lvert\int \varphi_k d\mu_1-\Phi^{*}(\mu_1)\right\rvert
		< 3\varepsilon.
	\end{eqnarray*}
	Then, $\lim_{i\rightarrow \infty}\frac{1}{n_i}\phi_{n_i}(x)=\Phi^{*}(\mu_1)$. Analogously, $\lim_{i\rightarrow \infty}\frac{1}{m_i}\phi_{m_i}(x)=\Phi^{*}(\mu_2)$. Therefore, $x \in I(\Phi,f)$ and $\Phi \in \hat{\mathfrak{C}}^0(f,X) $.

\end{proof}

\textbf{\textit{Proof of the Theorem B}}
\begin{proof}
	We fix $\varepsilon>0$ and $\Phi \in \mathfrak{\hat{C}}^0(f,X)$. By Variational Principle \ref{eq variational principle}, we choose a ergodic measure $\mu$ such that $h_{\mu}(f)+\Phi^{*}(\mu)>P_f(\Phi)-\varepsilon$. Choose $\theta\in (0,1)$ close to 1 satisfying
	\[\theta(h_{\mu}(f)+\Phi^{*}(\mu))>P_f(\Phi)-\varepsilon,\]
	and $(1-\theta)\|\Phi\|<\varepsilon$, where $\|\Phi\|$ is a positive constant such that $\|\Phi\|\geq \Phi^{*}(\mu) $ for all $\mu \in \mathcal{M}_fX$, see item 2, Proposition A1 of \cite{Feng2010}. For $\Upsilon \in D$, by Lemma \ref{lemma.inf.sup}, there is an invariant measure $\mu_{\Upsilon}$ such that $\Upsilon^{*}(\mu_{\Upsilon})\neq \Upsilon^{*}(\mu)$. Take the measure $\nu_{\Upsilon}=\theta \mu+(1-\theta)\mu_{\Upsilon}$. Then

	\begin{eqnarray*}
		h_{\nu_{\Upsilon}}(f)+\Phi^{*}(\nu_{\Upsilon})&=&\theta h_{\mu}(f)+(1-\theta)h_{\mu_{\Upsilon}}(f)+\theta\Phi^{*}(\mu)+(1-\theta)\Phi^{*}(\mu_{\Upsilon})\\
		&\geq& \theta(h_{\mu}(f)+\Phi^{*}(\mu))-(1-\theta)\|\Phi\|> P_f(\Phi)-2\varepsilon.
	\end{eqnarray*}
	Remember that $\Phi^{*}(\mu)=\displaystyle
	\lim_{n\rightarrow \infty}\int \frac{\phi_n(x)}{n}\mu(x)$. Then, by product topology, two  asymptotically additive sequence $\Psi:=(\psi_n)_n$ and $\Upsilon:=(\upsilon_n)_n$ are close if the sequences of continuous functions $\displaystyle\frac{\psi_n}{n}$ and $\displaystyle\frac{\upsilon_n}{n}$ are close for each $n$. Therefore, by continuity of sequences, for each $\Upsilon \in D$ there exist an neighborhood $\mathcal{V}_{\Upsilon}\subseteq \mathfrak{C}^0(f,X) $ of $\Upsilon$ such that for each $\Psi \in \mathcal{V}_{\Upsilon}$ we have $\Psi^{*}(\mu)\neq \Psi^{*}(\nu_{\Upsilon})$. So $\{\mathcal{V}_{\Upsilon}: \Upsilon \in D\}$ forms an cover of $D$. We can take a countable subcover $\{\mathcal{V}_{\Upsilon_i}\}_{i\in \mathbb{N}}$ of $D$ because $C^0(X)$ has countable topological basis and then the product topological  $\mathfrak{C}^0(f,X)$ also has(second axiom of countability). Then, for each $\Psi \in D$, there is $i\geq 1$ such that $\Psi \in \mathcal{V}_{\Upsilon_i}$ and satisfying
	\[\Psi^{*}(\mu)\neq \Psi^{*}(\nu_{\Upsilon_i}).\]

	Put $\nu_i:= \nu_{\Upsilon_i}$ and define the sequence of measures $\eta_i:=\theta_i\mu+(1-\theta_i)\nu_i$, where $(\theta_i)_i\subset (0,1)$ is a increasing sequence converging the 1 such that $\theta_1\geq \theta$. Then, for each $\Psi\in D$, there is $i\geq 1$ such that $ \Psi^{*}(\mu)\neq \Psi^{*}(\eta_{i})$. Note that

	\begin{eqnarray*}
		h_{\eta_i}(f)+\Phi^{*}(\eta_{i})&\geq& \theta_i(h_{\mu}(f)+\Phi^{*}(\mu))+(1-\theta_i)h_{\nu_i}(f)-(1-\theta_i)\|\Phi\|\\
		&\geq& \min\left\{\theta(h_{\mu}(f)+\Phi^{*}(\mu)),h_{\mu}(f)+\Phi^{*}(\mu)\right\}-(1-\theta)\|\Phi\|\\
		&>&P_f(\Phi)-2\varepsilon.
	\end{eqnarray*}

	Define the set $K:=\{\mu\}\cup \bigcup_{i=1}^{\infty}\{t\eta_i+(1-t)\eta_{i+1}, t \in [0,1]\}$. By fact that $\eta_i$ converge for $\mu \in K$ this implies that $K$ is compact and connected set and that every $\nu \in K$ satisfies $h_{\nu}(f)+\Phi^{*}(\nu)>P_f(\Phi)-2\varepsilon$.
	Since $f$ is sequential saturated, then
	\[P_f(G_K,\Phi)=\inf\{h_{\nu}(f)+\Phi^{*}(\nu): \nu \in K\}\geq P_f(\Phi)-2\varepsilon.\]
	For finishing the proof of Theorem, we show that $G_K\subseteq \bigcap_{\Psi \in D}I(\Psi,f)$. Let $x\in G_K$ and $\Psi \in D$. Then $\Psi^{*}(\mu)\neq \Psi^{*}(\eta_i)$ for some $i\geq 1$. We have that $\mathcal{M}_xf=K\supseteq \{\mu, \eta_i\}$. Then, there two sequences $(n_j)_j$ and $(m_j)_j$ such that $\frac{1}{n_j}\sum_{l=0}^{n_j-1}\delta_{f^l(x)}=\mu$ and $\frac{1}{m_j}\sum_{l=0}^{m_j-1}\delta_{f^l(x)}=\eta_i$. And as $\Psi$ is asymptotically additive, we have

