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Coastal societies must adapt to the reality of rising sea levels and more frequent coastal flooding, argues Dr John Church.
Today there are more than 140 million people and a trillion dollars in infrastructure in the first one metre above high tide level around the world.
Coastal areas (people, infrastructure and the environment) are already affected by extreme events such as coastal flooding and coastal erosion. For example, Hurricane Katrina caused over $100 billion of damage and the loss of over 1,000 lives along the USA Gulf coast and cyclone Nargis resulted in the loss of well over 100,000 lives in Burma.
And just last week, Cyclone Yasi caused widespread damage and storm surge flooding along north Queensland's coast.
These events remind us that coastal societies in both developed and developing nations are vulnerable to extreme coastal sea levels. This vulnerability will increase with rising sea level as a result of anthropogenic climate change.
With greenhouse gas emissions continuing to rise, it is becoming increasingly clear that modern coastal societies will need to adapt to rising sea levels. To cost effectively adapt, we need to know what to adapt to. How, why and where sea level has changed in the past, how it is changing now, and how it will change in the future.
via The rising oceans › Opinion (ABC Science).


text:
\section{Introduction}
The study of the complexity or chaotic behavior is a central topic
in topological dynamics. Starting from the work of Li and York
\cite{LY75} various authors introduce a lot of definitions of chaos
according to their understanding of the phenomena. Among them,
Li-Yorke chaos, Denavey chaos \cite{D89} and positive entropy
\cite{Block1992} are popular ones. It is important to understand
their relationships. Recently, it has been shown that for a general
topological dynamical system, Devaney chaos implies Li-Yorke chaos
\cite{HY02} and positive entropy implies Li-Yorke chaos
\cite{BGKM02}.

\medskip

In the study of the so called ``local entropy theory" (for a survey
see \cite{GY09}), a lots of notions are introduced to describe
dynamical properties. It is not clear the relationship of those
properties (related to entropy) with the chaotic behaviors for a
given space. The purpose of the current paper is to study the
relationship in the case when the given space is a closed interval.
We believe that many results of the paper hold for a graph map even
more general spaces.

\medskip

To state our results, we introduce some notations first. Let $I$ be
the closed unit interval $[0,1]$ and $C(I,I)$ denote the class of
continuous maps of $I$ to itself. For $f\in C(I,I)$, let $f^0$ be
the identity, and for $n\in \mathbb N$, let $f^{n+1}=f^n\circ f$,
where $\mathbb{N}$ stands for the set of positive integers.

\medskip

A point $x\in I$ is called a {\it periodic point} of $f$ with {\it
period} $n$ if $f^n(x) = x$, $f^k(x)\neq x$ for $1\leq k<n$.  A
periodic point with period $1$ is called a {\it fixed point}. The
{\em $\omega$-limit set of $x$}, denoted by $\omega_f(x)$, is the set of
limit points of $\{f^i(x)\}_{i=0}^\infty$. A set $W\subset I$ is
called some {\em $\omega$-limit set} for $f$, if there exists an $x\in I$
such that $W=\omega_f(x)$. Denote the collection of all
$\omega$-limit sets for $f$ by $\omega_f$.

\medskip

A point $x\in X$ is called (1) a {\em recurrent point}, if for every neighborhood $U$ of $x$, there exists some
$n>0$, such that  $f^n(x)\in U$; (2) a {\em strongly recurrent point}, if for every neighborhood $U$ of $x$,
there exists some $N>0$, such that if $f^m(x)\in U$ then $f^{m+k}(x)\in U$ for some $k$ with $0<k\leq N$;
(3) a {\em regularly recurrent point}, if for every neighborhood $U$ of $x$, there exists some
$N>0$, such that $f^{kN}(x)\in U$ for all $k>0$.

Denote $Per(f)$, $Rec(f)$, $SR(f)$ and $RR(f)$ by the set of periodic points, recurrent points,
strongly recurrent points and regularly recurrent points,
respectively. It is well known that $$ Per(f)\subset RR(f)\subset
SR(f)\subset Rec(f).$$

The terminology ``chaos'' was first introduced by Li and Yorke
\cite{LY75}  to describe the complex behavior of trajectories. A
pair $\langle x,y\rangle\in I^2$ is called {\em proximal} if
$\liminf_{n\to \infty}|f^n(x)-f^n(y)|=0$ and is called {\em asymptotic} if
$\lim_{n\to \infty}|f^n(x)-f^n(y)|=0$. A {\em scrambled pair} or {\em Li-Yorke pair} is one that
is proximal but not asymptotic. A pair $\langle x,y\rangle$ is called {\em proper} if $x\neq y$.

