text:
\section{Introduction}\label{sec:introduction}

Let $M$ be a compact Riemannian manifold of dimension $d$, and $f:M\rightarrow M$ be a diffeomorphism. We call an $f$-invariant Borel probability measure $\mu$ a \emph{physical measure} if the set $B(\mu)$ defined as
\begin{displaymath}
B(\mu)=\{x\in M:\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x))=\int\varphi d\mu,\textrm{for all}\ \varphi\in C^0(M)\}
\end{displaymath}
has positive Lebesgue measure. The set $B(\mu)$ is called the \emph{basin of $\mu$}. These notions were introduced by Sinai in \cite{S}, where it was proved the existence of physical measures for Anosov diffeomorphisms. Subsequently, in \cite{R} and \cite{BR}, Bowen and Ruelle showed that for hyperbolic diffeomorphisms and flows, there exist a finite number of physical measures whose basins are all essentially open (i.e., open module a set of null Lebesgue measure), and the union of these basins cover a full Lebesgue measure subset of the ambient manifold. For non-hyperbolic diffeomorphisms, in \cite{P} Palis conjectured that every system can be approximated by systems with a finite number of physical measures whose basins unite to cover a full Lebesgue measure subset of the whole manifold. In this term, as a positive step, in \cite{BV} and \cite{ABV}, Alves, Bonatti and Viana proved the existence and finiteness of physical measures for a large class of non-hyperbolic diffeomorphisms, namely partially hyperbolic systems with mostly contracting or mostly expanding center.

For hyperbolic systems, basins of physical measures are essentially open. However, the boundaries of the basins are often fractal sets (e.g., see \cite{DGOY}). In \cite{K}, Kan constructed for the first time examples of partially hyperbolic endomorphisms defined on the 2-dimensional cylinder with exactly two physical measures whose basins are \emph{intermingled}. Here we say two physical measures $\mu$ and $\nu$ have \emph{intermingled basins} or their basins are \emph{intermingled} if for any open set $U$ in the manifold, we have $\operatorname{Leb}(B(\mu)\cap U)>0$ and $\operatorname{Leb}(B(\nu)\cap U)>0$.

Kan's examples robustly admit two physical measures with intermingled basins within boundary preserving systems. His construction was extended by Ilyashenko, Kleptsyn and Saltykov\cite{IKS}. Kan also constructed partially hyperbolic examples on the thickened torus with two physical measures whose basins are intermingled. Then in \cite{KS}, Kleptsyn and Saltykov proved that the boundaries of basins of physical measures in Kan's examples have Hausdorff dimension strictly smaller than the dimension of the phase space.

One would ask whether there exist more than two physical measures with their basins intermingled. In \cite{DVY}, Dolgopyat, Viana and Yang constructed examples on $\mathbb{T}^2\times\mathbb{S}^2$ admitting arbitrarily finitely many physical measures with intermingled basins. And recently, Bonatti and Potrie (\cite{BP}) give examples on $\mathbb{T}^3$ with arbitrarily finitely many physical measures whose basins are intermingled. Their examples are partially hyperbolic but not strongly partially hyperbolic, that is, the center stable bundle $E^{cs}$ can not be split to two invariant subbundles, which is different from the previous examples.

It is well known that Kan's examples can be extended to boundaryless manifolds such as $\mathbb{T}^3$, following the same arguments. In this case, however, Ures and Vasquez proved recently in \cite{UV} that Kan's examples can not be robust. In fact, they proved for $C^r(r\geq2)$-partially hyperbolic, dynamically coherent diffeomorphisms of $\mathbb{T}^3$ with compact center leaves, if two physical measures are intermingled, then they must both support in periodic tori which are $s,u$-saturated. Thus these systems can not be accessible, and the number of physical measures with intermingled basins can not be larger than two. Recall that in \cite{BHHTU}, Burns, Herts, Herts, Talitskaya and Ures proved that accessibility property is $C^1$ open and $C^{\infty}$ dense among partially hyperbolic diffeomorphisms with one-dimensional center. This implies that partially hyperbolic diffeomorphisms on $\mathbb{T}^3$ with intermingled basins can not form a $C^r$ open set.

It's also a natural question to ask whether Kan's examples can be transitive or robustly transitive. In Chapter 11 of \cite{BDV}, it is showed that with two more assumptions, Kan's examples on 2-dimensional cylinder can be transitive. And recently, in \cite{O}, Okunev proved that for the set of $C^r$ partially hyperbolic diffeomorphisms which are skew products over transitive Anosov diffeomorphisms and fibered by $\mathbb{S}^1$, there exists a residual subset $\mathcal{R}$, such that for every $f\in\mathcal{R}$ either it is transitive or its non-wandering set has zero Lebesgue measure.

In this paper, we append some extra constructions to Kan's examples on $\mathbb{T}^3$, which lead them to be robustly transitive. First, we need to give the precise definitions of Kan's examples on $\mathbb{T}^2\times[0,1]$ and $\mathbb{T}^3=\mathbb{T}^2\times \mathbb{S}^1$, where in this paper $\mathbb{S}^1$ will be identified with $\mathbb{R}/2\mathbb{Z}$. These definitions were abstracted from Kan's original examples (see \cite{BDV}), which will guarantee the existence of two physical measures with intermingled basins. In this paper, we will use ${\operatorname{Diff}^{1+}(M)}$ to denote the set of diffeomorphisms on $M$ with H$\rm\ddot{o}$lder derivative.

\begin{definition}\label{def:Kan's example with boundary}
Let $M=\mathbb{T}^2\times[0,1]$. $K\in\operatorname{Diff}^{1+}(M)$ is called a \emph{Kan's example} if it is a skew product diffeomorphism $K(x,\theta)=(\mathbb{A}x,\phi(x,\theta))$, where $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a hyperbolic automorphism, and $\phi:M\rightarrow[0,1]$ is $C^{1+}$, satisfying the following conditions:
\begin{itemize}
\item[(K1)] For every $x\in\mathbb{T}^2,\phi(x,0)=0$ and $\phi(x,1)=1.$
\item[(K2)] For $r,s\in\mathbb{T}^2$, fixed points of $\mathbb{A}$, the map $\phi(r,\cdot)$ (resp. $\phi(s,\cdot)$) is Morse--Smale and has exactly two fixed points, a source (resp. sink) at $\theta=1$ and a sink (resp. source) at $\theta=0$.
\item[(K3)] For every $(x,\theta)\in M$, $\parallel \mathbb{A}^{-1}\parallel^{-1}<|\partial_{\theta}\phi(x,\theta)|<\parallel \mathbb{A}\parallel.$
\item[(K4)] $\int_{\mathbb{T}^2}\log|\partial_{\theta}\phi(x,0)|dx<0$ and $\int_{\mathbb{T}^2}\log|\partial_{\theta}\phi(x,1)|dx<0$.
\end{itemize}
The set of $C^{1+}$ Kan's examples on $\mathbb{T}^2\times[0,1]$ will be denoted by $\mathscr{K}^{1+}(\mathbb{T}^2\times[0,1]).$
\end{definition}

For the boundaryless case, we have naturally the following definition (see also \cite{UV}).

\begin{definition}\label{def:Kan's example on boundaryless manifolds}
Let $M=\mathbb{T}^3=\mathbb{T}^2\times \mathbb{S}^1$. $K\in\operatorname{Diff}^{1+}(M) $ is called a \emph{Kan's example} if it is a skew product diffeomorphism $K(x,\theta)=(\mathbb{A}x,\phi(x,\theta))$, where $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a hyperbolic automorphism, and $\phi:M\rightarrow\mathbb{S}^1$ is $C^{1+}$, satisfying the following conditions:
\begin{itemize}
\item[(K1)] For every $x\in\mathbb{T}^2,\phi(x,0)=0$ and $\phi(x,1)=1.$
\item[(K2)] For $r,s\in\mathbb{T}^2$, fixed points of $\mathbb{A}$, the map $\phi(r,\cdot)$ (resp. $\phi(s,\cdot)$) is Morse--Smale and has exactly two fixed points, a source (resp. sink) at $\theta=1$ and a sink(resp. source) at $\theta=0$.
\item[(K3)] For every $(x,\theta)\in M$, $\parallel \mathbb{A}^{-1}\parallel^{-1}<|\partial_{\theta}\phi(x,\theta)|<\parallel \mathbb{A}\parallel.$
\item[(K4)] $\int_{\mathbb{T}^2}\log|\partial_{\theta}\phi(x,0)|dx<0$ and $\int_{\mathbb{T}^2}\log|\partial_{\theta}\phi(x,1)|dx<0$.
\end{itemize}
The set of $C^{1+}$ Kan's examples on $\mathbb{T}^3$ will be denoted by $\mathscr{K}^{1+}(\mathbb{T}^3).$
\end{definition}

Noting that Kan's aim is to show the existence of intermingle property, we may generalize Kan's examples as in \cite{UV} to the following Kan--like diffeomorphisms.

\begin{definition}[Kan--like diffeomorphisms]\label{def:Kan-lian diffeo}
Let $M$ be some compact Riemannian manifold. A $C^{1+}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ is called a \emph{Kan--like diffeomorphism} if it admits at least two hyperbolic physical measures such that their basins are intermingled. The set of Kan--like diffeomorphisms is denoted by $\mathscr{K}_l^{1+}(M)$.
\end{definition}

We have mentioned that a beautiful description of Kan--like diffeomorphisms on $\TT^3$ has been given in \cite{UV}. They proved that every partially hyperbolic Kan--like diffeomorphism on $\TT^3$ must admit two $s,u$-saturated periodic tori as supports of physical measures.

Our example is $C^{\infty}$ smooth, and can be got by an arbitrarily small perturbation of $\mathbb{A}\times (-\operatorname{Id}_{\mathbb{S}^1})$, where $\mathbb{A}$ is a hyperbolic automorphism on $\mathbb{T}^2$, and $-\operatorname{Id}_{\mathbb{S}^1}$ is an orientation reversing isometry on $\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$, which admits $0$ and $1(\equiv-1)$ as two fixed points.

\begin{theorem}\label{theo:main theorem}
Let $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$ be a hyperbolic automorphism. Then there exists a $C^{\infty}$-arc $\{K_t\}_{t\in[0,1]}$ of $C^{\infty}$-diffeomorphisms on $\mathbb{T}^3=\mathbb{T}^2\times \mathbb{S}^1$, such that
\begin{enumerate}
\item $K_0=\mathbb{A}\times (-\operatorname{Id}_{\mathbb{S}^1})$.
\item For every $t\in(0,1]$, $K_t\in\mathscr{K}^{1+}(\mathbb{T}^3)$ has two physical measures with intermingled basins and is $C^1$-robustly transitive, but not topologically mixing.
\end{enumerate}

Moreover, for every $t\in(0,1]$, there exists a $C^1$-neighborhood $\mathcal{U}_t$ of $K_t$, such that for every  $g\in\mathcal{U}_t\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$,
\begin{itemize}
\item either $g$ has two physical measures with intermingled basins and $g\in\mathscr{K}_l^{1+}(\mathbb{T}^3)$,
\item or $g$ has a unique physical measure and $g$ is robustly topologically mixing.
\end{itemize}
\end{theorem}

\begin{remark}
A diffeomorphism is $C^r$-robustly transitive if it has a $C^r$-neighborhood consisting of transitive diffeomorphisms. All the known examples of $C^1$-robustly transitive diffeomorphisms are topologically mixing. So our example is the first $C^1$-robustly transitive diffeomorphism that are not topologically mixing, which gives a negative answer to a question in \cite{AC}.
\end{remark}
\begin{remark}
We would like to point out here that if $f\in\mathscr{K}^{1}(\mathbb{T}^3)$ preserves the orientation of the center fibers, then $f$ can not be robustly chain transitive. To see this, notice that there are two open sets $U=\mathbb{T}^2\times(-1,0)$ and $V=\mathbb{T}^2\times(0,1)$ that are both $f$-invariant. Define $\mathcal{X}$ to be a Morse--Smale vector field on $\mathbb{S}^1$ with a unique sink at $\theta=-0.5$ and a unique source at $\theta=0.5$. Then $f_t=(\operatorname{Id}_{\mathbb{T}^2}\times\mathcal{X}_t)\circ f$ is a family of diffeomorphisms isotopic to $f$, such that $\overline{f_t(U)}\subset U$ for every $t>0$. This means that $U$ is a trapping region of $f_t$ for every $t>0$, and it follows that $f_t$ is not chain transitive.
\end{remark}

To prove Theorem~\ref{theo:main theorem}, we will firstly give the constructions in Section~\ref{sec:construction}, and then prove the corresponding properties via a sequence of propositions in Section~\ref{sec:proofs}. In Proposition~\ref{prop:robust transitivity}, we prove the examples we constructed are robustly transitive. In Proposition~\ref{prop:two physical measures}, we prove that if the perturbed system has two physical measures, then their basins are intermingled. And in Proposition~\ref{prop:unique physical measure}, we prove that if the perturbed system has a unique physical measure, then the system is robustly topologically mixing. We should point out here that instead of proving Proposition~\ref{prop:two physical measures}, we will actually prove the following theorem which is more general than Proposition~\ref{prop:two physical measures}.

\begin{theorem}\label{theo:two-pm}
For any $f\in\mathscr{K}^{1+}(\mathbb{T}^3)$, there exists a $C^1$-neighborhood ${\cal U}$ of $f$, such that for any $g\in{\cal U}\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$, if $g$ has more than one physical measures, then $g\in\mathscr{K}_l^{1+}(\mathbb{T}^3)$.
\end{theorem}

\begin{remark}
Generally, this theorem doesn't work for $f\in\mathscr{K}_l^{1+}(\mathbb{T}^3)$ since our proof of Theorem~\ref{theo:two-pm} relies on property (K2) in the definition of Kan's examples.
\end{remark}

Another byproduct of our main result is the following corollary.

\begin{corollary*}
There exists a partially hyperbolic skew product diffeomorphism $g\in\operatorname{Diff}^{\infty}(\mathbb{T}^3)$ such that $g$ is robustly topologically mixing, but neither the stable nor the unstable foliation of $g$ is minimal.
\end{corollary*}

\begin{remark}\label{remark}
Since $g$ in the above corollary is a skew product diffeomorphism over a linear Anosov automorphism of the torus, it must have normally hyperbolic periodic center compact leaves. Moreover, $g$ can also be constructed such that it preserves the orientation of all three invariant bundles of $g$'s partially hyperbolic splitting. However, for an open dense subset of such kind systems, Theorem 1.6 of \cite{BDU} showed that both the stable and the unstable foliations are robustly minimal.
\end{remark}

Notice that Kan constructed in \cite{K} an endomorphism on the cylinder admitting two physical measures with intermingled basins. One can define Kan's map on the cylinder like Definition \ref{def:Kan's example with boundary} (see \cite{BDV}). Recently, Gan and Shi \cite{GS} proved that all these Kan's maps are robustly transitive in the setting of boundary preserving case.