	\begin{eqnarray*}
		\left\lvert\frac{1}{n_i}\psi_{n_i}(x)-\Psi^{*}(\mu)\right\rvert&\leq& \left\lvert\frac{1}{n_i}\psi_{n_i}(x)-\frac{1}{n_i}S_{n_i}\varphi_k(x)\right\rvert + \left\lvert\frac{1}{n_i}S_{n_i}\varphi_k(x)-\int \varphi_k d\mu_1\right\rvert\\
		&+&\left\lvert\int \varphi_k d\mu_1-\Psi^{*}(\mu)\right\rvert
		< 3\varepsilon.
	\end{eqnarray*}
	Therefore $\lim_{i\rightarrow \infty}\frac{1}{n_i}\psi_{n_i}(x)=\Psi^{*}(\mu_1)$. Analogously, $\lim_{i\rightarrow \infty}\frac{1}{m_i}\psi_{n_i}(x)=\Psi^{*}(\eta_i)$. Then, $x \in I(\Psi,f)$. As $x \in G_K$ and $\Psi \in D$ are arbitrary, the Theorem it's proved.

\end{proof}

\section{Applications and examples}\label{aplication}
In this section, we given some examples and applications of our results.
\begin{example}
	Our result apply to the  asymptotically additive sequences  of the examples \ref{Ban-Cao-Hu2010} and \ref{Barreira1996}.

\end{example}

\subsection{Cocycles  under shift of the finite type}

We will consider cocycles under shift of the finite type in \cite{FengKaenmaki2011}. For more details, see \cite{Barreira-survey}.

\begin{example}
	Let $\sigma:\Sigma \rightarrow \Sigma$ be the shift map on the space $\Sigma=\{1,...,m\}^{\mathbb{N}}$, $m\geq 2$ doted with the metric $d(x,y)=2^{-\min\{j\geq 1;x_j\neq y_j\}}$ where $x=(x_1x_2...)$ and $y=(y_1y_2...)$.

	Consider matrices $M_1,...,M_m \in \mathcal{M}_{d\times d}(\mathbb{C})$ such that for each $n\geq 1$ there exists $i_1,...,i_n \in \{1,...,m\}^{n}$ such that $M_{i_1}\cdot ... \cdot M_{i_n}\neq 0$. Then, the topological pressure function is well defined with
	\[P(t)=\lim_{n\rightarrow \infty}\frac{1}{n}\log \sum_{w\in \{1,...,m\}^{n}}\|M_{i_1}\cdot ... \cdot M_{i_n}\|^t,\]
	where $w=(i_1,...,i_n)$.

	We define a class of functions that are obtained via a product of matrices. For each $t\geq 0$, $n \in \mathbb{N}$ and $w=(i_1,...,i_n) \in \{1,...,m\}^{n}$, we consider the locally constant functions $\psi^t_{w}:\Sigma \rightarrow \mathbb{R}^+$
	\[\psi^t_{w}(x)=\|M_{i_1}\cdot ... \cdot M_{i_n}\|^t.\]
	We define a sequence of functions $\Phi^t=(\phi^t_n:\Sigma \rightarrow \mathbb{R})_n$ of the following form
	\[\phi^t_n(x)=\sup_{w'\in C(x)}\log \psi_{w'}^t(x)=\sup_{w'\in C(x)}\log \|M_{i_1}\cdot ... \cdot M_{i_n}\|^t,\]
	where $w'=(i_1,...,i_n)$ and $C(x)$	is the set of blocks of $n$ elements  that are equal to the first elements of $x$.

	In \cite{Feng2009}, it show that there exists $C>0$ and $k\in \mathbb{N}$  such that $w, w' \in \bigcup_{n\in \mathbb{N}}\{1,...,m\}^n$ there exists $\overline{w}\in \sum_{j=1}^{k}\{1,...,m\}^k$ for which

	\begin{eqnarray}\label{condition distortion}
	\|M_w M_{\overline{w}} M_{w'}\|\geq C \|M_w\|\cdot \|M_{w'}\|.
	\end{eqnarray}
	The property in (\ref{condition distortion}) ensures that the sequence $\Phi^t$ is almost additive(see \cite{Feng2009},\cite{Barreira-survey}), then is asymptotically additive. Then, by Theorem B, we have
	\[P_{\sigma}\left(\bigcap_{s\geq 0}I(\Phi^s),\Phi^t\right)=P_{\sigma}(\Phi^t)\]
	for each $t\geq0$.

\end{example}

\subsection{Nonconformal repellers}

We describe a class of nonconformal repellers considered by Barreira and Gelfert in \cite{Barreira-Gelfert}. For more details, see also \cite{Barreira-survey}.

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a $C^1$ map and let $\Lambda\subset \mathbb{R}^2$ a compact $f$-invariant set. We say that $f$ is expanding map on $\Lambda$ and that $\Lambda$ is a repeller of $f$ if there exist constant $C>0$ and $\beta>1$ such that
\[\|Df^n(x)v\geq C \beta^n\|v\|\]
for each $x\in \Lambda, n \in \mathbb{N}$ and $v\in T_xM$.
We assume that there a open set $\mathcal{U}\supset \Lambda$ such that $\Lambda=\bigcap_{n\in \mathbb{N}}f^n(\mathcal{U})$ and that $f$ is  topologically mixing on $\Lambda$.

Given a number $\gamma<\frac{1}{2}$ and a subspace 1-dimensional $E(x)\subset \mathbb{R}^2$, we considered the cone

\[C_{\gamma}(x):=\{(u,v)\in E(x)\oplus E(x)^{\perp};\|v\|\leq \gamma \|u\|\}.\]

We say that differential map $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ satisfies the ``cone condition" on a compact set $\Lambda \subset \mathbb{R}^2$ if there exist $\gamma<1$
and for each $x \in \Lambda$ a 1-dimensional subspace $E(x)\subset \mathbb{R}^2$ varying continuously with $x$ such that

\[Df(x)C_{\gamma}(x)\subset\{0\}\cup \mbox{int}C_{\gamma}(f(x)).\]

Let $\Phi_i$ be the almost additive sequence obtained as follows: Let the singular values of a $2\times 2$ matrix $A$, and

\[\sigma_1(A)=\|A\|\;\;\; \mbox{and}\;\;\; \sigma_2(A)=\|A^{-1}\|^{-1}.\]