For $\delta>0$, a pair $\langle x,y\rangle$ is said to be
{\em $\delta$-scrambled}  if
$$\liminf_{n\to \infty} |f^n(x)-f^n(y)|=0 \text{ and } \limsup_{n\to \infty}|f^n(x)-f^n(y)|\geq \delta.$$
A set $C\subset I$ is called {\em scrambled} (resp. {\em $\delta$-scrambled}) if
any proper pair $\langle x,y\rangle \in C^2$  is scrambled (resp.
$\delta$-scrambled). The map $f$ is called {\em Li-Yorke chaotic} (resp.
{\em $\delta$-Li-Yorke chaotic}) if there exists an uncountable scrambled
set (resp. $\delta$-scrambled set).

In \cite{LY75}, Li and Yorke proved that for an interval map period $3$ implies Li-Yorke chaos. In \cite{JS86}, Jankova and Smital generalized this result as follows: if an interval map has positive entropy, then it is Li-Yorke chaotic.

\medskip
The converse of this result is not true: Xiong \cite{X86} and Smital
\cite{S86} constructed  some interval maps with zero entropy which are
Li-Yorke chaotic.

\medskip
In \cite{S86}, Smital also built some useful tools for zero entropy
interval maps: the periodic portion of an $\omega$-limit set and
$f$-nonseparable points. See \cite{Block1992} for another approach
to the periodic portion of an $\omega$-limit set.
\medskip

In this paper, we discuss those various chaotic properties and their
relationships for  interval maps. In section 2, for preparation we
recall some basic definitions and results for a general dynamical system.
In section 3, we review the structure of the $\omega$-limit set and
build a new approach to the periodic portion of an $\omega$-limit
set.

In section 4,  we deal with zero entropy interval maps. First,
we show that  the proximal relation is an equivalence relation for a zero entropy
interval map. Second, some properties of $f$-nonseparable pairs are
obtained. Third, we discus the relationship between Li-Yorke chaos
and $f$-nonseparable pair. Finally, after
reviewing some recent results on the sequence entropy pair, we show
that for a zero entropy interval map a pair is a sequence entropy
pair if and only if it is $f$-nonseparable.

In section 5, we obtain some
equivalent conditions of positive entropy which relate to the number
``$3$" and show that strongly mixing is equivalent to topological K
for interval maps. In section 6, we show that for an interval map if
it is null, then the pattern entropy of every open cover is of
polynomial order, which give an positive answer for a problem in
\cite{HY09} for interval maps.

\section{Preliminaries}
In this section we briefly review some basic definitions and results for a general dynamical system.
By a {\em topological dynamical system} (TDS for short), we mean a pair $(X, T)$,
where $X$ is a  compact metric space with metric $d$ and $T:X \to X$
is a continuous map.

\begin{defn}
Let $(X, T)$ be a TDS. The system $(X,T)$ (or the map $T$) is called
\begin{enumerate}
\item {\em transitive} if for every two nonempty open subsets $U, V$ of $X$, there exists some $n\geq 0$
such that $T^n U \cap V\neq \emptyset$.
\item {\em weakly mixing} if $T\times T$ is transitive on $X\times X$.
\item {\em strongly mixing} if for every two nonempty open subsets $U, V$ of $X$,
there exists some $N\geq 0$ such that $T^n U \cap V\neq \emptyset$ for all $n\geq N$.
\item {\em minimal} if there is no non-trivial subsystem.
\end{enumerate}
\end{defn}

\begin{defn}
Let  $(X, T)$ be a TDS. The system $(X,T)$ (or the map $T$) is called
\begin{enumerate}
\item {\em sensitive on initial conditions} (or just {\em sensitive}) if there exists $\varepsilon >0$
such that for every nonempty open subset $U$ of $X$, there are $x, y\in U$ and $n\geq 0$ such that $d(T^n(x), T^n(y))\geq \varepsilon$.
\item {\em Devaney chaotic}, if it is transitive and the set of periodic points is dense in $X$.
\end{enumerate}
\end{defn}