~

\noindent\emph{Outline of the paper}: Our main results are all stated in Section~\ref{sec:introduction}. In Section~\ref{sec:pre}, we recall some materials that will be necessary for the proofs. In Section~\ref{sec:construction}, we show how to construct the examples we need. The core step of this section is the construction of blender-horseshoes, which is kind of technical and complicated. The readers could skip this part for the first reading, because we will summarize the main properties of our examples at the beginning of Section~\ref{sec:proofs}. Finally in the remaining of Section~\ref{sec:proofs}, we will give the proofs of our results.

\section{Preliminaries}\label{sec:pre}

In this section, we will give some definitions and notations that will be used in this paper. Firstly, we will introduce the definition of Gibbs $u$-states, which was introduced in \cite{PS}, and has been proved to be a powerful tool for studying physical measures. Then, we will define a special class of diffeomorphisms, now called systems with mostly contracting center. These systems are firstly introduced and studied by Bonatti and Viana\cite{BV}. In a word, they have shown the existence and finiteness of physical measures for such systems.

Secondly, we will give a brief introduction to blender-horseshoes and blenders. The conception of blenders is firstly introduced by Bonatti and D\'iaz \cite{BD1}, and it has been proved to be a remarkable idea and a powerful tool for proving persistence of cycles and transitivity (see \cite{BD1,BD2}). Then in \cite{BD3}, they raise another conception, called blender-horseshoes, and prove it to be a special class of blenders.

\subsection{Gibbs \boldmath{$u$-}states and systems with mostly contracting center}
Given a compact Riemannian manifold $M$, a diffeomorphism $f:M\rightarrow M$ is called \emph{partially hyperbolic} if the tangent bundle admits a dominated splitting $TM=E^s\oplus E^c\oplus E^u,$ such that $Df|_{E^s}$ is uniformly contracting, $Df|_{E^u}$ is uniformly expanding and $Df|_{E^c}$ lies between them:
\begin{displaymath}
\parallel Df|_{E^s(x)}\parallel<\parallel Df^{-1}|_{E^c(f(x))}\parallel^{-1},\quad \parallel Df|_{E^c(x)}\parallel<\parallel Df^{-1}|_{E^u(f(x))}\parallel^{-1},\quad\textrm{for all}\ x\in M.
\end{displaymath}

The \emph{stable bundle} $E^s$ and \emph{unstable bundle} $E^u$ are automatically integrable: there exists a unique foliation $\mathcal{F}^u$ (resp. $\mathcal{F}^s$) tangent to $E^u$ (resp. $E^s$) at every point. $\mathcal{F}^u$ and $\mathcal{F}^s$ are called \emph{unstable foliation} and \emph{stable foliation}, respectively. A subset $\Lambda$ of $M$ is called \emph{$u$-saturated} (resp. \emph{$s$-saturated}) if it consists of entire strong unstable (resp. strong stable) leaves. Any compact disk embedded in a leaf of $\mathcal{F}^u$ is called a \emph{$u$-disk}.

An $f$-invariant probability measure $\mu$ is called a \emph{Gibbs $u$-state} if the conditional probabilities of $\mu$ along $\mathcal{F}^u$ are absolutely continuous with respect to the Lebesgue measures on the leaves.

The following proposition summarizes the basic properties of Gibbs $u$-states.

\begin{proposition}[\cite{BDV,PS}]\label{prop:gibbs ustate basic properties}
\begin{itemize}
\item[(1)] For every $u$-disk $D$, every accumulation point of the sequence of probability measures
$$\mu_n=\frac{1}{n}\sum_{i=0}^{n-1}f^i_*\left(\frac{m_D}{m_D(D)}\right)$$
is a Gibbs $u$-state, where $m_D$ is the Lebesgue measure restricted to the disk $D$.
\item[(2)] The support of every Gibbs $u$-state is $u$-saturated.
\item[(3)] Every ergodic component of a Gibbs $u$-state is still a Gibbs $u$-state.
\item[(4)] For Lebesgue almost every point $x$ in any $u$-disk, every accumulation point of $\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}$ is a Gibbs $u$-state.
\item[(5)] Every physical measure is a Gibbs $u$-state. Conversely, every ergodic Gibbs $u$-state whose center Lyapunov exponents are all negative is a physical measure.
\end{itemize}
\end{proposition}

We say a $C^{1+}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ has \emph{mostly contracting center} if for any $u$-disk $D^u$, we have
\begin{displaymath}
\limsup_{n\rightarrow\infty}\frac{1}{n}\log\parallel Df^n(x)|_{E^c(x)}\parallel<0
\end{displaymath}
for every $x$ in a subset $D_0^u\subset D^u$ with positive Lebesgue measure.

\begin{proposition}[\cite{A,VY}]\label{prop:mcc equivalent property}
A diffeomorphism $f$ has mostly contracting center if and only if the center Lyapunov exponents of all ergodic
Gibbs $u$-states of $f$ are negative.
Moreover, if $f$ has mostly contracting center, then
\begin{itemize}
\item there exist finitely many ergodic Gibbs $u$-states for $f$, and each of them is an ergodic physical measure.
\item the supports of the ergodic Gibbs $u$-states of $f$ are pairwise disjoint.
\end{itemize}
\end{proposition}

Proposition~\ref{prop:mcc equivalent property} gives the equivalent characterization of systems with mostly contracting center. Next lemma shows that mostly contracting systems are $C^1$ open.

\begin{lemma}[\cite{Y}, Theorem~B]\label{lem:openness of mcc}
Let $M$ be some compact Riemannian manifold. Assume $f$ is any $C^{1+}$-partially hyperbolic diffeomorphism on $M$ having mostly contracting center, then there exists a $C^1$-neighbourhood $\mathcal{U}$ of $f$, such that every $g\in\mathcal{U}\cap\operatorname{Diff}^{1+}(M)$ also has mostly contracting center.
\end{lemma}

\subsection{\boldmath{Blender-horseshoes and $cu$-}blenders}
Suppose $f:M\rightarrow M$ is a $C^1$-diffeomorphism, where $M$ is an $n$-dimensional compact Riemannian manifold, $n\geq3$. In this subsection, we write $n=n_s+1+n_u,$ where $n_s,n_u\geq1.$ Consider a $C^1$-embedding $\varphi: D^s\times[-1,1]\times D^u\rightarrow M$, where $D^s=[-1,1]^{n_s}$ and $D^u=[-1,1]^{n_u}$. Denote by $\Gamma$ the image $\varphi(D^s\times[-1,1]\times D^u)$. In $\Gamma$, we take the coordinate $(x_s,x_c,x_u), x_s\in D^s, x_c\in[-1,1], x^u\in D^u.$ In the manifold $M$, we take a metric $\parallel\cdot\parallel$ that induces in the cube $\Gamma$ the product of the usual euclidean metrics in $D^s,[-1,1]$ and $D^u$. Denote four specific parts of $\partial \Gamma$ as follows:
\begin{displaymath}
\begin{array}{rcl}
\partial^u\Gamma&=&D^s\times\partial([-1,1]\times D^u).\\
\partial^{uu}\Gamma&=&D^s\times[-1,1]\times\partial D^u.\\
\partial^{ss}\Gamma&=&(\partial D^s)\times[-1,1]\times D^u.\\
\partial^c\Gamma&=&D^s\times\{-1,1\}\times D^u.
\end{array}
\end{displaymath}

Given a $k$-dimensional plane $\Pi\subset\mathbb{R}^n$ and $\varepsilon>0$, define the \emph{$\varepsilon$-cone around $\Pi$} by $C_{\varepsilon}(\Pi)=\{u\in \mathbb{R}^n: u=v+w,w\in\Pi,w\in\Pi^{\bot},\parallel w\parallel\leq\varepsilon\parallel v\parallel\}$.

For $\varepsilon>0,$ define $\mathcal{C}^{uu}_{\varepsilon},\mathcal{C}^u_{\varepsilon}$ and $\mathcal{C}^{ss}_{\varepsilon}$ to be  conefields of size $\varepsilon$ around the tangent space of the families of disks $\{x_s,x_c\}\times D^u$, $\{x_s\}\times[-1,1]\times D^u$ and $D^s\times\{x_c,x_u\}$, respectively.

Now fix an $\varepsilon>0$. We say that an $n_u$-disk(resp. $n_s$-disk) $\Delta$ in $\Gamma$ is a \emph{vertical disk}(resp. horizontal disk) if $\Delta$ is tangent to $\mathcal{C}^{uu}_{\varepsilon}$(resp. $\mathcal{C}^{ss}_{\varepsilon}$) and its boundary is contained in $\partial^{uu}\Gamma$(resp. $\partial^{ss}\Gamma$).

Let $\Delta\subset \Gamma$ be a horizontal disk such that $\Delta\cap\partial^u\Gamma=\varnothing$. Then there are two different homotopy classes of vertical disks through $\Gamma$ disjoint from $\Delta$. A vertical disk that does not intersect $\Delta$ is \emph{at the right of $\Delta$} if it is in the homotopy class of $\{0_s\}\times\{1\}\times D^u$. Otherwise we say that it is \emph{at the left of $\Delta$}.

Let $\Delta$ be some horizontal disk. A \emph{vertical strip} (\emph{at the right of $\Delta$}) is defined to be an embedding $\Phi:[0,1]\times D^u\rightarrow\Gamma$ such that for every $t\in[0,1]$, $\Phi(\{t\}\times D^u)$ is a vertical disk (at the right of $\Delta$) and $\Phi([0,1]\times D^u)$ is tangent to $\mathcal{C}^{u}_{\varepsilon}$. We denote $\Phi([0,1]\times D^u)$ by $\mathcal{S}$. The width of a vertical strip $\mathcal{S}$, denoted by $\operatorname{Wd}(\mathcal{S})$, is defined to be the minimum of the length of arcs contained in $\mathcal{S}$ and connecting $\Phi(\{0\}\times D^u)$ and $\Phi(\{1\}\times D^u)$.

\begin{definition}[Blender-horseshoes]\label{def:blenderhorseshoe}
We call $\Lambda=\cap_{i\in\mathbb{Z}}f^i(\Gamma)$, the maximal invariant set in $\Gamma$, a \emph{blender-horseshoe} if the following conditions hold(with respect to some local coordinate system)
\begin{itemize}
\item[(1)] $f(\Gamma)\cap(\mathbb{R}^s\times\mathbb{R}\times[-1,1]^u)$ consists of two components, denoted by $f(\tilde{\Gamma}_1)$ and $f(\tilde{\Gamma}_2)$. Moreover, $f(\tilde{\Gamma}_1)\cup f(\tilde{\Gamma}_2)\subset(-1,1)^s\times\mathbb{R}\times[-1,1]^u$, and $(\tilde{\Gamma}_1\cup\tilde{\Gamma}_2)\cap\partial^{uu}\Gamma=\varnothing.$
\item[(2)] There exists $\varepsilon_0>0$ such that the conefields $\mathcal{C}^{uu}_{\varepsilon_0},\mathcal{C}^u_{\varepsilon_0}$ and $\mathcal{C}^{ss}_{\varepsilon_0}$ are $Df$-invariant. That is, for $x\in f(\tilde{\Gamma}_1)\cup f(\tilde{\Gamma}_2)$, $Df^{-1}(\mathcal{C}^{ss}_{\varepsilon_0}(x))\subsetneqq \mathcal{C}^{ss}_{\varepsilon_0}(f^{-1}x)$, and for $x\in\tilde{\Gamma}_1\cup\tilde{\Gamma}_2$, $Df(\mathcal{C}^{uu}_{\varepsilon_0}(x))\subsetneqq \mathcal{C}^{uu}_{\varepsilon_0}(f(x))$ and $Df(\mathcal{C}^{u}_{\varepsilon_0}(x))\subsetneqq \mathcal{C}^{u}_{\varepsilon_0}(f(x))$. Moreover, $\mathcal{C}^{ss}_{\varepsilon_0}$ and $\mathcal{C}^{u}_{\varepsilon_0}$ are uniformly contracting and expanding under $Df$, respectively.
\item[(3)] Denote by $\Gamma_i=f^{-1}(f(\tilde{\Gamma}_i)\cap\Gamma), i=1,2.$ Then $\Gamma_1\cup\Gamma_2$ is disjoint from $\partial^u\Gamma$ and $f(\Gamma_1)\cup f(\Gamma_2)$ is disjoint from $\partial^{ss}\Gamma$. It can be deduced that $\{\Gamma_1,\Gamma_2\}$ forms a Markov partition of $\Lambda$. Denote by $P_1$ and $P_2$ to be the unique hyperbolic fixed points contained in $\Gamma_1$ and $\Gamma_2$, respectively.
\item[(4)] For any $x\in\Lambda$, define $W^{\sigma}_{loc}(x)$ to be the connected component of $W^{\sigma}(x)\cap \Gamma$ containing $x$, where $\sigma=s,u,uu$. Then for any vertical disks $\Delta$ and $\Delta'$ such that $\Delta\cap W^s_{loc}(P_1)\neq\varnothing$ and $\Delta'\cap W^s_{loc}(P_2)\neq\varnothing$, we have $\Delta\cap\partial^c\Gamma=\Delta'\cap\partial^c\Gamma=\Delta\cap\Delta'=\varnothing$.
\item[(5)] If $\Delta$ is a vertical disk at the right (resp. left) of $W^s_{loc}(P_i)$, then $f(\Delta\cap\Gamma_i)$ is a vertical disk at the right (resp. left) of $W^s_{loc}(P_i)$, where $i=1,2$. If $\Delta$ is at left of $W^s_{loc}(P_1)$ or $\Delta\cap W^s_{loc}(P_1)\neq\varnothing$, then $f(\Delta\cap\Gamma_2)$ is a vertical disk at the left of $W^s_{loc}(P_1)$. If $\Delta$ is at right of $W^s_{loc}(P_2)$ or $\Delta\cap W^s_{loc}(P_2)\neq\varnothing$, then $f(\Delta\cap\Gamma_1)$ is a vertical disk at the right of $W^s_{loc}(P_2)$.
\item[(6)] If $\Delta$ is a vertical disk in between $W^s_{loc}(P_1)$ and $W^s_{loc}(P_2)$, then either $f(\Delta\cap\Gamma_1)$ or $f(\Delta\cap\Gamma_2)$ is a vertical disk in between $W^s_{loc}(P_1)$ and $W^s_{loc}(P_2)$.
\end{itemize}
\end{definition}