Given a $C^1$ map $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$. The sequences of functions $\Phi_i=(\phi_{i,n})_n$, $i=1,2$, is given by
\[\phi_{i,n}(x)=\log \sigma_i(Df^n(x)).\]
\begin{proposition}[\cite{Barreira-Gelfert}, Proposition 4]
	Let $\Lambda$ be a repeller of a $C^1$ map $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$. If $f$ satisfies a cone condition on $\Lambda$, then $\Phi_i$ is almost additive sequence for $i=1,2$.
\end{proposition}
Let $\delta>0$ be such that for each $x\in \Lambda$ the map $f$ is invertible on $B(x,\delta)$. For each $x\in \Lambda$ and $n\in \mathbb{N}$ we define
\[B(x,n,\delta)=\bigcap_{l=0}^{n-1}f^{-l}(B(f^l(x),\delta).\]
We say that $f$ have \textit{ bounded distortion}  on $\Lambda$ if there exist $\delta>0$ such that

\[\sup\left\{\|Df^n(y)(Df^n(z))^{-1}\|;x \in \Lambda\;\; \mbox{and}\;\; y,z \in B(x,n,\delta)\right\}< \infty.\]

We will see the relationship between the sequences $\Phi_i$ and the Lyapunov exponents. Given a differentiable transformation $f:M\rightarrow M$. By Oseledet's multiplicative ergodic theorem, for each finite $f$-invariant measure $\mu$ on $M$ there exist a $\mu$-full set $X\subset M$  such that $x\in X$, there exists numbers $\lambda_1(x),...,\lambda_{s(x)}$ and subspaces $M=V_1(x)\supset V_2(x)\supset...\supset V_{s(x)}(x)\supset V_{s(x)+1}(x)=\{0\}$ such that
\[\lim_{n\rightarrow \infty}\frac{1}{n}\log\|Df^n(x)v\|=\lambda_i(x),\]
for each $v\in V_{i}(x)\backslash V_{i-1}(x)$ and $i=1,...,s(x)$. The numbers $\lambda_1(x),...,\lambda_{s(x)}$ are the Lyapunov exponents of $\mu$.
In particular, for $f$ above($M=\mathbb{R}^2$) and for $x\in X$ we have

\[\lim_{n\rightarrow \infty}\frac{\phi_{i,n}(x)}{n}=\lim_{n\rightarrow \infty}\frac{1}{n}\log\sigma_i(Df^n(x))=\lambda_i(x)\;\;\;\mbox{for each}\;\;\; i=1,2.\]
Barreira and Gelfert, \cite{Barreira-Gelfert}, proved that if a $C^1$ map $f:\mathbb{R}\rightarrow \mathbb{R}$ has bounded distortion and satisfy a cone condition on $\Lambda$ then $(f,\Phi_i)$ has a unique equilibrium states $\mu_i$. Then, by Theorem B, ensures that
\begin{eqnarray*}
	P_f\left(I(\Phi_1)\cap I(\Phi_2),\Phi_i\right)
	&=&P_f(\Phi_i)=h_{\mu_i}(f)+\int_{\Lambda}\lambda_i(x)d\mu_i(x)\\
	&=&h_{\mu_i}(f)+\lim_{n\rightarrow \infty}\frac{1}{n}\int_{\Lambda}\log \sigma_i(Df^n(x))d\mu_i(x)
\end{eqnarray*}
for each $i=1,2$.

\hspace{1.6cm}

\textbf{Acknowledgments}: This work was partially supported by a INCT-Mat/Capes-Brazil postdoctoral
fellowship at University of Bahia. The author is grateful to P.
Varandas  and V. Ramos for reading a preliminary version of this paper and making important suggestions that helped improve the presentation of the text.  The author  is also grateful to anonymous referees for giving us a simpler proof for the Theorem A and suggestions that will help improve the text.
\vspace{2cm}

redpajama_set_name: RedPajamaArXiv


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Artspace is located in the Manchester area of Richmond, Virginia, in the Plant Zero Art Center. Plant Zero is 3 blocks south of the James River at the 14th Street Bridge and is bounded by East 3rd St., Hull St., East 4th St., and Decatur St.
On the north side of the river, 14th Street divides two popular tourist and entertainment areas of Richmond, Shockoe Slip and Shockoe Bottom. The Canal Walk, a heron rookery and boat launch are all located on the 14th Street Bridge, also known as the Mayo Bridge.
Artspace can be accessed from within Plant Zero as well as from a separate entrance physically located at 31 East 3rd St. The 3rd Street door is used for loading/unloading artwork and is only open to the public during Openings and Performances.
If entering Plant Zero from the E. 4th Street door, follow main hallway to the end, turn right and follow ramp to bottom, then turn left. artspace is ahead on the right.
Enter from E. 3rd Street directly into the gallery during Openings and Performances or for loading/unloading artwork.
Street parking is available as well as a fenced and lighted Pay Lot which faces the E. 4th St. entrance and can be accessed from Hull St. or Decatur. The Pay Lot is FREE on Fourth Fridays. The lot at the corner of East 3rd and Decatur is a Private Lot. Do not park there or you will be towed.


At AATec we pride ourselves on delivering quality services and offer our clients solutions that meet and often exceed their expectations. We recognise that the delivery of quality service is dependent on the quality of our people. In our view the quality team we propose for each and every engagement provides a distinct competitive advantage to deliver quality services. The proposed core team is comprised of professionals with a wide range of relevant skills and with years of experience in their respective fields.
Tjaart is an accomplished project manager and has been involved in the manufacturing sector for more than 20 years. He has developed a highly sought after set of skills and as a recognised technical innovator, has vast experience in the fields of both heavy and light engineering. Tjaart has for the past 10 years been involved primarily in managing technical companies across all functions and has developed his skills in communicating, negotiating, project management, systems (EPR), maintenance, risk assessment, computers and finances. Tjaart's analytical and decision-making abilities are his strengths and he has learnt to manage teams of people to perform at a high level for sustainable lengths of time through people management, attitude and culture change.