In 1965, Adler, Konheim and McAndrew introduced topological entropy for a
TDS. Let $\mathcal C_X^o$ be  the set of finite open covers of $X$.
Given two open covers $\mathcal U$ and $\mathcal V$, let
$$\mathcal U \vee \mathcal V =\{U\cap V: U\in \mathcal U, V\in \mathcal V\}.$$
We define $N(\mathcal U)$ as the minimum cardinality of subcovers of
$\mathcal U$.  The topological entropy of $T$ with respect to
$\mathcal U$ is
$$h(T,\mathcal U)=\lim_{n\to\infty}\frac{1}{n}\log N\left(\bigvee_{i=0}^{n-1}T^{-i}\mathcal U\right).$$
The {\em topological entropy} of $T$ is
$$h(T)=\sup_{\mathcal U\in \mathcal C_X^o}h(T,\mathcal U).$$

In 1974, Goodman introduced sequence topological entropy.
For an $A=\{0\leq t_1<t_2<\cdots \}\subset \mathbb{Z}_+$ and an open
cover $\mathcal U$ of $X$,  the topological sequence entropy of $T$
with respect to $\mathcal U$ and $A$ is
$$h_A(T,\mathcal U)=\lim_{n\to\infty}\frac{1}{n}\log N\left(\bigvee_{i=0}^{n-1}T^{-t_i}\mathcal U\right).$$
The {\em topological sequence entropy} of $(X,T)$ along $A$ is
$$h_A(T)=\sup_{\mathcal U\in \mathcal C_X^o}h_A(T,\mathcal U).$$

Recently, local entropy theory has aroused great interesting, see
\cite{GY09} for a survey on this topic. The notions of
entropy tuple and sequence entropy tuple of length $n$ were defined
in paper \cite{HY06}. Originally, entropy tuple and sequence entropy
tuple are defined using open covers, now we state the equivalence
definition using the notion of independence set (see \cite{HY06, KL07}).
Recall that a TDS $(X, T)$ is {\em tame} if the cardinal
number of its enveloping semigroup is not greater than the cardinal
number of $\mathbb R$ \cite{G06}, and a TDS $(X,T)$ is {\em null} if
$\sup_A h_A(T)=0$.

\begin{defn}
Let $(X, T)$ be a TDS. For a tuple $\tilde{A}=(A_1,\ldots,A_k)$ of subsets of
$X$, a subset $J \subset \mathbb{Z}_+$  is called an {\em independence
set} for  $\tilde{A}$ if for every nonempty finite subset $I
\subset J$, we have
$$\bigcap_{i\in I} T^{-i}A_{s(i)}\neq \emptyset$$
for all $s\in \{1,\ldots, k\}^I$.
\end{defn}

In \cite{KL07}, Kerr and Li defined IE-tuple, IT-tuple and IN-tuple
(standing for entropy, tame and null, respectively) as follows.

\begin{defn}
A tuple $\tilde{x}=\langle x_1, \ldots, x_k\rangle \in X^k$ is called
\begin{enumerate}
\item an {\em IE-tuple} if for every product neighborhood $U_1 \times\cdots\times U_k$ of $\tilde{x}$
the tuple $(U_1,\ldots, U_k)$ has an independence set of positive density.
\item an {\em IT-tuple} if for every product neighborhood $U_1 \times\cdots\times U_k$  of  $\tilde{x}$
the tuple $(U_1,\ldots, U_k)$ has an infinite independence set.
\item an {\em IN-tuple} if for every product neighborhood $U_1 \times\cdots\times U_k$ of $\tilde{x}$
the tuple $(U_1,\ldots, U_k)$  has arbitrarily long finite independence sets.
\end{enumerate}
\end{defn}