\begin{remark}
The cube $\Gamma$ is called the \emph{reference cube} of the blender-horseshoe $\Lambda$. And $P_1,P_2$ are called the \emph{reference saddles}. For more details of Definition~\ref{def:blenderhorseshoe}, see \cite{BD3}.
\end{remark}

\begin{definition}[$cu$-blenders]\label{def:blender}
Let $f:M\rightarrow M$ be a $C^1$-diffeomorphism and $\Lambda$ be a transitive hyperbolic set with stable index $n_s$. Then $(\Lambda,f)$ is called a \emph{$cu$-blender} if there are a $C^1$-neighbourhood $\mathscr{U}$ of $f$ and a $C^1$-open set $\mathcal{D}$ of embeddings of $n_u$-dimensional disks $\Delta$ into $M$ such that for every $g\in\mathscr{U}$ and every $\Delta\in\mathcal{D}$, $\Delta\cap W^s_{loc}(\Lambda_g)\neq\varnothing$. Then $\mathcal{D}$ is called the \emph{superposition region} of the blender, and $\mathbb{D}=\cup_{\Delta\in\mathcal{D}}\Delta$ is called the \emph{superposition domain} of the blender. We also say $\Lambda$ is a $cu$-blender if there is no ambiguity.
\end{definition}

\begin{lemma}\label{lem:blenderhorseshoe and cu blender}
$(1)$(\cite{BD3},Lemma~3.9) Let $f:M\rightarrow M$ be a $C^1$-diffeomorphism and $\Lambda$ be a blender-horseshoe with reference cube $\Gamma$ and reference saddles $P_1$ and $P_2$. Then there exists a $C^1$- neighbourhood $\mathcal{U}$ of $f$ such that for any $g\in\mathcal{U}$, $\Lambda_g$ is a blender-horseshoe with reference cube $\Gamma$ and reference saddles $P_{1,g}$ and $P_{2,g}$.

$(2)$(\cite{BD3},Remark~3.10) The blender-horseshoe $\Lambda$ is also a $cu$-blender, and the superposition region contains the set of the vertical disks lying in between $W^s_{loc}(P_1)$ and $W^s_{loc}(P_2)$.
\end{lemma}

The following lemma can be deduced from the definition of $cu$-blenders and Palis' $\lambda$-lemma.

\begin{lemma}\label{lem:blender}
Suppose $(\Lambda,f)$ is a $cu$-blender, $P\in\Lambda$ is a hyperbolic fixed point, and $Q$ is a hyperbolic fixed point of index $n_s+1$ such that $W^u(Q)$ contains a vertical disk that belongs to the superposition region $\mathcal{D}$, then there exists a $C^1$-neighbourhood $\mathcal{U}$ of $f$, such that for any $g\in\mathcal{U}$, $W^s(Q_g,g)\subset\overline{W^s(P_g,g)}$.
\end{lemma}
\begin{proof}
Fix $\mathcal{U}$ small enough such that for any $g\in\mathcal{U}$, $\Lambda_g,P_g,Q_g$ are all well defined and $(\Lambda_g,g)$ is a $cu$-blender. Then since $\Lambda_g$ is transitive, $\Lambda_g\subset \overline{W^s(P_g,g)}$. Hence, we have that $W^s_{loc}(\Lambda_g,g)\subset \overline{W^s(P_g,g)}$. Since $W^u(Q)$ contains a vertical disk, denoted by $\Delta$, that belongs to the superposition region $\mathcal{D}$, then $W^u(Q_g,g)$ also contains a vertical disk, denoted by $\Delta_g$, that belongs to $\mathcal{D}$, by the continuity of (the compact part of) $W^u(Q_g,g)$ with respect to $g$  and the openness of $\mathcal{D}$.

Now for any $x\in W^s(Q_g,g)$, and any open neighbourhood $V$ of $x$, there exists $N>0$ large enough such that $g^N(V)$ contains a vertical disk $\Delta'_g$ close enough to $\Delta_g$ such that $\Delta'_g\in\mathcal{D}$. Therefore, $g^N(V)\cap W^s_{loc}(\Lambda_g,g)\neq\varnothing$, which implies $g^N(V)\cap \overline{W^s(P_g,g)}\neq\varnothing$. Now the conclusion follows by using the $g$-invariance of $W^s(P_g,g)$.
\end{proof}

\section{Construction of the example}\label{sec:construction}
We will firstly construct our example on $\mathbb{T}^2\times[0,1]$, and then give the construction on $\mathbb{T}^3$ using a reversing technique.

Suppose $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$ has four fixed points, namely, $p,q,r$ and $s$. Let $\lambda>3$ be one of the two eigenvalues of $\mathbb{A}$. Denote by
\begin{displaymath}
P=(p,0.5),P_i=(p,i), Q_i=(q,i), R_i=(r,i), S_i=(s,i), i=0,1.
\end{displaymath}
From now on in this paper, we will take $\mathbb{T}^2$ to be the base space, and $\mathbb{T}^2\times[0,1]$ and $\mathbb{T}^3=\mathbb{T}^2\times\mathbb{S}^1$ will be viewed as the product spaces over $\mathbb{T}^2$. The points on $\mathbb{T}^2$ will be denoted by lower case letters, such as $p,q,r,s,x$. And the points on the product spaces will be denoted by corresponding capital letters, such as $P,Q,R,S,X$.

First of all, we need a local chart around point $p$ on $\mathbb{T}^2$ given by the next lemma.
\begin{lemma}\label{lem:adapted domain in 2d}
Consider the system $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$. There exist a homoclinic point of $p$, denoted by $a$, and a local chart centered at $p$, say $(U(p);(x_s,x_u))$, such that
\begin{itemize}
\item[(1)] $\mathbb{A}(x_s,x_u)=(\lambda^{-1}x_s,\lambda x_u),$ for every $(x_s,x_u)\in[-3,3]_s\times[-3,3]_u$.
\item[(2)] The coordinate of $p$ is $(0_s,0_u)$, and the coordinate of $a$ is $(0_s,1_u)$. There exists $n_0>0$ such that the coordinate of $\mathbb{A}^{2n_0}(a)$ is $(1_s,0_u)$.
\item[(3)] Denote $[-2,2]_s\times[-2,2]_u$ by $C$. Then $C\cap\mathbb{A}^{-2n_0}(C)$ contains two connected components, namely $C_1=[-2,2]_s\times[-2\lambda^{-2n_0},2\lambda^{-2n_0}]_u$ and $C_2=[-2,2]_s\times[1-2\lambda^{-2n_0},1+2\lambda^{-2n_0}]_u$. Moreover, $\mathbb{A}^{2n_0}(x_s,x_u)=(1_s+x_s\lambda^{-2n_0},(x_u-1_u)\lambda^{2n_0})$, for every $(x_s,x_u)\in C_2$.
\item[(4)] $C,\mathbb{A}(C_2),\ldots,$ $\mathbb{A}^{2n_0-1}(C_2)$ are pairwise disjoint.
\end{itemize}
\end{lemma}

\begin{proof}
We firstly choose a sufficiently small neighbourhood $U(p)$ of $p$ and some local chart on $U(p)$ such that item(1) is satisfied. This is trivial since $\mathbb{A}$ is linear Anosov. Now choose some homoclinic point $b$ of $p$, such that $b\in W_{loc}^u(p)$ and $\mathbb{A}^{2m}(b)\in W_{loc}^s(p)$, $m>0$. By taking iterations of $b$ and taking a linear transformation of the local chart, $b$ and $m$ can be chosen such that the following conditions are satisfied simultaneously
\begin{itemize}
\item $b=(0_s,1_u),\mathbb{A}^{2m}(b)=(1_s,0_u)$;
\item $\mathbb{A}(b),\ldots,\mathbb{A}^{2m-1}(b)\notin[-2,2]_s\times[-2,2]_u$.
\end{itemize}
Then for every $1\leq i\leq 2m-1$, we choose some open neighbourhood $U(\mathbb{A}^i(b))$ of $\mathbb{A}^i(b)$ such that $U(\mathbb{A}(b)),\ldots,U(\mathbb{A}^{2m-1}(b))$ and $[-2,2]_s\times[-2,2]_u$ are pairwise disjoint. Let $\varepsilon>0$ be some small positive number such that for every $1\leq i\leq 2m-1$, $U(\mathbb{A}^i(b))$ contains an $su$-box centered at $\mathbb{A}^i(b)$ of size $\varepsilon$, i.e.,
$$\bigcup_{x\in W^u_{\varepsilon}(\mathbb{A}^i(b))}W^s_{\varepsilon}(x)\subset U(\mathbb{A}^i(b)).$$
Lastly, take $a=\mathbb{A}^{-2m_1}(b)\in W^u_{\frac{\varepsilon}{2}}(p)$, $\mathbb{A}^{2n_0}(a)=\mathbb{A}^{2m_1+2m+2m_2}(b)\in W^s_{\frac{\varepsilon}{2}}(p)$, and by another linear transformation of the local chart, we assume $a=(0_s,1_u),\mathbb{A}^{2n_0}(a)=(1_s,0_u)$. Define $C,C_1$ and $C_2$ as in item(3), and we claim that item(4) is satisfied. To see this, it is sufficient to notice that $\mathbb{A}^i(C_2)$ is contained in the $su$-box centered at $\mathbb{A}^i(a)$ of size $\varepsilon$ for every $1\leq i\leq 2n_0-1$.
\end{proof}
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{domains.pdf}
\caption{Perturbation domains.}
\end{figure}
It can be easily deduced from the proof of the lemma that we can choose the local chart appropriately such that the Lebesgue measure of $\cup_{i=1}^{2n_0-1}\mathbb{A}^i(U(C_2))$ is smaller than any given positive number. Now we fix some $\varepsilon>0$ small enough, and choose a set of open sets contained in $\mathbb{T}^2$ such that
\begin{itemize}
\item[(1)] $U(x)\ni x$, and $\operatorname{Leb}(U(x))<\varepsilon$, where $x=q,r,s.$
\item[(2)] $U(C)\supset C$ and $U(C_2)\supset C_2$, where $C,C_2$ are defined as in Lemma~\ref{lem:adapted domain in 2d}, and $U(C_2)\cap C_1=\varnothing$. Moreover, $$\operatorname{Leb}(U(C)\bigcup\cup_{i=1}^{2n_0-1}\mathbb{A}^i(U(C_2)))<\varepsilon.$$
\item[(3)] $U_1\subset U_2,$ and $\operatorname{Leb}(U_1)>0.5$.
\item[(4)] $U(r),U(s),U(q),U(C),U_2,\mathbb{A}(U(C_2)),\ldots,\mathbb{A}^{2n_0-1}(U(C_2))$ are pairwise disjoint.
\end{itemize}

\begin{remark}
The open sets above are the perturbation domains. In fact, we will perturb $\mathbb{A}\times\operatorname{Id}:\mathbb{T}^2\times[0,1]\rightarrow\mathbb{T}^2\times[0,1]$ in $U(r)\times[0,1]$ and $U(s)\times[0,1]$ to get two fibers as in Definition~\ref{def:Kan's example with boundary} (K2). And perturbations will be made in $U_2\times[0,1]$ to guarantee (K4) in Definition~\ref{def:Kan's example with boundary}. As to  $U(C)\times[0,1]$ and $U(C_2)\times[0,1]$, they are used to construct a blender-horseshoe, which is the most important part in our example. Lastly, we will make perturbations in $U(q)\times[0,1]$ to get a hyperbolic fixed point which activates the blender constructed beforehand.
\end{remark}

Now we define a smooth vector field $\mathcal{X}$ on $\mathbb{T}^2\times[0,1]$, such that
\begin{equation}\label{equation:x}
\mathcal{X}(x,\theta)=\left\{\begin{array}{ll}
                             \alpha_1(x)\sin(\pi\theta)\frac{\partial}{\partial \theta},&x\in U(s),\\
                             -\alpha_1(x)\sin(\pi\theta)\frac{\partial}{\partial \theta},&x\in U(r),\\
                             -\alpha_1(x)\sin(2\pi\theta)\frac{\partial}{\partial\theta},&x\in U_2\cup U(C),\\
                             -\alpha_1(x)\beta_1(\theta)\sin(2\pi\theta)\frac{\partial}{\partial\theta},&x\in\cup_{i=0}^{2n_0-1}\mathbb{A}^i(U(C_2)),\\
                             0,&\textrm{otherwise},
                             \end{array}\right.
\end{equation}
where $\alpha_1\in C^{\infty}(\mathbb{T}^2)$ and $\beta_1\in C^{\infty}([0,1])$ are two bump functions satisfying
\begin{displaymath}
\alpha_1(x)\left\{\begin{array}{ll}
                 =1,&x\in\{r,s\}\cup U_1\cup C\bigcup\cup_{i=0}^{2n_0-1}\mathbb{A}^i(C_2),\\
                 =0,&x\notin U(r)\cup U(s)\cup U_2\cup U(C)\bigcup\cup_{i=0}^{2n_0-1}\mathbb{A}^i(U(C_2)),\\
                 \in[0,1],&\textrm{otherwise},
                 \end{array}\right.
\end{displaymath}
and
\begin{displaymath}
\beta_1(\theta)=\left\{\begin{array}{ll}
                       1,&\theta\in(0.5-\varepsilon,0.5+\varepsilon),\\
                       0,&\theta\in[0,\varepsilon)\cup(1-\varepsilon,1],\\
                       \in[0,1],&\textrm{otherwise.}
                       \end{array}\right.
\end{displaymath}

Denote by $\widehat{U(p)}=U(p)\times(0.5-\varepsilon,0.5+\varepsilon)$, and let $(\widehat{U(p)};(x_s,x_u,x_c))$ be a local chart defined on $\mathbb{T}^2\times[0,1]$  such that
\begin{itemize}
\item $(\widehat{U(p)};(x_s,x_u,x_c))$ is compatible with $(U(p);(x_s,x_u))$.
\item The coordinate of $P$ is $(0_s,0_u,0_c)$.
\item The vector field $\mathcal{X}$ can be wrote as $\mathcal{X}(x_s,x_u,x_c)=x_c\frac{\partial}{\partial x_c},$ for every $(x_s,x_u,x_c)\in[-3_s,3_s]\times[-3_u,3_u]\times[-3_c,3_c]$.
\end{itemize}

\begin{figure}[htbp]
\centering\includegraphics[width=10cm]{blender.pdf}
\caption{Construction of blender-horseshoes.}
\end{figure}

Then define $\Gamma$ to be a subset of $\widehat{U(p)}$ such that it is identified with $[-2,2]_s\times[-2,2]_u\times[-2,2]_c$ under the local chart.