I am still recovering from a week in the El Paso sun and heat. We had a great week doing sports camp and building a home for the Padilla family. It did not come without its challenges. The plumbing had to be moved (which means breaking out concrete – thanks Mark and Nathan for your work to get that done!). The slab was less than square (that is an understatement similar to saying that the Grand Canyon is hole in the ground). But everyone pitched in and did their part! It truly was a blessing to see how God worked in and through our team.
On the last day, we prayed with the family and handed over the keys. We said our good-bye's, especially to the kids, and it was in that moment that I realized more clearly than at any point during the week, the impact we had in the life of that family – an impact of showing God's love and compassion. We may never see all that will come from last week this side of heaven – but really, when you think about it, God does not ask us to do anything more than to be faithful with what we have. God has blessed us with talents and resources and all we have to do is find ways to use them for God's glory and in God's service. God will take care of the rest.
Mission trips are always packed with wonderful experiences – long days and hard work. This one has been no different so far. We have caught up with old friends and made new ones. We have begun to grow closer as a group and most importantly, we have seen God in action! We are working with a family in west El Paso, building them a new home and it has been a real challenge so far. Today, day 3, we began to see a little headway with relating to the family.
I covet your prayers as we strive to not only build a home for this family but also build a relationship with them as well. We also ask for your prayers as our students minister to local children through our sports camp ministry.
It is exhausting but it is such a rush to be out of my comfort zone truly putting my faith into action. If you have never done anything like participating in a mission trip, what are you waiting for? You will never regret it!
Follow me on Twitter to see the progress we are making each day in real time. Isn't technology incredible!
Thanks for your prayers and for reading my ramblings.
I am about to go out and load all the tools into the trailer that we will need on our mission trip. I am really excited about what God is going to do through our teams as we head to El Paso to build a home and do a sports camp. We covet your prayer as our teams go to minister. We have an amazing team that God has assembled to do both sides of this trip. Our sports camp team will be led by Sam Griffin. He will do an incredible job – this will be his third sports camp so he is quickly becoming a pro.
I will be leading the construction team and I have to tell you, nothing gets me going like walking up on a slab on the first day knowing that in 4 days, we will walk away from a home! Maybe its a man thing! (Actually, the fact that it always happens is definitely a God thing!) We are taking fourteen guys to begin this build. Some of them will have to return home on Sunday leaving five of us to finish but the experience each will have in just two days will be life-changing – both for each of them and for the family to which they minister.
One of our goals for the fall is to put together a men's ministry at our church and we want to kick it off with a men's retreat. I think this mission trip may just be the lead in for that ministry. Please pray that God will use our experience to move our church in that direction.
We will be doing something new for us with this trip. I encourage you to follow both myself – jbrianhill, and Sam – samgriffin, on Twitter. We will be posting real time updates throughout the week. Of course you can go to www.fbclevelland.com to check out more pics and details each day but if you want to be connected in real time – look for us on Twitter.
He discussed his excitement about the growing opportunities for the church in the social community networks, ie. Twitter, FaceBook, Linked In, etc. Even though the media has been around for a while, the church (as is normally the case) has been slow to jump in. So the opportunities are still brand new. Opportunities such as having the majority of the church using Twitter (Idealistic? Probably) But think of the possibilities – a person has a medical emergency and a friend Tweets about it. Within seconds, prayer is being offered up from all over the city. People are there in minutes to do what they can. Now, online communities must never replace face to face time. But the real value here is the added layer of community that social media provides.
Here is my question for thought (and please comment if you have any ideas): How do we help our people make the transition to truly value social media and commit to use it? Any thoughts?
We, as a church, are gearing up to spend 7 days in the El Paso area on a mission trip. I am really excited about how this trip is coming together. There are people signed up for this trip who have never been on a trip like this. I think this experience will be life-changing for all involved.
We finished up a study on the book of James a couple of months ago. The overall thrust of that book in the Bible deals with living out our faith. Mission trips are a great way to do just that – getting out of our comfort zone and doing for others; not for what we get out of it but rather for what we can give.
If we truly have faith in God, then the outflow of that faith should, no must be compassion for others. The Bible says that if we say we love God but it doesn't show in our actions toward others then we are liars. Hey, I didn't say it! Look at it here.
If we have real faith, it will affect how we relate to those around us. It will come out in love. Is it possible to serve others without this faith? Sure – its called social work and there is absolutely nothing wrong with that – in fact, it is one of the most honorable and rewarding career paths I know.
Is it possible to have faith and no visible outflow of compassion toward others? Some say it is – it looks like legalism and empty spirituality. Based on the Bible, I don't think it is possible to have an authentic faith and not have visible signs of that faith acted out in the way we relate to others. Now is it possible to get so wrapped up in our lives and in good things and neglect the best thing? Sure. In fact, I think that is what is wrong with the Church (big C) today.
I believe if our lives are to be full and meaningful and if the Church is to grow and flourish, we have to refocus our lives on living out our faith on a daily basis.
What does that look like on a practical level in your life? I would love to hear your comments.
It reminded me that even in the midst of turmoil – whatever that turmoil might be – God is with us and offering a peace that we can't even understand. How often do we forget that God is in control. We get so worked up over the negative all around us and fail to remember the bigger picture.
So no matter what you are going through today or this week, just remember, God is with you and offering a peace that doesn't even make sense. It doesn't mean that your turmoil will disappear but it does mean that it can never overcome you. Take heart!
Ahh! The weekend. Anyone excited? I am wrapping up some things at the office this morning and then plan to head up to Capitan, NM, for the evening to be with our kids at Pre-teen camp at Fort Lone Tree. Tonight is family night and all the kids put on skits for the rest of the campers and the parents that show up. I am looking forward to the change of pace.
This week has been insane! We took at least one direct lightening strike on Wednesday and it took out a phone line, some phones, 5 air conditioners, our lighting system in the auditorium and who knows what we haven't even found yet. Yesterday we had workers here from 5 different companies working on things. I am ready for a break!
I think God knew what we needed (an understatement) when the sabbath was created. We all need rest. It is when we rest that God refreshes us to accomplish what we are called to do. I know the weekend is usually spent running kids around and trying to catch up on projects around the house, but this weekend, set aside a little time to rest – at least some time to rest in God through prayer. I sure plan to! Have a great weekend – and thanks for reading.


U.S. stocks rallied after yesterday's loss with the S&P 500 continuing to gain and closing at yet another record high. ADP (ADP) announced today that the private sector added only 179,000 jobs in May, failing to meet expectations. Many economists were predicting that there would be approximately 215,000 jobs added by the private sector in May. The Fed reported today that the U.S. economy grew in all 12 Federal Reserve bank districts in the past month and a half. The Beige Book noted the improvement over the previous report which only showed 10 regions growing.
U.S. economic activity continues at same pace.
Private sector added only 179k jobs in May.
McDonald's (MCD) will support minimum wage hike.
U.S. Services activity climbed in May.
Former Skype CEO hired as GoPro's new President.
Pinterest begins testing paid ads.
The Fresh Market (TFM) rises on insider buying.
Inside the mind of the world's greatest investor.
Investing lessons of Warren Buffett.
Private Sector Adds 179k Jobs in May; U.S. Economy Grew in all 12 Bank Districts. InvestorGuide.com. WebFinance, Inc. http://www.investorguide.com/article/14942/private-sector-adds-179k-jobs-in-may-u-s-economy-grew-in-all-12-bank-districts-igd/ (access:April 24, 2019).