Recall that a tuple $\langle x_1, \ldots, x_k\rangle \in X^k$ is
said to be {\em essential} if $x_i\neq x_j$  for all $1\leq i<j\leq k$ and
{\em non-diagonal} if there are $1\leq i<j\leq k$ such that $x_i\neq x_j$.
\begin{defn}
Let $(X, T)$ be a TDS. The system $(X,T)$ is called
\begin{enumerate}
\item {\em uniformly positive entropy} if every essential pair $\langle x_1,  x_2\rangle \in X^2$ is an IE-tuple.
\item {\em topological K} if every essential $k$-tuple $\langle x_1, \ldots, x_k\rangle \in X^k$
is an IE-tuple for all $k\geq 2$.
\end{enumerate}
\end{defn}

\begin{thm} \cite{HY06,KL07} Let $(X, T)$ be a TDS.
\begin{enumerate}
\item A tuple is an entropy tuple iff it is a non-diagonal IE-tuple. In particular
the system $(X, T)$ has zero entropy iff every IE-pair is diagonal.
\item A tuple is a sequence entropy tuple iff it is a non-diagonal IN-tuple.
In particular the system  $(X, T)$ is null iff every IN-pair is diagonal.
\item  The system $(X, T)$ is tame iff every IT-pair is diagonal.
\end{enumerate}
\end{thm}

\section{The structure of $\omega$-limit sets}

We first recall some classical results on the structure of $\omega$-limit sets for interval maps.
The following result is well known, see \cite{Block1992} for example.

\begin{thm}\label{interval-zero-entropy}
Let $f\in C(I,I)$. The following conditions are equivalent:
\begin{enumerate}
\item $h(f)=0$;
\item the period of every periodic point is a power of $2$;
\item every $\omega$-limit set can not properly contain a periodic orbit.
\end{enumerate}
\end{thm}

The following result first appeared in \cite{Sh66}, see also in \cite{BBHS96}.
\begin{thm}\label{interval-sequence-limit-set}
Let $f\in C(I,I)$.
\begin{enumerate}
\item If $\omega_1$ and $\omega_2$ are two $\omega$-limit sets and $a\in \omega_1\cap \omega_2$ is
a limit point from the left (resp., from the right) of both  $\omega_1$ and $\omega_2$,
then $\omega_1\cup \omega_2$ is also an $\omega$-limit set of $f$.
\item If $\omega_1\subset \omega_2\subset \cdots$ is a sequence of $\omega$-limit sets,
then $\overline{\bigcup_{i=1}^\infty \omega_i}$ is also an $\omega$-limit set of $f$.
\end{enumerate}
\end{thm}

On the basis of the above result and Zorn's Lemma we have:
\begin{prop}
Let $f\in C(I,I)$ and $\omega_f$ partially ordered by the inclusion relation.
Then each maximal chain in  $\omega_f$ has a maximal element.
\end{prop}

\begin{lem}\label{interval-maximal-omega-set}
Let $f\in C(I,I)$ with $h(f)=0$. If $\omega_f(x)$ and $\omega_f(x)$
are  two maximal $\omega$-limit  sets, then $\omega_f(x)$ and
$\omega_f(x)$ either coincide or are disjoint.
\end{lem}
\begin{proof}
Assume that $P=\omega_f(x)\cap\omega_f(y)\neq \emptyset$. If $P$ is
finite, then $P$ contains  a periodic point since $f(P)\subset P$.
By Proposition \ref{interval-zero-entropy}(3), both
$\omega_f(x)$ and $\omega_f(y)$ are periodic orbits, which implies
$\omega_f(x)=\omega_f(y)$. If $P$ is infinite, then any limit point
of $P$  is a limit point from the left or from the right of both $\omega_f(x)$ and
$\omega_f(y)$. By Proposition \ref{interval-sequence-limit-set}(1),
$\omega_f(x)\cup \omega_f(y)$ is also an $\omega$-limit set. Then the maximality of $\omega_f(x)$ and
$\omega_f(y)$ imply that they coincide.
\end{proof}

\begin{prop}\cite[Lemma VI.14]{Block1992} \label{interval-periodic-portion-2}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose $\omega_f(x)$
is infinite.  For every $k\geq 1$ and $i=0,1,\ldots, 2^k-1$, let
$$J_k^i=[\min \omega_{f^{2^k}}(f^i(x)), \max \omega_{f^{2^k}}(f^i(x))]. $$
Then
\begin{enumerate}
\item $f(J_k^i)\supset J_k^{i+1(\text{mod } 2^k)}$,
\item the closed intervals $(J_k^i)_{0\leq i < 2^k}$ are pairwise disjoint,
\item $J_{k+1}^i\cup J_{k+1}^{2^k+i}\subset J_k^i$ for $0\leq i<2^k$. Both $J_{k+1}^i$
and $J_{k+1}^{2^k+i}$ have an endpoint in common with $J_k^i$,
\item for every $0\leq i<2^k$, $J_{k}^i$ contains a periodic point of period $2^k$,
but no periodic point of period less than $2^k$.
\end{enumerate}
\end{prop}