Now consider a family of $C^{\infty}$diffeomorphisms $\{f_{t,\nu}\}_{t,\nu\geq0}$ satisfying
\begin{itemize}
\item $f_{t,\nu}(x,\theta)=(\mathbb{A}\times\operatorname{Id})\circ\mathcal{X}_{t}(x,\theta)$, for all $x\in U(r)\cup U(s)\cup U(C)\cup U_2$, where $\mathcal{X}_t$ is the time $t$ map of $\mathcal{X}$.
\item $f_{t,\nu}^{2n_0}(x_s,x_u,x_c)=T_{\nu}\circ (\mathbb{A}\times\operatorname{Id})^{2n_0}\circ\mathcal{X}_{2n_0t}(x_s,x_u,x_c),$ for all $(x_s,x_u,x_c)\in C_2\times(0.5-0.5\varepsilon,0.5+0.5\varepsilon)$, where $T_{\nu}$ is the translation defined by $T_{\nu}(x_s,x_u,x_c)=(x_s,x_u,x_c-\nu).$
\item $f_{t,\nu}$ is skew product for all $t\geq0$ and $\nu\geq0$, that is, every $f_{t,\nu}$ preserves the center foliation $\mathscr{L}=\{\{x\}\times[0,1]:x\in\mathbb{T}^2\}$.
\item $f_{t,\nu}(x,\theta)=(\mathbb{A}x,\theta)$, for every $x\in\bigcup_{i=0}^{2n_0-1}\mathbb{A}^i(U(C_2))$ and $\theta\in[0,\varepsilon)\cup(1-\varepsilon,1]$.
\end{itemize}

\begin{lemma}\label{lem:parameters of blenderhorseshoe}
Let $\nu_t=e^{2n_0t}-1$, and $\Lambda_t=\cap_{i\in\mathbb{Z}}f^{2n_0i}_{t,\nu_t}(\Gamma)$. Then for $t>0$ small enough, $(\Lambda_t,f^{2n_0}_{t,\nu_t})$ is a blender-horseshoe with reference cube $\Gamma$ and reference saddles $P=(0_s,0_u,0_c)$ and $O=(1_s/(1-\lambda^{-2n_0}),1_u/(1-\lambda^{-2n_0}),1_c)$.
\end{lemma}
\begin{remark}
It is worthwhile pointing out that here the reference cube and reference saddles of the blender-horseshoes are independent of $t$.
\end{remark}

Define $\Gamma_1=[-2,2]_s\times[-2\lambda^{-2n_0},2\lambda^{-2n_0}]_u\times[-2e^{-2n_0t},2e^{-2n_0t}]_c$, $\Gamma_2=[-2,2]_s\times[1-2\lambda^{-2n_0},1+2\lambda^{-2n_0}]_u\times[1-3e^{-3n_0t},1+e^{2n_0t}]_c$. Then $f^{-2n_0}_{t,\nu_t}(\Gamma)\cap\Gamma=\Gamma_1\cup\Gamma_2,$ and
\begin{displaymath}
f^{2n_0}_{t,\nu_t}(x_s,x_u,x_c)=\left\{\begin{array}{lc}
                             (\lambda^{-2n_0}x_s,\lambda^{2n_0}x_u,e^{2n_0t}x_c),&(x_s,x_u,x_c)\in\Gamma_1.\\
                             (\lambda^{-2n_0}x_s+1,\lambda^{2n_0}(x_u-1),e^{2n_0t}(x_c-1)+1),&(x_s,x_u,x_c)\in\Gamma_2.\end{array}\right.
\end{displaymath}

Then it is easy to check that $P\in\Gamma_1$ and $O\in\Gamma_2$ are two hyperbolic fixed points of $f^{2n_0}_{t,\nu_t}$.

We will not go into details of the proof of Lemma~\ref{lem:parameters of blenderhorseshoe}, and refer the reader to Proposition~5.1 of \cite{BD3}. Instead, we will prove the following lemma just to exhibit the main properties of the blender-horseshoe.
\begin{lemma}
For every $t>0$ small enough, $W^s(P,f^{2n_0}_{t,\nu_t})$ intersects every strip $\mathcal{S}$ in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$.
\end{lemma}

\begin{proof}
For any $t>0$ small enough, we choose the corresponding conefields $\mathcal{C}^s_{\varepsilon_t},\mathcal{C}^u_{\varepsilon_t},\mathcal{C}^{uu}_{\varepsilon_t}$ of size $\varepsilon_t$ sufficiently small, such that for any $v\in\mathcal{C}^u_{\varepsilon_t}$, $\parallel Df^{2n_0}_{t,\nu_t}(v)\parallel>(e^{2n_0t}+1)/2\parallel v\parallel.$ Denote by $\lambda'_t=(e^{2n_0t}+1)/2$.

To prove the lemma, we only have to show that for any vertical strip $\mathcal{S}$ in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$, $f^{2n_0}_{t,\nu_t}(\mathcal{S}\cap(\Gamma_1\cup\Gamma_2))$ contains a vertical strip $\mathcal{S}'$, such that either $\operatorname{Wd}(\mathcal{S}')>\lambda'_t\operatorname{Wd}(\mathcal{S})$ or $\mathcal{S}'\cap W^s_{loc}(P,f^{2n_0}_{t,\nu_t})\neq \varnothing$.

To show the main ingredients of the proof, we firstly consider the vertical strips $\mathcal{S}$ in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$ of the type $\{x_s\}\times[-2,2]_u\times[t_1,t_2]_c,$ where $0<t_1<t_2< 1$. If $t_2\leq e^{-2n_0t}$, denote $f^{2n_0}_{t,\nu_t}(\Gamma_1\cap\mathcal{S})$ by $\mathcal{S}'$, then $\mathcal{S}'=\{\lambda^{-2n_0}x_s\}\times[-2,2]_u\times[e^{2n_0t}t_1,e^{2n_0t}t_2]_c$ is a vertical strip in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$, with width $e^{2n_0t}\operatorname{Wd}(\mathcal{S})>\lambda'_t\operatorname{Wd}(\mathcal{S})$.

If $e^{-2n_0t}<t_2< 1$, denote $f^{2n_0}_{t,\nu_t}(\mathcal{S}\cap\Gamma_2)$ by $\mathcal{S}'$, then there are two subcases. If $e^{2n_0t}(t_1-1)+1\geq0,$ then $\mathcal{S}'=\{\lambda^{-2n_0}x_s+1\}\times[-2,2]_u\times[e^{2n_0t}(t_1-1)+1,e^{2n_0t}(t_2-1)+1]_c$ is a vertical strip in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$, with width $e^{2n_0t}\operatorname{Wd}(\mathcal{S})>\lambda'_t\operatorname{Wd}(\mathcal{S})$. Otherwise, noting that $e^{2n_0t}(t_2-1)+1>0,$ we deduce that $\mathcal{S}'$ is a vertical strip intersecting $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$.

Now consider a general vertical strip $\mathcal{S}$ in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$, i.e., it is the image of an embedding $\Phi:[0,1]\times D^u\rightarrow \Gamma$ such that $\Phi(\{t\}\times D^u)$ is a vertical disk in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$ for every $t\in[0,1]$, and $\Phi([0,1]\times D^u)$ is tangent to $\mathcal{C}^u_{\varepsilon_t}$.

If $\mathcal{S}\cap\Gamma_1\neq\varnothing$, and $\mathcal{S}\cap\{x\in\Gamma:x_c\geq e^{-2n_0t}\}=\varnothing$, then it is obvious to see that $\mathcal{S}'=f^{2n_0}_{t,\nu_t}(\mathcal{S}\cap\Gamma_1)$ is a vertical strip in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$. Moreover, $\operatorname{Wd}(\mathcal{S}')>\lambda'_t\operatorname{Wd}(\mathcal{S})$ since $\mathcal{S}$ is tangent to $\mathcal{C}^u_{\varepsilon_t}$.

Otherwise, there exists $s_0\in(0,1)$, such that $\Delta_{s_0}=\Phi(\{s_0\}\times D^u)\subset\{x\in\Gamma:x_c\geq0.5\}$, and $\Delta'_{s_0}=f^{2n_0}_{t,\nu_t}(\Delta_{s_0}\cap\Gamma_2)$ is a vertical disk in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$. Now define $I$ to be the maximal subinterval of $[0,1]$ containing $s_0$ such that for every $s\in I,$ $\Delta'_{s}=f^{2n_0}_{t,\nu_t}(\Delta_s\cap\Gamma_2)$ is a vertical disk in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$. It is immediate that $I$ has nonempty interior by the continuity of $\Phi(\{\cdot\}\times D^u)$. If $I$ equals $[0,1]$, we are done. Otherwise, at lease one of the extremes of $I$, say $s_1$, is not contained in $I$. Then $f^{2n_0}_{t,\nu_t}(\Delta_{s_1})$ must have nonempty intersection with $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ or $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$. The second subcase can not happen, since if $f^{2n_0}_{t,\nu_t}(\Delta_{s_1})\cap W^s_{loc}(O,f^{2n_0}_{t,\nu_t})\neq\varnothing,$ then $\Delta_{s_1}\cap W^s_{loc}(O,f^{2n_0}_{t,\nu_t})\neq\varnothing$ by computation, which contradict with the assumption that $\Delta_{s_1}$ is in between $W^s_{loc}(P,f^{2n_0}_{t,\nu_t})$ and $W^s_{loc}(O,f^{2n_0}_{t,\nu_t})$. Then the first subcase must hold, i.e., $f^{2n_0}_{t,\nu_t}(\Delta_{s_1})\cap W^s_{loc}(P,f^{2n_0}_{t,\nu_t})\neq\varnothing.$ The proof of the lemma has now been finished.
\end{proof}

From now on, we will denote $f_{t,\nu_t}$ by $f_t$ for simplicity. Define $\Gamma'=\{x\in\Gamma:0<x_c<1,-2<x_s<2\}$. Then it is easy to check that $\Gamma'$ is contained in the superposition domain $\mathbb{D}_t$ of the blender-horseshoe $(\Lambda_t,f^{2n_0}_{t,\nu_t})$ for every $t>0$ small enough. Since $\mathbb{A}$ is Anosov, then $W^u(\Gamma',\mathbb{A}\times\operatorname{Id})\cap \bar{q}$ contains an interval with nonempty interior, where $\bar{q}$ stands for the fiber $\{q\}\times[0,1]$. Choose such an interior point $Q=(q,\theta_0)$, and we remark here that $Q$ can be chosen very close to $(q,0.5)$. Then there exists a $C^1$-neighbourhood $\mathcal{U}$ of $\mathbb{A}\times\operatorname{Id}$, such that for every $g\in\mathcal{U}$, $W^{uu}(Q,g)$ contains a vertical disk that belongs to the superposition region $\mathcal{D}_t$ of $(\Lambda_t,f^{2n_0}_{t,\nu_t})$.

Let $\gamma_t$ be a segment in $W^{uu}(Q,f_t)$ containing $Q$ and a vertical disk contained in the superposition region $\mathcal{D}_t$. Denote by $\gamma'_t$ the connected component of $\gamma_t\cap(U(q)\times[0,1])$ containing $Q$, and let $U'(q)\subset U(q)$ be a neighbourhood of $q$ such that $\gamma_t\cap (U'(q)\times[0,1])=\gamma'_t.$

Now let $\alpha_2:\mathbb{T}^2\rightarrow[0,1]$ be a smooth function such that $\alpha_2(q)=1$ and $\alpha_2(x)=0$ for all $x\notin U'(q).$ Let $\beta_2:[0,1]\rightarrow\mathbb{R}$ be a smooth function satisfying
\begin{itemize}
\item $\beta_2(0)=\beta_2(1)=\beta_2(\theta_0)=0.$
\item $\beta_2(\theta)>0$, when $0<\theta<\theta_0$, and $\beta_2(\theta)<0$, when $\theta_0<\theta<1$.
\item $\beta'_2(\theta)=1,$ when $\theta\in[0,\varepsilon)\cup(1-\varepsilon,1].$
\end{itemize}

\begin{figure}[htbp]\label{fig:global}
\centering
\includegraphics[width=10cm]{fig2.pdf}
\caption{Robustly transitive Kan's example on $\mathbb{T}^3$.}
\end{figure}

Define on $\mathbb{T}^2\times[0,1]$ a smooth vector field $\mathcal{Y}(x,\theta)=\alpha_2(x)\beta_2(\theta)\frac{\partial}{\partial\theta}$.

\begin{proposition}\label{prop:construction of kan on boundary manifolds}
For any $t>0$ small enough, we have $\tilde{f}_t=f_t\circ\mathcal{Y}_{t}\in\mathscr{K}^{\infty}(\mathbb{T}^2\times[0,1]).$
\end{proposition}
\begin{proof}
Denote by $\tilde{f}_t(x,\theta)=(\mathbb{A}x,\phi_{x,t}(\theta))$. Notice that the supports of $\mathcal{X}$ and $\mathcal{Y}$ are disjoint, the fiber diffeomorphisms $\phi_{x,t}$ can be written as the following for $\theta\in[0,\varepsilon)\cup(1-\varepsilon,1]$
\begin{equation}\label{equation:phi}
\phi_{x,t}(\theta)=\left\{\begin{array}{ll}
                         \mathcal{X}_{t}(x,\theta),& \textit{when}\ x\in U(r)\cup U(s)\cup U(C)\cup U_2\\
                         \mathcal{Y}_{t}(x,\theta),& \textit{when}\ x\in U(q).\\
                         \theta,&\textrm{otherwise}.
                         \end{array}\right.
\end{equation}
Now we are going to check the conditions $(K1)$--$(K4)$ in Definition~\ref{def:Kan's example with boundary} one by one.