That Ka'b bin Malik told him that in the lifetime of Allah's Apostle he demanded his debt from Ibn Abu Hadrad in the Mosque. Their voices grew louder till Allah's Apostle heard them while he was in his house. So he lifted the curtain of his room and called Ka'b bin Malik saying, "O Ka'b!" He replied, "Labbaik! O Allah's Apostle!" He beckoned to him with his hand suggesting that he deduct half the debt. Ka'b said, "I agree, O Allah's Apostle!" Allah's Apostle then said (to Ibn Abu Hadrad), "Get up and pay him the rest."


text:
Q: Confused over using IOC container, service locator and factory Suppose I have a BaseForm which depends on an ILogger or IResourceManager or something like that. Currently it resolves the correct implementation of the required service using  the service locator which I know is an anti-pattern.

*

*Is using the constructor injection the right way to resolve this kind of dependency?

*Do I have to register my BaseForm (and its' derived types) in the container in order to create instances of them with resolved dependencies? Doesn't that complicate everything?

*Is it bad to use a static factory wrapped around a service locator?

*Unit-testing aside, will I really be punished because of using service locator anti-pattern?

Sorry about asking many questions at once. I've read the following SO questions and many others but reading them only added to my confusion:

*

*How to use Dependency Injection and not Service Locator

*What's the difference between the Dependency Injection and Service Locator patterns?

*How to avoid Service Locator Anti-Pattern?

A: If possible, you should always go with dependency injection, since it has a few clear strength. With UI technologies however, it is not always possible to use dependency injection, since some UI technologies (in .NET space, Win Forms and Web Forms, for instance) only allow your UI classes (forms, pages, controls, etc) to have a default constructor. In that case you will have to fall back to something else, which is service locator.
In that case I can give you the following advice:

*

*Only fall back to Service Locator for UI classes that can't be created by your container using dependency injection, and for stuff that you aren't unit testing anyway.

*Try to implement as less logic as possible in those UI classes (as Humble objects with only view related stuff). This allows you to unit test as much as possible.

*Wrap the container around a static method to hide the container from the rest of the application. Make sure that a call to this static method fails, when the dependency cannot be resolved.

*Resolve all dependencies in the (default) constructor of that type. This allows the application to fail fast when one of its dependencies cannot be resolved when that type is created, instead of later on when some button is clicked.

*Check during app start-up (or using a unit test), if all those UI types can be created. This saves you from having to go through the whole application (by opening all forms) to see if there is an error in the DI configuration.

*When types cannot be built by the container, there is no reason to register them in the container. If they can be created by the container (such as with ASP.NET MVC Controller classes), it can be useful to register them explicitly, because some containers allow you to verify the configuration up front, which will detect configuration errors in those types right away.

Besides unit testing, there are two other important arguments against the use of the Service Locator, which are given by Mark Seemann in his famous blog post Service Locator is an Anti-Pattern:

*

*Service Locator "hides a class' dependencies, causing run-time errors instead of compile-time errors"

*Service Locator is "making the code more difficult to maintain"

redpajama_set_name: RedPajamaStackExchange


text:
Minsk hosts Belarus Winter Swimming Championships 2017
For three days Minsk will host Belarus Winter Swimming Championships 2017. Belarusian winter swimmers are thoroughly prepared for the opening of the season. The water temperature is about 8ºC. About 100 athletes from Russia, Ukraine, Poland and China will compete in freestyle, butterfly and breaststroke. There are several age categories. The youngest swimmer is 12 years old. The national championship was designed to form a national team for the World Cup in Latvia.
Japanese cranes moved to new home in Minsk
Utilities services switching to heavy duty operation
Selskaya Gazeta turns 100 years
Frosts up to 29 degrees expected