We call those intervals $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ are a
{\em periodic portion} of  the $\omega$-limit set $\omega_f(x)$. The
periodic portion of an $\omega$-limit set does not depend on the
choose of the base point, i.e. if $\omega_f(x)=\omega_f(y)$, then
they have the same periodic portion.

\begin{thm}\cite[Lemma VI.16, VI.18]{Block1992}\label{interval-structure-of-limit-set}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose $\omega_f(x)$
is infinite.  Let $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic
portion of $\omega_f(x)$ and
$$C(x)=\bigcap_{k=1}^\infty \bigcup_{i=0}^{2^k-1} J_k^i. $$
Then $C(x)$ is closed, $f(C(x))=C(x)$ and $C(x)\cap Per(f)=\emptyset$.

Moreover, for every nested sequence $J_1^{i_1}\supset
J_2^{i_2}\supset \cdots$,  put $K= \bigcap_{k=1}^\infty J_k^{i_k}$,
then exactly one of the following alternatives holds:
\begin{enumerate}
\item $K=\{y\}\subset\omega_f(x)$ and $y$ is regularly recurrent,
\item $K=[y,z]$, $K\cap\omega_f(x)=\{y,z\}$ and both endpoints of $K$ are strongly recurrent but not regularly recurrent,
\item $K=[y,z]$, $K\cap\omega_f(x)=\{y,z\}$ and one endpoint of $K$ is regularly recurrent and the other is not recurrent.
\end{enumerate}
In every case, if $y\in \omega_f(x)$, then $y$ is regularly recurrent iff $\lim_{n\to\infty}f^{2^n}(y)= y$.
\end{thm}

\begin{thm}\cite[Theorem VI.30]{Block1992}\label{interval-adding-machine}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. If $Y=\omega_f(x)$ is
infinite, then there  exists a continuous map $\phi$ from $Y$ onto
the adding machine $J$ such that except at most countable points in
$J$ which have two preimages, other points have exact one preimage
and
$$\phi\circ f(y)= \tau \circ \phi(y), \ \forall y\in Y.$$
Moreover, $\phi$ maps $Y$ homeomorphically onto $J$ iff every point $y\in Y$ is regularly recurrent iff $(Y,f)$ is a minimal subsystem.
\end{thm}

\begin{lem}\label{interval-key-lemma}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose $\omega_f(x)$
is infinite.  Let $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic
portion of $\omega_f(x)$. For every $k\geq 1$, if $J_k^{r_1}$ and $J_k^{r_2}$
are two intervals in $(J_k^i)_{0\leq i <2^k}$,
then  there exists some periodic point between $J_k^{r_1}$ and $J_k^{r_2}$.
\end{lem}
\begin{proof} Recall that for every $k\geq 1$ there exists a periodic point of
periodic $2^k$ in  $J_k^i$ but no periodic point of periodic $2^j$ for any $j<k$ (see Proposition \ref{interval-periodic-portion-2}).

We use induction to show the result.

If $k=1$, let $J_0=[\min \omega_{f}(x), \max \omega_{f}(x)]$. Since
$f(J_0)\supset J_0$,  there is a fixed point $p$ in $J_0$, but
neither $J_1^0$ or $J_1^1$ can contain a fixed point. Then $p$ lies
between $J_1^0$ and $J_1^1$.

Assume for all $k\leq n$, the conclusion is established.

Let $k=n+1$,  we have two cases.

Case 1: $J_{n+1}^{r_1}$ and $J_{n+1}^{r_2}$ are contained in the same
$J_{n}^{r_3}$.  Since $f^{2^n}(J_n^{r_3})\supset J_n^{r_3}$, there
exists a periodic point $p$ in $J_n^{r_3}$ of periodic $2^n$, but
neither $J_{n+1}^{r_1}$ nor $J_{n+1}^{r_2}$ can contain a periodic point
of periodic $2^n$. Then $p$ lies between $J_{n+1}^{r_1}$ and $J_{n+1}^{r_2}$.