By definition, for $i=0,1$, $\mathcal{X}(x,i)=0$ and $\mathcal{Y}(x,i)=0$, so $\phi_{x,t}(i)=i$. Hence condition $(K1)$ holds.

$\mathcal{X}(r,\theta)=-\sin(\pi\theta)\frac{\partial}{\partial \theta}$, so $\phi_{r,t}$ has exactly two fixed points, a sink at $\theta=0$ and a source at $\theta=1$. Similarly, $\mathcal{X}(s,\theta)=\sin(\pi\theta)\frac{\partial}{\partial \theta}$, so $\phi_{s,t}$ has exactly two fixed points, a sink at $\theta=1$ and a source at $\theta=0$. Hence condition $(K2)$ holds.

When $x\in U(r)\cup U(s)\cup U(C)\cup U_2$, $|\frac{\partial}{\partial\theta}\phi_{x,t}(\theta)|=\exp\int_0^t\frac{\partial}{\partial\theta}\mathcal{X}(x,\phi_{x,s}(\theta))ds\sim\exp(Ct)\rightarrow1(t\rightarrow0).$ Similar results hold when $x\in U(q)$. Thus $\tilde{f}_t$ is partially hyperbolic when $t$ is close to $0$. Hence condition $(K3)$ holds.

\begin{align}\nonumber
\int_{\mathbb{T}^2}\log\left|\frac{\partial}{\partial\theta}\phi_{x,t}(0)\right|dx&=\int_{U(r)}-\pi\alpha(x)tdx+\int_{U(s)}\pi \alpha(x) tdx+\int_{U_2\cup U(C)}-2\pi\alpha(x)tdx\\
&<\pi t(\operatorname{Leb}(U(s)-2\operatorname{Leb}(U_1)))<\pi t(\varepsilon-1)<0.\nonumber
\end{align}
Similarly, $\int_{\mathbb{T}^2}\log\left|\frac{\partial}{\partial\theta}\phi_{x,t}(1)\right|dx<\pi t(\varepsilon-1)<0$. Hence condition $(K4)$ holds. Now the proof of the proposition has been completed.
\end{proof}

Lastly, we will define a family of $C^{\infty}$-diffeomorphisms $\{K_t:\mathbb{T}^3\rightarrow\mathbb{T}^3\}_{t\geq0}$ by a reversing skill. Recall that $\mathbb{T}^3=\mathbb{T}^2\times\mathbb{S}^1$, and $\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$, we extend $\tilde{f}_t$ to $\hat{f}_t:\mathbb{T}^3\rightarrow\mathbb{T}^3$ by
\begin{equation}\label{equation:hat f}
\hat{f}_t(x,\theta)=\left\{\begin{array}{ll}
                 \tilde{f}_t(x,\theta)=(\mathbb{A}x,\phi_{x,t}(\theta)),&\theta\in[0,1]\\
                 (\mathbb{A}x,-\phi_{x,t}(-\theta)),&\theta\in[-1,0]
                 \end{array}\right.
\end{equation}
It is easy to see that $\hat{f}_t$ is the odd extension of $\tilde{f}_t$. And we remark here that $\hat{f}_t(t\geq0)$ are all smooth diffeomorphisms. This is because the odd extensions of $\mathcal{X}$ and $\mathcal{Y}$(denoted by $\hat{\mathcal{X}}$ and $\hat{\mathcal{Y}}$ respectively) are both smooth vector fields on $\mathbb{T}^3$, and $\hat{f}_t$ can be viewed as the smooth perturbations(supported on $\bigcup_{i=1}^{2n_0-1}\mathbb{A}^i(U(C))\times(0.5-\varepsilon,0.5+\varepsilon)$) of $\hat{\mathcal{X}}_t\circ\hat{\mathcal{Y}}_t$. At last, we define our examples $K_t(t\geq0)$ as the composition of $R=\operatorname{Id}_{\mathbb{T}^2}\times-\operatorname{Id}_{\mathbb{S}^1}:\mathbb{T}^2\times\mathbb{S}^1\rightarrow\mathbb{T}^2\times\mathbb{S}^1$ and $\hat{f}_t(t\geq0)$, i.e., $K_t=R\circ\hat{f}_t$. According to equation~\ref{equation:hat f}, we have
\begin{equation}\label{equation:K_t}
K_t(x,\theta)=\left\{\begin{array}{lr}
                     (\mathbb{A}x,-\phi_{x,t}(\theta)),&0\leq\theta\leq1.\\
                     (\mathbb{A}x,\phi_{x,t}(-\theta)),&-1\leq\theta\leq0.
                     \end{array}\right.
\end{equation}

From the arguments above and Proposition~\ref{prop:construction of kan on boundary manifolds}, we get the following proposition as wanted.
\begin{proposition}\label{prop:construction of kan on boundaryless manifold}
For any $t>0$ small enough, $K_t\in\mathscr{K}^{\infty}(\mathbb{T}^3)$.
\end{proposition}

\section{Proof of the Theorem~\ref{theo:main theorem}}\label{sec:proofs}
Before we move on to the detailed proofs, we summarize here the main properties of the examples $\{K_t\}_{t\geq0}$ constructed in the previous section(see figure~\ref{fig:global}):
\begin{enumerate}
\item $K_0=\mathbb{A}\times-\operatorname{Id}_{\mathbb{S}^1}.$ And for any $t>0$, $K_t\in\mathscr{K}^{\infty}(\mathbb{T}^3)$, i.e. $K_t$ satisfies item(K1)--(K4) in Definition~\ref{def:Kan's example on boundaryless manifolds}.
\item For any $t>0$, $K_t$ is a skew product reversing the orientation of the fibers, see equation~\ref{equation:K_t}.
\item For any $t>0$, $K_t$ admits a fixed fiber $\bar{p}=\{p\}\times\mathbb{S}^1$ which is Morse--Smale with exactly four fixed points, two sinks $P_0=(p,0),P_1=(p,1)$ and two sources $P=(p,0.5),P'=(p,-0.5)$. Moreover $K_t$ admits a blender-horseshoe with reference cube $\Gamma$(a small cube around $P$, contained in $\mathbb{T}^2\times(0,1)$) and reference saddles $P,O$, where $\Gamma,P,O$ are all independent of $t>0$. Symmetrically, $K_t$ admits a blender-horseshoe with reference cube $\Gamma'$(a small cube around $P'$, contained in $\mathbb{T}^2\times(-1,0)$) and reference saddles $P',O'$, where $\Gamma',P',O'$ are all independent of $t>0$.
\item For any $t>0$, $K_t$ admits a fixed fiber $\bar{q}=\{q\}\times\mathbb{S}^1$ which is Morse--Smale with exactly four fixed points, two sources $Q_0=(q,0),Q_1=(q,1)$ and two sinks $Q=(q,\theta_0),Q'=(q,-\theta_0)$. Moreover, $W^u(Q,K_t)$ contains a $u$-disk which belongs to the superposition region of the $cu$-blender $\Lambda_t=\cap_{n\in\mathbb{Z}}K_t^n(\Gamma)$. Symmetrically, $W^u(Q',K_t)$ contains a $u$-disk which belongs to the superposition region of the $cu$-blender $\Lambda'_t=\cap_{n\in\mathbb{Z}}K_t^n(\Gamma')$.

\end{enumerate}
\subsection{Robust transitivity}
In this subsection, we will prove that the examples constructed in the previous section is robustly transitive. The following criterion will be used.
\begin{lemma}[\cite{BDV}, Lemma~7.3]\label{lem:criterium of robust transitivity}
Let $f:M\rightarrow M$ be a diffeomorphism, and $p$ be a hyperbolic periodic point. Suppose that invariant manifolds of $\operatorname{Orb}(p)$ are both (robustly) dense in $M$. Then $f$ is (robustly) transitive.
\end{lemma}

The following lemma can be deduced from Theorem~7.1 of \cite{HPS} directly.
\begin{lemma}[\cite{HPS}]\label{lem:openness of skew product}
Let $M,N$ be compact boundaryless manifolds, and $F:M\times N\rightarrow M\times N, (x,y)\mapsto(f(x),\phi_x(y))$ be a normally hyperbolic skew--product $C^1$-diffeomorphism. Assume that $f:M\rightarrow M$ is Anosov, then there exists a $C^1$-neighbourhood $\mathcal{U}$ of $F$ in $\operatorname{Diff}^1(M\times N)$, such that for any $G\in\mathcal{U}$,
\begin{enumerate}
\item $G$ is conjugate by $\Phi_G$ to a skew--product $G^*:(x,y)\rightarrow(f(x),\phi^G_x(y))$, which depends continuously on $G$.
\item For any $x\in M$, $\Phi_G^{-1}(\{x\}\times N)$ is diffeomorphic to $N$, and $C^1$-close to $\Phi_F^{-1}(\{x\}\times N)=\{\phi^{-1}(x)\}\times N$.
\end{enumerate}
\end{lemma}

Recall that for any $x\in\mathbb{T}^2$, we define $\bar{x}$ to be the fiber $\{x\}\times\mathbb{S}^1$. Using Lemma~\ref{lem:openness of skew product}, there exists a $C^1$-neighbourhood $\mathscr{U}_t$ of $K_t$ such that for any $g\in\mathcal{U}_t$ and any $x\in\mathbb{T}^2$, we can define the continuation of $\bar{x}$ to be $\Phi_g^{-1}(\{x\}\times\mathbb{S}^1)$, and denote it by $\bar{x}_g$.

\begin{proposition}\label{prop:robust transitivity}
Let $\{K_t\}_{t\geq0}$ be as above. Then for every $t>0$ small enough, $K_t$ is robustly transitive, i.e., there exists a $C^1$-neighbourhood $\mathcal{U}_t$ of $K_t$, such that every $g\in\mathcal{U}_t$ is transitive.
\end{proposition}
\begin{proof}
We will prove the proposition by using the argument of Lemma~\ref{lem:criterium of robust transitivity}. In fact, we only have to show that
\begin{claim}
$W^u(\operatorname{Orb}(P),K_t)$ and $W^s(\operatorname{Orb}(P),K_t)$ are both robustly dense in $\mathbb{T}^3$.
\end{claim}
\noindent\emph{Proof of the claim:} Denote $P'=(p,-0.5)$, then $\operatorname{Orb}(P)=\{P,P'\}$. Similarly, denote $Q'=(q,-\theta_0)$, then $\operatorname{Orb}(Q,K_t)=\{Q,Q'\}$. Firstly, note that
$$\overline{W^u(\operatorname{Orb}(P),K_t)}=\overline{W^u(P,K_t)}\cup\overline{W^u(P',K_t)}=\overline{W^u(\bar{p},K_t)}=\mathbb{T}^3.$$
Since $K_t$ is normally hyperbolic skew product diffeomorphism, we can use Lemma~\ref{lem:openness of skew product} to conclude that there exists a $C^1$-neighbourhood $\mathcal{U}_t$ of $K_t$ such that $\overline{W^u(\bar{p}_g,g)}=\mathbb{T}^3$ still holds for every $g\in\mathcal{U}_t,$ and hence $\overline{W^u(\operatorname{Orb}(P_g),g)}=\mathbb{T}^3$. Analogous argument shows that $\overline{W^s(\bar{q}_g,g)}=\mathbb{T}^3$ holds for every $g\in\mathcal{U}_t,$ and hence $\overline{W^s(\operatorname{Orb}(Q_{g}),g)}=\mathbb{T}^3.$

According to the construction, $(\Lambda_t,K_t^{2n_0})$ is a blender-horseshoe with reference cube $\Gamma$ and reference saddles $P$ and $O$. And $W^u(Q,K_t^2)$ contains a vertical disk that belongs to the superposition region $\mathcal{D}_t$. Then by using of Lemma~\ref{lem:blenderhorseshoe and cu blender} and Lemma~\ref{lem:blender}, we have $W^s(Q_{g},g^2)\subset \overline{W^s(P_g,g^2)}.$ Symmetrically, we have $W^s(Q'_{g},g^2)\subset\overline{W^s(P'_g,g^2)}.$ Therefore,
\begin{align}
\overline{W^s(\operatorname{Orb}(P_g),g)}&=\overline{W^s(P_g,g^2)}\cup\overline{W^s(P'_g,g^2)}\nonumber\\
                           & \supset\overline{W^s(Q_{g},g^2)}\cup\overline{W^s(Q'_{g},g^2)}\nonumber\\
                           &=\overline{W^s(\operatorname{Orb}(Q_{g}),g)}=\mathbb{T}^3.\nonumber
\end{align}

The proof of the claim has now been finished, thus the proof of the proposition has been finished too.
\end{proof}

\subsection{Having two physical measures means Kan--like}
In this subsection, we will prove Theorem \ref{theo:two-pm}. First we need a lemma for Kan's examples on $\mathbb{T}^3$.

\begin{lemma}\label{lem:trans-inters}
For every $f\in\mathscr{K}^{1+}(\mathbb{T}^3)$, denote $R_0=(r,0)$ and $S_1=(s,1)$ to be two fixed point of $f$ with indices 2 from the definition (K2) of Kan's examples. Then there exists a $C^1$-neighborhood ${\cal U}$ of $f$, and a constant $L>2$, such that for any $g\in{\cal U}$ and $X\in\mathbb{T}^3$, we have
$$
W^{uu}_L(X,g)\pitchfork W^s_L(R_{0,g},g)\neq\varnothing, \qquad or \qquad
W^{uu}_L(X,g)\pitchfork W^s_L(S_{1,g},g)\neq\varnothing.
$$
\end{lemma}

\begin{proof}
From the definition of $f$, we only need to take the constant $L$ large enough, then at least one of these two properties holds. Since the transverse intersection is an open property, and the stable(unstable) manifolds of uniform size $L$ vary continuously with respect to the diffeomorphism, we have proved this lemma.
\end{proof}

\begin{lemma}\label{lem:kan is mcc}
For every $f\in\mathscr{K}^{1+}(\mathbb{T}^3)$, $f$ has mostly contracting center.
\end{lemma}

\begin{proof}
This is a well-known result, and we write down its proof for completeness. Let $\mu_0$ and $\mu_1$ be the normalized  Lebesgue measures restricted to the invariant tori $T_0=\mathbb{T}^2\times\{0\}$ and $T_1=\mathbb{T}^2\times\{1\}$, respectively. Since $\mathbb{A}:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a hyperbolic automorphism, $\mu_0$ and $\mu_1$ are the unique ergodic Gibbs $u$-states supported on $T_0$ and $T_1$, respectively. Recall that the center Lyapunov exponents of $\mu_0$ and $\mu_1$ are both negative, according to $(K4)$, so they are both physical measures by Proposition~\ref{prop:gibbs ustate basic properties}. According to Proposition~\ref{prop:mcc equivalent property}, to show $f$ has mostly contracting center, it suffices to show that $f$ has no other ergodic Gibbs $u$-states.