redpajama_set_name: RedPajamaCommonCrawl


text:
OT: Cool interview with Hans Zimmer and Radiohead
echoopera
November 2017 in Off-topic
The beauty of randomness and generative goodness
Cool, I love Radiohead. I was slightly distracted by Thom's trousers though.
RUST( i )K
I love Hans.
So deep in his internal sound and perception.
I would love to have an hour to speak with him.
JohnnyGoodyear
@echoopera Thank you. Off to roll some dice....
ExAsperis99
Unpopular opinion, but this piece is much less than the sum of its parts. (It's also one of the most unprepossessing Radiohead tracks.) And Vox is kind of grating on my nerves with its series of music wonkery, as well done as they are. But I guess with the death of record stores, the cleverer-than-thou postcollegiate music snob — come on, aleatoric music? Conway's game of life? — has to emerge somewhere.
Wow, wrong side of the bed today!
tomato_juice
I'm happy to see them experimenting when it's all so high profile. Thanks for the link.
Great vid.
Refreshing approach to composition after all of the relentless 4/4 grid based stuff made by the collective electrosheep .
Nice to hear great artists making music.
@ExAsperis99 hrmmm sounds familiar
BF12E90C-8687-464D-AC28-3B8F4C1D8427.jpeg 55K
@echoopera said:
Hahaha. Guilty.
syrupcore
@ExAsperis99 said:
Looks like Thom agrees with you at about 2:15.
Love this stuff and to echo what others have said, love that some of it is going to be exposed to a whole lot of new-comers via this video.
@syrupcore said:
My inner curmudgeon. God, I love Radiohead, but have you ever seen the absolutely miserable documentary "Meeting People Is Easy"? FUN. And same with this live set of "Kid A." Seriously, how on earth does this band play festivals with such gloriously weird songs?
Because gloriously weird people spend their money and go because they are a great band and sometimes it's not the best, but when it is it is.
Meeting People Is Easy is the sort of Camus novel of Rock documentaries. Unending misery. Ol' Thom's come a long way from drowning in an astronaut helmet to this though:
@JohnnyGoodyear said:
I just meant to say that they're somewhat dour. It's usually pretty great.
Just to watch a sea of people clapping along to "Everything in Its Right Place" — a song in 10/4! I feel like we, the gloriously weird, got one in under the radar, and that will never happen again. Maybe Kendrick Lamar. Maybe.
Pretty credible cover of "EIIRP" featuring DJ Spooky. Bonus points: What's the app he's using?
You are not to blame for
Bittersweet distractors...
Hahaha! I was hoping you'd catch the reference!
I kind of agree with you - I love all the experimentation and resulting music, but I find the dissection and lectures a bit well....boring. A bit like discussions on modern art. Just do it, and let the work speak for itself.
I love the 'meeting people' doc, and the weirder Radiohead are the better for me. All part of the glorious soup.
Thom, in funny pants: "if you think about it, everything is really just like triangles in the ocean. Whyyyy? ewwww ohhhhh"
Jonny: "Yeah, I can play that."
Phil, Ed, Colin: "Yeah, we can hold that shit down."
Radiohead is often about Thom and Jonny (for good reason) but I think a critical reason why they not lost in the stupid is Phil, Colin and Ed. They, almost literally, 'hold it down'.
I just listened to "There There" three times in a row, like a teenager. Still amazing. Amazing I tell you! T and J bring the magic, for sure, but P, E and C make it listenable on repeat.
I mean, in other words (I love Radiohead and I don't mean this as a diss on T or J), imagine you could just let your wildest crazy music making fantasy go indulgently wild knowing you had Phil, Colin and Ed to ground it (amazingly) in reality.
+1 All 'proper' bands need a solid rhythm section, otherwise it's all mush at gig time. As a bass player, I was overjoyed to have a drummer I completely synced with in my last band. A lot of spacey jamming was held together as we had an almost psychic link of where we were going next. That rock solid base allows the rest to wig-out and do their psychedelic thang.
Looping_Loddar
@MonzoPro said:
(...), but I find the dissection and lectures a bit well....boring. A bit like discussions on modern art.
Where the discussions about it (and the upper class habitus in doing it) seem to be much more important than the music they have produced. Hans Zimmer seem to be quite normal and nice, but these Radiohead people... - okay, maybe they just had a bad day.
I got a strong feeling to add a loud 4-on-the-flour-beat to these music and some strange noises, and maybe a typical disco bass. This would be my part in the aleatoric composition...
Nice. It's exactly this that lets me know that all the lonesome noodling in the world (and there are many benefits) also misses the one-and-one-makes-three aspects of the collective. As for the interview, more than obvious it was a requirement (probably for the benefit of the Beeb). Certainly not to be dissected for any PhDs.....(although it will be...)


text:
Tag Archives: Dick Price
The Esalen Connection: Fifty Years of Re-Visioning Madness and Trying to Transform the World
This entry first appeared at Mad in America on December 12, 2013.
When Richard Price was a young man, he experienced extreme states for which he was labeled schizophrenic and forcibly 'treated' with psychiatric medications, ECT, and insulin shock. He suffered from residual effects from this for the rest of his life. In 1962, Price and Michael Murphy founded the Esalen Institute on the Big Sur coast of Northern California. From its beginning, Esalen worked to create sanctuary for people who, like Price, experienced extreme states. "Esalen was Price's revenge on the mental hospital!" says Murphy.
Both at the institute itself and through the creation of projects like the rigorously designed, NIMH-funded Agnews Project research, Esalen helped to create the contemporary model of madness sanctuary. Agnews, which yet stands as the largest randomly-assigned, double-blind study on first episode psychosis, showed a 70 percent lower re-hospitalization rate advantage from providing a med-free environment for people in initial extreme states; a result which provided impetus and support for John Weir Perry's med-free sanctuary Diabasis House, and the creation of the I-Ward sanctuary that I served at and wrote about here on MIA in "Remembering a Medication-Free Madness Sanctuary".
Dick Price
Dick Price also wanted Esalen to be the kind of outside-the-box think tank that could fundamentally re-vision the experience of madness away from the medical model vision. Price realized the applied practice of the medical model belief system was the cause of the greatest trauma in his life. Price said about his own experience that "The so-called 'psychosis' was an attempt towards spontaneous healing, and it was a movement towards health, not a movement towards disease." He believed his experience of madness was not pathological but was full of meaning and was transformative, even perhaps touched with mystical power. He saw Esalen as a refuge for people in extreme states, "A space where it's possible to live through experience rather than having it blotted out, a place where there aren't the same negative self-definitions of someone going through this type of experience."
In addition to Price's commitment that Esalen provide refuge for those in extreme states, Esalen's broader liberating influence on world culture would prove to be enormous, as pioneers in philosophy and psychology made it the birthplace and epicenter of the human potential movement. (Sascha Dubrul has written a great article here on MIA about the lasting relevance of the human potential movement for current activism.) The young and gifted Price and Murphy attracted figures like Alduous Huxley, Alan Watts, and Abraham Maslow, who helped put Esalen on the map with week-long and even month-long gatherings for re-visioning madness. Ongoing symposia with names like "The Value of Psychotic Experience" went on for the whole summer of 1968! Gregory Bateson, Murphy and Price, Alan Watts, RD Laing, Erik Erikson, Fritz Perls, John Weir Perry, Claudio Naranjo, Virginia Satir, Julian Silverman, Alan Ginsberg, Michael Harner, Joan Halifax, Stan Groff, and many others all contributed to a growing understanding of extreme states that continues to evolve, as it certainly always must. They saw a neglected possibility for humankind, rooted in a philosophy that has always served to bring a dimension of the sacred and numinous into view.
That ancient but always emerging mythic vision serves as a counterweight to the objectifying, pathologizing and materialistic world view that I believe makes the tragically narrow vision of psychiatry possible. Esalen's role in altering the views and approaches to madness, or extreme states, continues to this day.
Last month, following in the tradition of passionately focused Esalen conferences, I organized a week-long invitation-only gathering there; "Alternative Views and Approaches to Psychosis." This conference was preceded in 2011 and 2012 by a workshop and then a conference on alternative approaches to extreme states that began a revival of this aspect of Price's work, one that had been dormant at Esalen since his death in 1985, when he was struck by a boulder while tending to the Esalen grounds following a flood.
This year, the Esalen conference was attended by 40 people involved in one way or another in the mental health revolution that is daily chronicled here on Madinamerica. People with lived experience brought their invaluable perspective to a gathering that included peer counselors, psychiatrists, therapists, authors, film makers, researchers, mental health services administrators and family members. A great many of the 40 people who gathered felt a sense of urgency to come up with collaborative, strategic ways to unite against the ever-growing human rights abuses of forced treatment. Those human rights abuses are being justified more and more in the name of protecting society from the suspected danger of people who have been given a psychiatric label. That public demonizing of those of us with lived experience of extreme states is being fueled by wide-scale fear mongering and draconian laws that threaten all who experience extreme states.
A number of us at the Esalen gathering committed to work together to fight human rights oppression on many fronts, via media, public education, legal campaigns, and expanding on existing – and developing new – humane, alternative supports for those in extreme states.
Dick Price was a psychiatric survivor whose compassion still touches us now. I know I wouldn't be writing here on MIA if not for him, because the I-Ward medication free extreme state sanctuary I went to work at in 1980 would never have existed without the Agnews research Price made happen. John Perry's Diabasis House – that I did my doctoral research on – never would have existed either.
So, thank you Richard Price, for fatefully touching my life, but so much more for helping so many people in extreme states receive the love you were denied in your hour of need, and escape the soul-shrinking diminishment of self-worth, and the hopelessness that a psychiatric diagnosis can inflict. Esalen lives on as an example of what can happen – for individuals and for society at large – when we respond to the extremes that sometimes come with human life not with fear and control but with receptivity and encouragement. There's no knowing – ahead of time, at least – how much accrues to each of us and to society at large when we learn to listen to the voices – weak, tenuous, or frightening as they may sometimes be – of people who are struggling with the process of coming-to-be in a world that often silences or eliminates them before they have a chance. Dick Price's voice was silenced early but lives on in the place in which he came to live and die; a memory and evidence of what can happen when the best of mind, body, spirit and community are given a chance to come together.
Posted in Esalen | Tagged Agnew's Project, diabasis, Dick Price, Esalen Institute, extreme states, human potential movement, human rights, I-Ward, John Weir Perry, mad in america, Michael Murphy, psychiatric survivor, psychology, psychosis, Richard Price, sanctuary | 2 Comments


Are you engaged and looking forward to organising your own wedding but don't know where to start? Look no further! Planning a wedding while hard work can be fun too for the engaged couple who can end up with a wedding they have created from scratch. However, it is no small feat. Here are a few tips that can help you to not only organise a kickass wedding but also, save money on your wedding.
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Cadillac is giving its long-loved CTS-V sedan an aggressive makeover… in Japan. Designed exclusively to the Japanese market (though the U.S. has a similar package), the new Cadillac CTS-V Carbon Black Edition gets a meaner exterior, a handsomely finished interior, and is limited to just four examples.
The CTS-V in question is equipped with the same supercharged 6.2-liter V8 buyers get in the U.S. That means the super sedan is good for 640 horsepower (477 kilowatts), a 0-to-60 (96 kmh) sprint of just 3.7 seconds, and a top speed of 200 miles per hour (321 kmh).
The body is available in just two color options: Sable Black, and Crystal White Tri-Coat. The exterior finish is combined with a carbon black detailing package that includes a black chrome grille and a matching black front diffuser, side vents, and rear spoiler and diffuser. The wheels, meanwhile, are coated in a dark gunmetal finish.
In the cabin, the all-black-everything theme continues. Recaro 16-way adjustable performance seats are wrapped in Jet Black leather and Alcantara while Jet Black accents can be found throughout the dash, trim, and even on the steering wheel. At the heart of it all is Cadillac's CUE infotainment system.
Interested buyers can pick up the package for about ¥14,900,000 (equivalent to about $133,000 USD). But hurry, only four models will be built for Japan. If you're looking for something a little faster still, GM also made available a limited Corvette Grand Sport Heritage Edition in the region, which costs only ¥12,800,00 in comparison, or about $115,000.
Those in the U.S. looking to get their hands on a similar package can option the Carbon Black trim onto any CTS, CTS-V, ATS, or ATS-V beginning this summer. No word on pricing for the U.S. package.
Is This The 2018 Cadillac XTS?
General Motors Japan Co., Ltd. (Shinagawa-ku, Tokyo, President: Wakamatsu, abbreviation: GM Japan) has a maximum output of 477 kW (649 PS) / 6,400 rpm and a maximum torque of 855 N · m (87.2 kg · m) To the power, unveiled a limited model " Cadillac CTS-V Carbon Black Package " wearing an exceptionally feminine form, will be on sale on June 3, 2017 in the Cadillac regular dealer network nationwide.
This is the first time for the special limited model of Super Sports Sedan "V Series" to inherit the lineage of Cadillac Racing . Domestic sales limited to 4 rare models. The manufacturer suggested retail price (tax included) is 14,900,000 yen.
The base model is a super sports sedan "Cadillac CTS-V" equipped with Cadillac's LT 4 6.2 liter V8 supercharged engine. Acceleration of 0 - 60 mph (about 96.6 km / h) achieved an amazing power of 3.7 seconds. Based on the Cadillac · Luxury · Sedan which provides agile exercise performance and a comfortable and sophisticated driving experience, it is the ultimate super sports sedan pursuing high-level driving performance by cutting-edge technology directly under Cadillac Racing.
In this "carbon black package", based on Spec-B which is a high spec model of CTS-V equipped with carbon fiber aero package and RECARO 16 way performance sheet, exclusive black chrome grill and 19 inch after midnight dark finish Adopted aluminum wheels, low gloss carbon fiber interior trim. We are emphasizing a more daunting and lacy image.
Body color is available in two colors: Sable Black and Crystal White Tri Coat. Combined with carbon black package, the aggressive type which unified the entire body with black, and the type which makes individuality and sense which made accent of black part to white body stand out. Both have a vivid presence. In addition, the interior adopts "Jet Black / Jet Black Accent" interior color and V carbon fiber interior trim. Even in a luxurious atmosphere full of fine quality, it enhances the lacy driving pleasure.