By contradiction, assume there exists an ergodic Gibbs $u$-state $\nu\notin\{\mu_0,\mu_1\}$. Then there exists a $u$-disk $D$ which intersect $B(\nu)$ on a full Lebesgue measure subset $D_0\subset D$. Since $\nu$ is an invariant measure, this also works for $f^n(D)$. According to lemma \ref{lem:trans-inters}, there exists $n$ large enough, such that $f^n(D)\pitchfork W^s_L(R_0)\neq\varnothing$  or $f^n(D)\pitchfork W^s_L(S_1)\neq\varnothing$. Without loss of generality, we assume the first case happens. Again, since $\mu_0$ is an ergodic Gibbs $u$-state, there exists a $u$-disk $D'\subset T_0$ and its subset $D'_0$ with positive Lebesgue measure such that every $X\in D'_0$ belongs to $B(\mu_0)$ and has 2-dimensional Pesin local stable manifold $W^s_{loc}(X)$ with uniform size. Moreover, the Pesin local stable manifolds vary absolutely continuously according to the classical Pesin theory.

Since $W^u(R_0)$ is dense in $T_0$, and $f^n(D)$ is arbitrarily close to $W^u(R_0)$ as $n$ tends to infinity, there exists $n_0$ large enough such that $f^{n_0}(D)$ cuts $\cup_{X\in D'_0}W^s_{loc}(X).$ The intersection is contained in $B(\mu_0)$ (since $\cup_{X\in D'_0}W^s_{loc}(X)\subset B(\mu_0)$), and has positive Lebesgue measure by the absolute continuity of $\{W^s_{loc}(X)\}_{X\in D'_0}$. This means $D$ itself has a positive Lebesgue measure subset contained in $B(\mu_0)$, which contradicts with the fact that $D$ has a full Lebesgue measure subset $D_0$ such that $D_0\subset B(\nu)$.
\end{proof}

\begin{proposition}\label{prop:two physical measures}
Let $K_t$ and $\mathcal{U}_t$ be defined as in Proposition~\ref{prop:robust transitivity}. Then by shrinking $\mathcal{U}_t$ if necessary, for any $g\in\mathcal{U}_t\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$, if $g$ has two physical measures, then $g\in\mathscr{K}^{1+}_l(\mathbb{T}^3)$.
\end{proposition}

It is easy to see that Proposition~\ref{prop:two physical measures} can be deduced from Theorem~\ref{theo:two-pm} immediately. So we only have to prove Theorem~\ref{theo:two-pm}.

\begin{proof}[Proof of Theorem~\ref{theo:two-pm}]
We take the $C^1$-neighborhood ${\cal U}$ small enough, such that it satisfies Lemma \ref{lem:trans-inters} and every $g\in{\cal U}\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$ has mostly contracting center. Moreover, we require that every $g\in{\cal U}\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$ is still partially hyperbolic with uniformly compact center foliation.

Now assume $\mu_1$ and $\mu_2$ are two different ergodic Gibbs $u$-states of some $g\in{\cal U}\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$. Then $\operatorname{Supp} (\mu_1)$ and $\operatorname{Supp} (\mu_2)$ are two disjoint compact $u$-saturated sets, according to Proposition~\ref{prop:mcc equivalent property}. We have the following claim.

\begin{claim}\label{claim:unique intersection}
For every $x\in\mathbb{T}^2$, $\bar{x}_g$ intersects $\operatorname{Supp} (\mu_i)$ with exactly one point, for $i=1,2$.
\end{claim}

\begin{proof}[Proof of the claim]
Argue by contradiction. Assume that there exist $x\in\mathbb{T}^2$ and two points $X_1,X_2\in\bar{x}_g$, such that $X_1,X_2\in\operatorname{Supp} (\mu_1)$. Notice that for any point $Y\in \operatorname{Supp} (\mu_2)$, from Lemma \ref{lem:trans-inters}, we know that either $W_L^{uu}(Y,g)\pitchfork W_L^s(R_{0,g},g)\neq\varnothing$, or $W_L^{uu}(Y,g)\pitchfork W_L^s(S_{1,g},g)\neq\varnothing$. Without loss of generality, we assume the first case holds. Then by using of $\lambda$-lemma and the invariance of $\operatorname{Supp} (\mu_2)$, we have $W^{uu}(R_{0,g},g)\subset \operatorname{Supp} (\mu_2)$.

Since $X_1$ and $X_2$ are in the same center leaf, there exist $z\in\mathbb{T}^2$ and two points $Z_1,Z_2\in\bar{z}_g$, such that
\begin{displaymath}
Z_i\in W^{uu}(X_i,g)\cap W^{ss}(\bar{r}_g), \qquad i=1,2.
\end{displaymath}
However, noticing that $W^{ss}(R_{1,g},g)\cap\bar{z}_g$ consists of a unique point, we conclude that at least one of $Z_1$ and $Z_2$ belongs to $W^s(R_{0,g},g)$. Applying $\lambda$-lemma again, we have
$W^{uu}(R_{0,g},g)\subset \operatorname{Supp} (\mu_1)$. This is a contradiction since $\operatorname{Supp} (\mu_1)\cap \operatorname{Supp} (\mu_2)=\varnothing$.
\end{proof}

From the proof of the claim, we can see that $\operatorname{Supp} (\mu_i)$ can not intersects $W^s(R_{0,g},g)$ and $W^s(S_{1,g},g)$ simultaneously, where $i=1,2$. Therefore, following the same arguments as above, we can assume that $W^{uu}(R_{0,g},g)\subset \operatorname{Supp} (\mu_1)$, and $W^{uu}(S_{1,g},g)\subset \operatorname{Supp} (\mu_2)$.

Since $\operatorname{Supp} (\mu_i)$ is compact, the map $x\in\mathbb{T}^2\mapsto\operatorname{Supp}(\mu_i)\cap\bar{x}_g$ is lower semi-continuous, where $i=1,2$. However, Claim~\ref{claim:unique intersection} shows us each of the intersections in the maps consists of a unique point. This implies that the maps are both continuous. That is, $\operatorname{Supp} (\mu_1)$ and $\operatorname{Supp} (\mu_2)$ are two topological tori transverse to the center foliation. Combining with the assumption in the previous paragraph, we conclude
\begin{displaymath}
\overline{W^{uu}(R_{0,g},g)}=\operatorname{Supp} (\mu_1), \qquad {\rm and} \qquad \overline{W^{uu}(S_{1,g},g)}=\operatorname{Supp} (\mu_2).
\end{displaymath}

\begin{figure}[htbp]
\centering
\begin{overpic}{fig1.pdf}
\put(80,40){$Y_0$}
\put(85,50){$V$}
\put(20,50){$X_0$}
\put(76,47){$Y$}
\put(55,50){$X$}
\put(45,70){$W^c(W_{loc}^{ss}(X_0,g),g)$}
\put(30,85){$W^c(W_{loc}^{uu}(X_0,g),g)$}
\put(90,59){$\operatorname{Supp}(\mu_1)$}
\put(90,60){\vector(-1,1){5}}
\put(51,53){$\sigma$}
\put(50,53){\vector(-1,-1){5}}
\end{overpic}
\caption{$\operatorname{Supp}(\mu_1)$ is $s$-saturated.}
\end{figure}

\begin{claim}
Both $\operatorname{Supp} (\mu_1)$ and $\operatorname{Supp} (\mu_2)$ are $s$-saturated, thus they are $su$-tori.
\end{claim}

\begin{proof}[Proof of the claim]
We only prove this claim for $\operatorname{Supp} (\mu_1)$. Suppose $\operatorname{Supp} (\mu_1)$ is not $s$-saturated, then there exist a point $X_0\in \operatorname{Supp} (\mu_1)$ and a point $Y_0\in W^{ss}_{loc}(X_0,g)\cap \bar{y}_g$, such that $Y_0\notin \operatorname{Supp} (\mu_1)$.  Since $\operatorname{Supp}(\mu_1)$ is compact, there exists an open neighbourhood $U$ of $y$ in $\bar{y}_g$, such that $U\cap \operatorname{Supp} (\mu_1)=\varnothing.$

Let $\sigma=W^c(W^{ss}_{loc}(X_0,g),g)\cap \operatorname{Supp} (\mu_1)$, then it is a continuous segment. Now using the strong stable holonomy map, there exists a segment $l\subset \bar{y}_g$ such that $l=h^s_g(\sigma)$. Let $Z_0\in l\cap \mathscr{U}$ be an interior point of $l$. Now we can choose a sufficiently small neighbourhood $V$ of $Z_0$ in $\mathbb{T}^3$, such that for every point $Y\in V,$ there exists $X\in \operatorname{Supp} (\mu_1)\cap W^{ss}(Y,g)$. This is because $\operatorname{Supp} (\mu_1)$ is a $u$-saturated continuous surface.

Recall that both $W^s(R_{0,g},g)$ and $W^s(S_{1,g},g)$ are dense in $\mathbb{T}^3$, there exists two points $Y^0\in \mathscr{V}\cap W^s(R_{0,g},g)$, $Y^1\in \mathscr{V}\cap W^s(S_{1,g},g)$. So from $\lambda$-lemma, we know that both $R_{0,g}$ and $S_{1,g}$ belongs to $\operatorname{Supp} (\mu_1)$. Since $\operatorname{Supp} (\mu_2)$ is also $u$-saturated, so Lemma \ref{lem:trans-inters} and $\lambda$-lemma also implies at least one of $R_{0,g}$ and $S_{1,g}$ belongs to $\operatorname{Supp} (\mu_2)$, which implies $\operatorname{Supp} (\mu_1)\cap\operatorname{Supp} (\mu_2)\neq\varnothing$.

This contradiction means $\operatorname{Supp} (\mu_1)$ must be $s$-saturated.
\end{proof}

Now we move on to prove that the basins of $\mu_1$ and $\mu_2$ are intermingled. The ingredients of the proof are quite similar to Kan's examples, and we just sketch the main arguments.

Since $g$ also has mostly contracting center, the center Lyapunov exponent of $\mu_i$ is still negative. For any curve $\gamma$ which is transverse to the $E^{cs}_g$ direction and does not intersect with $\operatorname{Supp} (\mu_1)$ nor $\operatorname{Supp} (\mu_2)$, from Lemma \ref{lem:trans-inters}, up to take some forward iteration, that $\gamma$ crosses the local surfaces $W^s_{loc}(R_{0,g},g)$ and $W^s_{loc}(S_{1,g},g)$. Therefore,  for sufficient large $n$, $f^n(\gamma)$ will contain some disks sufficiently close to $\operatorname{Supp} (\mu_1)$ and $\operatorname{Supp} (\mu_2)$, and hence intersect with $B(\mu_1)$ and $B(\mu_2)$ both in a set of positive Lebesgue measure. Notice that $B(\mu_1)$ and $B(\mu_2)$ are both $g$-invariant, we conclude that $\gamma$ itself intersects with $B(\mu_1)$ and $B(\mu_2)$ both in a set of positive Lebesgue measure. Now the conclusion follows by using Fubini's Theorem.
\end{proof}

\subsection{Having a unique physical measure means mixing property}
In this subsection, we will prove that for the examples constructed in the previous section, if they admit a unique physical measure under perturbations, then the perturbed ones must be topologically mixing. Combined with Proposition \ref{prop:two physical measures}, the proof of Theorem~\ref{theo:main theorem} will be finished.

Recall that in the previous subsection, the perturbed systems remain to have two invariant tori. While in this subsection, we will show that if the perturbed systems admit a unique physical measure, then at least one of its invariant tori is broken. The following proposition provides us a characterization of the broken torus, which will be used later.

\begin{proposition}\label{prop:break invariant torus}
Let $K_t$ and $\mathcal{U}_t$ be defined as in Proposition~\ref{prop:robust transitivity}, $T$ be one of its two invariant tori, and $A\in\bar{a},B\in\bar{b}$ be any two hyperbolic periodic points contained in $T$. By shrinking $\mathcal{U}_t$ if necessary, assume that the continuation of $A$ and $B$ are well defined for any $g\in\mathcal{U}_t$. Then for any $g\in\mathcal{U}_t$,
\begin{itemize}
\item either $g$ admits an invariant torus $T_g$ which is $s,u$-saturated, and $T_g$ is close to $T$ in  Hausdorff metric,
\item or there exists $x\in W^u(a,\mathbb{A})\cap W^s(b,\mathbb{A})$ such that $W^{uu}(A_g,g)\cap\bar{x}_g\neq W^{ss}(B_g,g)\cap\bar{x}_g$.
\end{itemize}
\end{proposition}
\begin{remark}
If the first case happens, we call $T_g$ the \emph{continuation} of $T$. And if the second case happens, we say that the invariant torus $T$ is \emph{broken} under $g$.
\end{remark}
\begin{proof}
Assume that for every $x\in W^u(a,\mathbb{A})\cap W^s(b,\mathbb{A})$, we have $W^{uu}(A_g,g)\cap\bar{x}_g= W^{ss}(B_g,g)\cap\bar{x}_g$, we will prove that $g$ admits an invariant torus $T_g$ which is $s,u$-saturated and close to $T$ in  Hausdorff metric.

Denote $\Lambda_g=\overline{\{W^{uu}(A_g,g)\cap\bar{x}_g:x\in W^u(a,\mathbb{A})\cap W^s(b,\mathbb{A})\}}.$ For any $y\in\mathbb{T}^2$, we claim that $\#(\Lambda_g\cap \bar{y}_g)=1.$ Argue by contradiction, we suppose that there exists a point $y_0\in\mathbb{T}^2$, such that $\Lambda_g\cap\bar{y}_{0,g}$ contains at least two points, namely $Y^1$ and $Y^2$. By definition, there exist two sequence of points $\{x_n^i\in W^u(a,\mathbb{A})\cap W^s(b,\mathbb{A})\}_{n>0}(i=1,2)$ and $X_n^i\in\bar{x}_{n,g}^i\cap\Lambda_g$ such that $X_n^i$ tends to $Y^i$, $i=1,2$.
Suppose the center distance of $Y^1$ and $Y^2$ is bigger than some positive number, say $\delta$. We now choose $N\in\mathbb{N}$ large enough such that $\operatorname{d}(X_N^i,Y^i)$ is much smaller that $\delta$, $i=1,2$. Then the distance of $x_N^1$ and $x_N^2$ in $\mathbb{T}^2$ is much smaller than $\delta$, and we can thus assume that there exists a point $z\in W^u_{loc}(x_N^1)\cap W^s_{loc}(x_N^2)$. Then using leaf conjugacy property, we see $W^{uu}_{loc}(\bar{x}_{N,g}^1,g)\cap W^{ss}_{loc}(\bar{x}_{N,g}^2,g)=\bar{z}_g.$ Due to the absolute continuity of the strong stable and strong unstable holonomy of $g$, there exist two points $Z^1,Z^2\in\bar{z}_g$ satisfying
\begin{itemize}
\item[(1)] $\{Z^1\}=W^{uu}_{loc}(X_{N,g}^1,g)\cap\bar{z}_g=W^{uu}(A_g,g)\cap\bar{z}_g,$
\item[(2)] $\{Z^2\}=W^{ss}_{loc}(X_{N,g}^2,g)\cap\bar{z}_g=W^{ss}(B_g,g)\cap\bar{z}_g,$
\item[(3)] $\operatorname{d}_c(Z^1,Z^2)>\frac{\delta}{2}.$
\end{itemize}
This implies that $W^{uu}(A_g,g)\cap\bar{z}_g\neq W^{ss}(B_g,g)\cap\bar{z}_g$, which contradicts with our assumption. This contradiction implies our claim is true, i.e., $\#(\Lambda_g\cap \bar{y}_g)=1$ for every $y\in\mathbb{T}^2$. This means $\Lambda_g$ is actually a topological torus transverse to the center foliation, and we denote it by $T_g$ from now on. Bearing in mind $T_g$ contains a dense subset of $W^{uu}(A_g,g)$(resp. $W^{ss}(B_g,g)$), we get $T_g\supset W^{uu}(A_g,g)$(resp. $T_g\supset W^{ss}(B_g,g)$). Hence, $T_g$ is $u$-minimal(resp. $s$-minimal) and $u$-saturated(resp. $s$-saturated). $T_g$ is close to $T$ with respect to Hausdorff metric, because $\overline{W^{uu}(A_g,g)}$ varies lower semi-continuously as $g$ changes.
\end{proof}

Now we are ready to prove the main proposition of this subsection.
\begin{proposition}\label{prop:unique physical measure}
Let $K_t$ and $\mathcal{U}_t$ be defined as in Proposition~\ref{prop:robust transitivity}. Then by shrinking $\mathcal{U}_t$ if necessary, for any $g\in\mathcal{U}_t\cap\operatorname{Diff}^{1+}(\mathbb{T}^3)$, if $g$ has a unique physical measure, then $g$ is robustly topologically mixing.
\end{proposition}
\begin{proof}
By shrinking $\mathcal{U}_t$ if necessary, suppose that the continuations of $R_0,S_1,P_0,Q_0,P_1,$ and $Q_1$ are all well defined.
We first claim that at least one of the two invariant tori of $K_t$ is broken under $g$. Otherwise, the two invariant tori $T_1\ni R_0$ and $T_2\ni S_1$ both have a continuation, denoted by $T_{1,g}$ and $T_{2,g}$ respectively. Since $T_{i,g}$ are both $u$-saturated and disjoint to each other, each of them support a Gibbs $u$-state, which is also a physical measure. This contradicts with the assumption that $g$ admits a unique physical measure.

Suppose now the invariant torus $T:=\mathbb{T}^2\times\{0\}$ is broken. To prove $g$ is topological mixing, we will prove both $W^s(P_g,g^2)$ and $W^u(P_g,g^2)$ are dense in $\mathbb{T}^3$.

Now that $T$ is broken, for $P_0$ and $Q_0$, there exists a point $x\in W^u(p,\mathbb{A})\cap W^s(q,\mathbb{A})$ such that $W^{uu}(P_{0,g},g)\cap\bar{x}_g\neq W^{ss}(Q_{0,g},g)\cap\bar{x}_g$. Then $W^s(Q_{0,g},g)=W^{ss}(Q_{0,g},g)$ intersects transversely with either $W^u(P_g,g^2)$ or $W^u(P'_g,g^2)$. We suppose $W^s(Q_{0,g},g)\cap W^u(P'_g,g^2)\neq\varnothing.$ Then $W^s(P'_g,g^2)\subset\overline{W^s(Q_{0,g},g)}$, due to Palis' $\lambda$-lemma. Recall that the existence of blender-horseshoes implies $W^s(Q_{g},g^2)\subset \overline{W^s(P_g,g^2)}$ and $W^s(Q'_{g},g^2)\subset \overline{W^s(P'_g,g^2)}$. Noting that $W^s(Q_{0,g},g)\subset \overline{W^s(Q_{g},g)}$, we conclude that
\begin{displaymath}
\overline{W^s(P_g,g^2)}\supset\overline{W^s(Q_g,g^2)}\supset\overline{W^s(Q_{0,g},g)}\supset\overline{W^s(P'_g,g^2)}\supset\overline{W^s(Q'_g,g^2)}.
\end{displaymath}
Therefore,
\begin{displaymath}
\overline{W^s(P_g,g^2)}\supset\overline{W^s(Q_{g},g^2)}\cup\overline{W^s(Q'_{g},g^2)}=\overline{W^s(\bar{q}_g,g)}=\mathbb{T}^3.
\end{displaymath}
Hence $\overline{W^s(P'_g,g^2)}=\overline{g(W^s(P_g,g^2))}=\mathbb{T}^3$.

Again, since $T$ is broken under $g$, for $P_0$ and $S_0$, there exists a heteroclinic point $y\in W^u(p,\mathbb{A})\cap W^s(s,\mathbb{A})$ such that $W^{uu}(P_{0,g},g)\cap\bar{y}_g\neq W^{ss}(S_{0,g},g)\cap\bar{y}_g$. Then $W^s(S_{0,g},g)$ intersects transversely with either $W^u(P_g,g^2)$ or $W^u(P'_g,g^2)$. We suppose $W^s(S_{0,g},g)\cap W^u(P_g,g^2)\neq\varnothing.$  And again, using Palis' $\lambda$-lemma, we get $W^u(S_{0,g},g)\subset\overline{W^u(P_g,g)}$, which immediately implies $\overline{W^u(P_g,g)}=\mathbb{T}^3.$

To show that $g$ is robustly mixing, it is sufficient to notice that if the invariant torus $T$ of $K_t$ is broken under $g$, then it is robustly broken.
\end{proof}

Now we can prove the corollary.

\begin{proof}[Proof of Corollary]
Fix some $t_0>0$ small enough. From the proof of Proposition \ref{prop:break invariant torus}, we see that we can take some sooth perturbation $g$ of $K_{t_0}$ such that one $su$-torus of $K_{t_0}$ has a continuation, and the other one is broken. Then $g$ is robustly mixing, according to Proposition~\ref{prop:unique physical measure}. However, the existence of an $su$-torus implies that neither the strong stable foliation nor the strong unstable foliation of $g$ is minimal.
\end{proof}

\begin{remark}
For the case where we require the diffeomorphism to preserve the orientation of the center foliation(see Remark~\ref{remark}), it is a little bit tricky. We just sketch the main ingredients of the proof. According to equation~\ref{equation:hat f}, for any $t>0$, $\hat{f}_t$ is a smooth function on $\mathbb{T}^3$ preserving the orientation of the center foliation. We now take $g$ to be a smooth perturbation of some $\hat{f}_t$ such that
\begin{itemize}
\item $g^{-1}\circ \hat{f}_t$ is a skew product diffeomorphism over $\operatorname{Id}_{\mathbb{T}^2}:\mathbb{T}^2\rightarrow\mathbb{T}^2$;
\item $g$ preserves the orientation of the center foliation;
\item $g^{-1}\circ \hat{f}_t$ is supported on a small neighbourhood of the invariant torus $\mathbb{T}^2\times\{0\}$ of $\hat{f}_t$;
\item $W^s(Q_{0,g},g)\pitchfork W^u(P'_g,g)\neq\varnothing$ and $W^s(S_{0,g},g)\pitchfork W^u(P_g,g)\neq\varnothing$.
\end{itemize}
Now by the same arguments as in Proposition~\ref{prop:unique physical measure}, we can show that both $W^u(P_g,g)$ and $W^s(P_g,g)$ are robustly dense in $\mathbb{T}^3$. Hence $g$ is robustly topologically mixing. However, since $g$ still admits an invariant $su$-torus $\mathbb{T}^2\times\{1\}$, we know that neither the strong stable foliation nor the strong unstable foliation of $g$ is minimal.
\end{remark}


text:
GRS Group Ranked on 2018 Top Company Cultures List Presented by ENTREPRENEUR and ENERGAGE
by GRS Group | Sep 20, 2018 | News & Events
Irvine, CA—September 20, 2018 GRS Group was recently ranked on Entrepreneur's Top Company Cultures list, a comprehensive ranking of U.S.-based businesses exhibiting high-performance cultures created in partnership with employee engagement platform and service...
David Larkin Joins GRS Title Services, LLC as Business Development Director
by GRS Group | Aug 23, 2018 | News & Events, Uncategorized
Career Sales Veteran Brings over 25 Years of Experience Richmond, VA, – August 20, 2018, GRS Title Services, LLC has announced that David Larkin has joined its team as Director, Business Development. Based in New Jersey, Mr. Larkin has a degree in Civil...
GRS Title Services Expands to Kansas City, Announces Hiring of Title Officer
by GRS Group | Aug 20, 2018 | News & Events
Richmond, VA, – August 20, 2018 GRS Title Services, LLC has announced that it has opened an office in Kansas City, Missouri and has hired Tammy L. Stribling as a Director. Ms. Stribling has previously worked as a commercial title officer for First American in...
Alex Rodriguez Joins GRS Group
by GRS Group | Jul 9, 2018 | News & Events
Los Angeles, CA, – July 9, 2018, GRS Group is pleased to announce that Alex Rodriguez has joined its transaction services team as Director of Business Development. "It's an exciting time to come on board at GRS Group", said Mr. Rodriguez. "They not only offer a...
Timothy Van Zant Joins GRS Group
by Michael Gerard | Jun 11, 2018 | News & Events
Los Angeles, CA, – June 11, 2018, GRS Group is pleased to announce that Timothy Van Zant has joined its team as Zoning Operations Manager. Based in Oklahoma City, Mr. Van Zant has been in the real estate industry since the mid-1980s. Over the past 11 years, he...
Kenneth Kuehm Joins GRS Group
by Michael Gerard | Jun 7, 2018 | News & Events
Los Angeles, CA, – June 6, 2018, GRS Group is pleased to announce that Kenneth Kuehm has joined its team as Survey Operations Manager. Based in Aurora, OH, Mr. Kuehm brings over 20 years of experience in commercial real estate due diligence, construction, and...