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Storm's death toll, humanitarian crisis grow in Puerto Rico
DANICA COTO / Associated Press
SAN JUAN, Puerto Rico — A humanitarian crisis grew Saturday in Puerto Rico as towns were left without fresh water, fuel, power or phone service following Hurricane Maria's devastating passage across the island.
A group of anxious mayors arrived in the capital to meet with Gov. Ricardo Rossello to present a long list of items they urgently need. The north coastal town of Manati had run out of fuel and fresh water, Mayor Jose Sanchez Gonzalez said.
"Hysteria is starting to spread. The hospital is about to collapse. It's at capacity," he said, crying. "We need someone to help us immediately."
The death toll from Maria in Puerto Rico was at least 10, including two police officers who drowned in floodwaters in the western town of Aguada. That number was expected to climb as officials from remote towns continued to check in with officials in San Juan.
Authorities in the town of Vega Alta on the north coast said they had been unable to reach an entire neighborhood called Fatima, and were particularly worried about residents of a nursing home.
"I need to get there today," Mayor Oscar Santiago told The Associated Press. "Not tomorrow, today."
Rossello said Maria would clearly cost more than the last major storm to wallop the island, Hurricane George in September 1998. "This is without a doubt the biggest catastrophe in modern history for Puerto Rico," he said.
A dam upstream of the towns of Quebradillas and Isabela in northwest Puerto Rico was cracked but had not burst by Saturday afternoon as the water continued to pour out of rain-swollen Lake Guajataca. Federal officials said Friday that 70,000 people, the number who live in the surrounding area, would have to be evacuated. But Javier Jimenez, mayor of the nearby town of San Sebastian, said he believed the number was far smaller.
Secretary of Public Affairs Ramon Rosario said about 300 families were in harm's way.
The governor said there is "significant damage" to the dam and authorities believe it could give way at any moment. "We don't know how long it's going to hold. The integrity of the structure has been compromised in a significant way," Rossello said.
The 345-yard (316-meter) dam, which was built around 1928, holds back a man-made lake covering about 2 square miles (5 square kilometers). More than 15 inches (nearly 40 centimeters) of rain from Maria fell on the surrounding mountains, swelling the reservoir.
Officials said 1,360 of the island's 1,600 cellphone towers were downed, and 85 percent of above-ground and underground phone and internet cables were knocked out. With roads blocked and phones dead, officials said, the situation may worsen.
"We haven't seen the extent of the damage," Rossello told reporters in the capital. Rossello couldn't say when power might be restored.
Maj. Gen. Derek P. Rydholm, deputy to the chief of the Air Force Reserve, said mobile communications systems were being flown in, but acknowledged "it's going to take a while" before people in Puerto Rico will be able to communicate with their families outside the island.
The island's electric grid was in sorry shape long before Maria struck. The territory's $73 billion debt crisis has left agencies like the state power company broke. It abandoned most basic maintenance in recent years, leaving the island subject to regular blackouts.
Rosello said he was distributing 250 satellite phones from FEMA to mayors across the island to re-establish contact.
At least 31 lives in all have been lost around the Caribbean, including at least 15 on hard-hit Dominica. Haiti reported three deaths; Guadeloupe, two; and the Dominican Republic, one.
Across Puerto Rico, more than 15,000 people are in shelters, including some 2,000 rescued from the north coastal town of Toa Baja.
Some of the island's 3.4 million people planned to head to the U.S. to temporarily escape the devastation. At least in the short term, though, the soggy misery will continue: Additional rain — up to 6 inches (15 centimeters) — is expected through Saturday.
In San Juan, Neida Febus wandered around her neighborhood with bowls of cooked rice, ground meat and avocado, offering food to the hungry. The damage was so extensive, the 64-year-old retiree said, that she didn't think the power would be turned back on until Christmas.
"This storm crushed us from one end of the island to the other," she said.
Hour-long lines formed at the few gas stations that reopened on Friday and anxious residents feared power could be out for weeks — or even months — and wondered how they would cope.
"I'm from here. I believe we have to step up to the task. If everyone leaves, what are we going to do? With all the pros and the cons, I will stay here," Israel Molina, 68, who lost roofing from his San Juan mini-market to the storm, said, and then paused. "I might have a different response tomorrow."


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40 Under 40: Blount County Sheriff's Deputy Jarrod Millsaps has a passion for outreach
Brenna McDermott
Jarrod Millsaps knew he would pursue a career in law enforcement from a young age. As a kid, he always wanted to play the cop, never the robber.
He was a Boy Scout Explorer by age 14, giving him hands-on experience with the Blount County Sheriff's Office. He started working at BCSO part-time at 18 and obtained his associates degree in criminal justice from Roane State Community College at the same time.
He's been there ever since.
40 Under 40 Gala:Get tickets here
Millsaps, 39, went from a deputy working patrol to the criminal investigations division, working as a domestic violence investigator.
He worked with victims and offered them resources. He decorated his office in a lighthouse theme.
More:Preventing domestic violence starts with listening
"It was just kind of symbolic that there's hope, there's someone that's there for you and looking out for you," he said.
He then became BCSO's first community outreach director, which involved crime prevention, hosting community events, managing volunteer programs and the Sherriff's Citizens Academy.
"It allowed me to be with community members and know what they needed from us, what they wanted and how to help them," Millsaps said.
In 2012, he obtained his bachelor's degree in criminal justice, something that was important to his father.
40 Under 40:Meet the class of 2018
He later became administrative sergeant and in 2014 became Deputy Chief of Administration. In that role, he is the public information officer, manages the office's $25 million budget, fleet, records, HR and technology.
Millsaps is also assistant commander of BCSO's SWAT team, president of the Fraternal Order of Police Lodge 9, president of Shop with a Cop Blount County, task force officer with the FBI and many more credits to his career.
"Just (getting) back to the basics of being a part of the community helps you stay grounded and (you) get to see the good things that happen instead of a lot of the negative things you see and have to deal with in law enforcement," he said.
Shop with a Cop is a nonprofit that pairs children in need with a police officer. The two go shopping for essentials and presents before the holidays.
"Seeing them happy and playing with toys while they're in the store, it's a jump start for me," he said.
Millsaps has an alter ego as well. He's been in Josh & Jared's Blues Brothers Tribute band with friend Josh Livingston since high school. It's a stress reliever, he said, to put on sunglasses, sing and dance.
About Jarrod:
Education/certifications: Walter State Regional Law Enforcement Academy; associate's in applied science criminal justice, Roane State Community College; bachelor's in criminal justice, Bethel University; multiple law enforcement certifications & instructor certifications
What do you like to do in your free time? I enjoy spending time with my family, offshore fishing, golfing, motorcycling and Jeeping.
What business person/leader in Knoxville do you look up to most? Sheriff James Lee Berrong, the Sheriff of Blount County. He focuses first and foremost on the safety of Blount County citizens and has a true heart for the community. He is proactive and aggressive in addressing the issues we face and he is dedicated to equip our deputies with the resources and technology they need to serve and protect our community.