redpajama_set_name: RedPajamaCommonCrawl


It's now been three and a half year since this project started in February, 2013. I do this 'year in review' article once a year in October as that is when I started collecting statistics for these hero builds. In 2015, we achieved a remarkable feat: 100 million total subscriptions. I persevered in updating the hero builds despite the system being completely system. Begun implementing a better hero builds tooltip system and established a proper policy in what direction and emphasis I want the hero builds to go towards.
2016 was challenging in and outside the project and the public recognition for it has been the highest it's ever been – with all of its appreciation and harsh criticism of character.
For those new to Dota 2, Hero Builds are an in-game guide system for users to learn on how to play a hero. These guides suggest in what order a hero's ability should be leveled, which items and in what order should be bought and guides can include tooltips to give contextual value on how best to use a hero's set of items or abilities throughout the game. This 2013 integral feature has been useful for many new players along with other Valve-released tools including the 'coaching' feature and robust tutorial system. This project was created to establish and maintain a 'standard' way of playing each and every hero in nearly all roles or forms.
Today we celebrate reaching over 170 million subscriptions across 150 guides (177,425,506 as of November 17, 2016). This article will take a look at the relevant statistics of our growth, discuss the continuing motivation to maintain these guides and the future goals and challenges for 2017.
In summary, this project has gained 70 million new subscriptions since last year. In fact, approximately 5 million a month since late-May 2014 (30 months consecutively). Hero builds have risen from 145 builds to 150 hero builds thanks to the release of the final DotA-Allstars hero: Arc Warden & Underlord.
In October 2013, we started with less than 5 million. Since 2014, our growth has been consistent of 4 to 6 million a month. In 2015, 37 guides have reached 1 million unique subscribers individually. That number has almost tripled with now 100 guides with their own 1 million unique subs.
Forecasts project that the project will achieve 200 million by February, 2017 and 234 million by October, 2017. Last year, we missed our prediction by a good margin and it might be safe to assume the same here. That said, we've achieved new milestones today with individual guides now breaking into the 2 million unique subscribers mark. Lifestealer, Sven, Juggernaut, Phantom Assassin, Phantom Lancer and Faceless Void have all recently hit their two million milestones and there are many other guides nearing it.
Observed is the incredible growth the Sven Hero Build achieved to reach and surpass Lifestealer Build's subscribers. Lifestealer achieved 1 million by August 2014 whereas Sven achieved similar by April 2015. By June 2016, both guides achieved two million simultaneously.
Though it isn't definitive, I firmly believe that because Sven has been more relevant in more patches and professional competitions than Lifestealer, its growth rate followed suit. This growth is not unique to a couple of builds. The average has grown from 716,000 to 1.16 million. The majority of hero builds have 1 million or more in unique subscribers. The distribution of subscribers is also much more ranged than before where guides can have as little as 150,000 subs. to as much as 2.4 million.
Finished overhauling the entire hero builds item and ability texts to better denote contextual value for each.
Transitioned the hero builds title system (Lane/Middle/Jungle) to less antiquated system (Core/Offlane/Jungle/Support) with sub-title specifications (Roaming, Safe Lane, Middle, etc.).
Released /r/HeroBuilds & PlayDota.com to better curate user feedback from more community hubs.
After years of problems, Valve took the initiative to fix some of the most glaring issues that outright halted the ability to make and publish hero builds. Even more so, they have been very responsive in any new critical bugs reported to them.
Before Valve, a user named Parnakra literally saved this project. I was on the brink of giving up before he helped not only fix the Dota 2 Hero Builds Cloud Server, but improve it in many ways.
Now with a direct line of communication with Valve and their promptness in fixing any new critical bugs, I hope to see more hero builds created and updated regularly without anything preventing users in doing so.
In terms of challenges, there are very few from the outside. There is still no real competition in terms of valuable hero builds currently in the database as the ones that are newly-created and maintained cannot break past some of the outdated versions from 2013 who continue to be rated up, regardless of accuracy or attention to relevance of the current patch (e.g.: heroes that get completely remade). Heroes such as Faceless Void don't have any updated hero builds in the year 2016 except for my own. In terms of competition, the top guide creators are still the same as last year with some emerging players such as EDJE and EZ MMR.
Across the board, we can see growth in all hero build collections from the most popular catalogs of hero builds. This can be indicative that there are just more or rotating new users subscribing to hero builds as a whole.
In December 2015, after much hesitation, I released a Patreon campaign for users who asked how else they can support the project. Generally, feedback was always preferred, but with the support of these great people, I can start to think about expansion into other medias to help players of all kind.
Thank you so much to the many people here for their generosity, support and care. They saw something that was made entirely for free and still wanted to provide support with the money they've earned. It's a great honor to receive this kind of support.
With the amount earned, I will be giving a portion to long-time feedbackers (though many have opted out or did not respond to request for their owed amount) and the rest will be going towards hiring a proper website designed to brand a website for the Hero Build Project and all the tools available. It will serve as a valuable hub to redirect users in being aware of all the hero builds available, tools, and statistics relevant to its growth and achievement. As always, no amount will be used for purposes other than to continue building this project.
In terms of future goals, the website hub is the first immediate one. Other cloud-level dreams would be expanding to content creation, meeting with the Valve Dota 2 team to suggest a variety of quality-of-life expansion to new player initiatives and/or newbie casting (this one's iffy – haven't casted since 2011 in StarCraft II).
In terms of motivations, 2016 turned out to be the year I receive the most recognition, appreciation, condescension and character criticism. Most of all, this year has been the most emotionally taxing where the hours of work remained consistently the same as last year's: 2-3 hours a day, 28 to 35 hours a week and 9 to 24 hours within two days when a new patch releases. This is nothing new and I've made my concerns about it in the past as I juggle work, a personal life and this project. Somewhere along the way, I've also noticed my anonymity was lost, where I would be recognized in-game and my words, actions (or inaction) would paint how people speak to me or about me.
My favourite flattery/criticism in recent times. Some others have been less than comedic and more personal.
If I was overly-critical towards someone in-game, it would persist past the game and try to demonize me as some two-faced person (rather than a person who does get mad when we lose and excited when we win). If a new patch hits and I did not update the guides by their expectations, the amount of spam and private messages received would range from begging to rude demands and threats (non-lethal versions). There is a lot of expectation for me to be a public model citizen and I suppose that comes with this newly-found recognition. I do not make a living off my personality, project or appearances – yet I have this new responsibility I was not prepared for and maybe selfishly feel I do not owe towards others. I don't think it's something I can rebel against, but I will be taking active measures to either not talk in-game or start using an alternate account for all my day-to-day test games. Come December, I may also slow down the amount of updates and testing, as I've recently found myself prioritizing the project over, say, enjoying an overdue vacation with significant partners (at one point, I was updating guides while on the bus to The Grand Canyon due to increased public pressure) or ruining my sleep schedule to keep up with the latest patch. In 2017, I will also be starting something very new in my career, which may cause an impasse on continuing the project at its current level of dedication or something less attentive.
Depsite this rant, I always go back and read previous topics, smile at the appreciation I receive and know there are people thankful. The numbers say it all and the words of the many drown out the negative few.
Without the help from the community at PlayDota.com, LiquidDota.com and /r/HeroBuilds, this project would not be able to be even remotely as close to its goal as it is now. I would never have the knowledge, expertise or understanding of the game that some of these people have provided throughout the years and for that, I thank them for completing this project in so many ways and keeping me on top of the most glaring inconsistencies with this collection of hero builds.
This project has been my pride and passion for the many years I've changed jobs, countries and lives. It kept me feeling personally successful no matter what low point I've hit. That kind of support is immeasurable and I sincerely thank you.
PS: thank you James Hu, Tam Vu and Foad Ghafor for the help with the website, graphs and this article.


text:
Snetterton Motor Racing Circuit är en brittisk racerbana, belägen utanför Norwich, England. Snetterton byggdes 1951. Banan är 3 141 meter lång och används bland annat till British Touring Car Championship.

Externa länkar
Officiell webbplats

Racerbanor i Storbritannien

redpajama_set_name: RedPajamaWikipedia


After Freifeld, when might a tenant be refused relief from forfeiture?
In Freifeld v West Kensington Court Limited  EWCA Civ 806, long-lessees had deliberately granted a future sub-lease of a commercial unit to a Chinese restaurant in breach of their alienation covenant not to sublet without landlord's consent. An initial application for relief from forfeiture failed, because the tenants had wilfully breached their alienation covenant, and because there was an extensive history of neglectful management by the tenants of their obligations under the headlease. The judge concluded that the relationship between the tenants and their landlord had become dysfunctional and that it should not be re-imposed on the landlord by the grant of relief to the tenants.
During the days in which the Judge was handing down Judgment, but before any order had been drawn, the tenants procured a surrender of the future sub-lease from the Chinese restaurant and made another application for relief from forfeiture on the grounds that the damage had been rectified and that they should be given six months to sell their Head-Lease, for otherwise the landlord would receive a windfall of £1 million to £2 million. The Judge dismissed that application too, on the grounds that the tenants were "… simply reaping what they [had] sowed," and that the head-lease was virtually valueless from the time that it was forfeit by service of the proceedings.
No challenge was mounted to the Judge's findings of fact, including his findings as to the dysfunctional relationship between the landlord and the tenants. Nevertheless, the Court of Appeal concluded that the Judge had misdirected himself as to the principles applicable to the determination of applications for relief from forfeiture and, re-exercising afresh the discretion whether or not to grant relief, the tenants were, effectively, granted relief from forfeiture, albeit on terms that they must sell the head-lease within six months.
Arden LJ gave the leading judgment with which Ryder and Briggs LJJ agreed. She started, as did the Court of Appeal in Magnic Ltd v Ul-Hassan  EWCA Civ 224 earlier this year, with the proposition that the purpose of a reservation of a right of re-entry is to provide the landlord with security for the performance of the tenant's covenants. The risk of forfeiture is not intended to operate as an additional penalty for breach. The wilfulness of the breach is certainly a relevant factor to be taken into account when deciding an application for relief, but the proportionality of the loss to the tenant (and the windfall to the landlord) when compared with the breach and the damage (if any) sustained to the landlord's reversion must also be considered. The Judge had misdirected himself in not taking that proportionality argument properly into account. He had further misdirected himself in attributing virtually no value to the head-lease. Although the Head-lease had been forfeit by the service of proceedings, it would not be a virtually value-less asset if the Court granted relief from forfeiture for the purposes of a sale, which is what it was being asked to do. The Judge had therefore misdirected himself.
The Court of Appeal decided to exercise the discretion to determine the relief application afresh. The sub-lease had been surrendered, and the head-lease was potentially valuable. They were able to protect the landlord's interests, without disproportionately penalising the tenants, by giving the tenants a period of 6 months within which to sell the head-lease; and requiring that a reputable managing agent be instructed to manage the property demised by head-lease during that 6 month period. If contacts for the sale of the head-lease were not exchanged within 6 months, then the Judge's order dismissing the application for relief would come into effect.
After Freifeld, when might a tenant be refused relief from forfeiture? Each case must, of course, be decided on its own facts, but some conclusions can be drawn. The extent to which the tenant's breaches were wilful will plainly be relevant, but will not be determinative. The same can be said for the value of the tenant's interest; the more valuable the tenant's interests, the more it can be said that forfeiture of that interest would be a disproportionate response to the breach. That element will therefore hold considerable sway where the leases concerned are 1954 Act protected leases, or (as in Freifeld) long-leases. Of most significance, however, will be the damage (if any) caused to the landlord's reversion by the tenant's behaviour and the extent to which such damage has been, or can be, remedied. In principle, most breaches can be remedied – alterations can be removed and the original configuration reinstated; unlawful sub-tenancies can be surrendered – and where the breaches have been remedied proportionality is likely to hold considerable sway. It is therefore in those cases where the damage caused is very considerable, or where the breaches cannot be remedied at all, that forfeiture, even of valuable leases, will remain a realistic outcome. The most common of such cases might, perhaps, be cases of unlawful subletting, ni which (unlike in Freifeld) a surrender of the unlawful sub-tenancy cannot be procured by the tenant in breach. In such a case, of course, forfeiture of the head-lease will cause the unlawful sub-tenancy to fall in; and it is probably in such circumstances that there is most chance of a tenant being refused relief from forfeiture.


The HERMA Group continued its unbroken expansion in the 2013 fiscal year, again achieving growth in sales in its business with self-adhesive materials, labels and labelling machines: the total revenues of 261.2 million euros represent a year-on-year increase of 6.4 % (from 245.6 million euros). "Having recorded the highest annual sales in company history to date in 2012, sustaining this success in 2013 was particularly challenging against the background of the weak economic recovery. We pulled it off and, with sales setting another record, we have again shown stronger growth than the market as a whole," commented managing directors Sven Schneller and Dr. Thomas Baumgärtner. In doing so, the self-adhesive specialist benefited from the comparatively acceptable economic conditions on the domestic market this time around. Whereas Germany's gross domestic product edged up slightly, by 0.4 %, 2013 as a whole was a year of slowdown in the eurozone and many other foreign markets important for HERMA. Nevertheless, the company based near Stuttgart, Germany, posted an upturn in exports of 5.9 % to 143.7 million euros. The domestic market grew by 6.9 %, to 117.5 million euros. Accordingly, the export ratio was 55.0 %, after 55.2 % in the previous year.
"Having recorded the highest annual sales in company history to date in 2012, sustaining this success in 2013 was particularly challenging against the background of the weak economic recovery."
"The 2013 pattern of business more than hit the mark in terms of our forecast, with results that were even slightly better than our target," stated the HERMA managing directors. "Our task now is once more to create the preconditions for renewed growth. We will continue to focus on enhanced efficiency in production and innovative strength in product development." Given the brighter outlook on the domestic and foreign markets, Mr. Schneller and Dr. Baumgärtner anticipate a 3 % rise in sales across the company in the current fiscal year.
In absolute figures the Self-adhesive Materials division contributed around half of the company's growth in 2013. Revenues climbed by 5.7 %, from 140.0 million euros to 147.9 million euros at year-end, although the division had to contend with currency-related contracting margins in the key British market in particular. In volume terms, the increase in sales of self-adhesive materials was around 11 %. This was based on the systematic expansion of HERMA's multi-layer technology, coupled with the continued introduction of new, innovative adhesives.
In percentage terms, the pattern of sales recorded by the Labels division was comparatively pleasing. Growth of 5.2 % boosted revenues from 74.5 million euros to 78.4 million euros. However, business in the office and consumer sector was still affected by an absence of market growth as a consequence of intensified digitisation of office jobs and falling numbers of children attending school. Parallel to this, classic wholesaling and retailing ceded market shares to online suppliers and large chains. In the industrial labels sector, further gains were made in business with large international customers and in the core unprinted labels segment.
In 2013 the biggest growth in percentage terms was generated by the Labelling Machines division. Sales rose by 12.2 % year on year, from 31.1 million euros to 34.9 million euros. "The growth strategy outlined in the previous year has been successfully put into operation," said the managing directors. Alongside ongoing product portfolio expansion, this includes stringent standardisation of the existing product range, as well as a strictly customer- and application-based approach to developing industry-specific solutions. The division also benefited from building up its international sales activities.


ICSC Corp. provides comprehensive web development and e-marketing solutions. We focus on designing advanced on-line presence campaigns combining corporate graphic design, programming, content authoring, and search engine marketing.
Joining the ICSC team will put you in a challenging environment that encourages personal development, business experience, and success. We are always looking for talented and driven individuals. If you are dedicated, self-motivated, and hard working, we would love to hear from you.
Experienced sales executive responsible for identifying new sales opportunities and securing new clients. Must have a good understanding of online marketing strategies, e-commerce, web site design, and search engine optimization.
2-5 years selling experience preferably in online media sales / business development.
We will accept resume submissions via info@icsc.vn only. Please include your name and the job title in all correspondence.