Case 2:  $J_{n+1}^{r_1}$ and $J_{n+1}^{r_2}$ are contained in
$J_{n}^{r_3}$ and $J_{n}^{r_4}$ respectively. By induction,
there exists a periodic point $p$ between $J_n^{r_3}$ and
$J_n^{r_4}$. However, $p$ also lies between $J_{n+1}^{r_1}$ and $J_{n+1}^{r_2}$.
\end{proof}
The following Lemma was proved in \cite{Ruette2002}, for completeness we provide a proof.
\begin{lem}\label{interval-Ruette-lemma}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose $\omega_f(x)$ is infinite.
\begin{enumerate}
\item If $J$ is an interval containing three distinct points of $\omega_f(x)$,  then $J$ contains a periodic point.
\item If $U$ is an open interval such that $U\cap \omega_f(x)\neq \emptyset$, then there exists $n\geq 0$ such that $f^n(U)$ contains a periodic point.
\end{enumerate}
\end{lem}
\begin{proof}
Let $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic portion of $\omega_f(x)$.

(1) Let $J\cap \omega_f(x)\supset \{x_1, x_2, x_3\}$. Without loss
of generality,  assume $x_1<x_2<x_3$. Then there exists a  nested
sequence $J_1^{i_{j_1}}\supset J_2^{i_{j_2}}\supset \cdots$ such
that $x_j\in \bigcap_{k=1}^\infty J_k^{i_{j_k}}$ for $j=1,2,3$. Let
$K_j= \bigcap_{k=1}^\infty J_k^{i_{j_k}}$, then $K_1\cap
K_3=\emptyset$, since $\#(K_j\cap \omega_f(x))\leq 2$ for $j=1,2,3$.
Then by Lemma \ref{interval-key-lemma} there exists a periodic point
$p$ between $K_1$ and $K_3$. Hence, $p\in J$ since $J$ is connected.

(2) Let $z\in U\cap \omega_f(x)$. If $U$ contains a periodic point,
then the proof is complete. Otherwise, let $V \supset U$ be the
maximal subinterval that contains no periodic point. By assumption,
there are $0<n_1<n_2<n_3$ such that $f^{n_j}(x)\in U$ for $j=1,2,3$.
The points $\{z, f^{n_3-n_1}(z), f^{n_2-n_1}(z)\}$ are distinct and
contained in $\omega_f(x)$. If  $\{z, f^{n_3-n_1}(z),
f^{n_2-n_1}(z)\} \subset V$, then the first part of the proof
implies that $V$ contains a periodic point, which contradicts the
definition of $V$. Thus, there exists $j\in \{2,3\}$ such that
$f^{n_j-n_1}(z)\not \in V$. The interval $f^{n_j-n_1}(U)$ contains
both $f^{n_j}(x)\in V$ and $f^{n_j-n_1}(z)\not\in V$. Therefore,
$f^{n_j-n_1}(U)$ must contains a periodic point by the maximality
of the interval $V$.
\end{proof}

\begin{rem}\label{interval-periodic-portion-3}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose $\omega_f(x)$
is infinite.  Let $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic
portion of $\omega_f(x)$. For every $k\geq 1$, since $(J_k^i)_{0\leq
i< 2^k}$ are pairwise disjoint closed intervals, let
$s_k=\min\{\frac{1}{k}, \frac{1}{4}d(J_k^i, J_k^{i+1}): 0\leq
i<2^k-1\}>0$ and $J_{k,s_k}^i=B(J_k^i, s_k)$, where $d(J_k^i,
J_k^{j})=\inf\{|x-y|:x\in J_k^i, y\in J_k^j\}$ and $B(J_k^i,
s_k)=\{x\in I: |x-y|<s_k $  for some $y\in J_k^i\}$. Then  $(J_{k,
s_k}^i)_{0\leq i< 2^k}$ are pairwise disjoint open intervals and
$d(J_{k,s_k}^i, J_{k,s_k}^{i+1})>s_k$ for $0\leq i<2^k-1$. For
convenience, we also call $(J_{k, s_k}^i)_{k\geq 1, 0\leq i< 2^k}$ is the
{\em periodic portion} of $\omega_f(x)$.
\end{rem}

\begin{lem}\label{interval-omega-lim-lem}
Let $f\in C(I,I)$ with $h(f)=0$ and  $x\in I$. Suppose $\omega_f(x)$
is infinite.  Let $(J_{k,s_k}^i)_{k\geq 1, 0\leq i <2^k}$ be the
periodic portion of $\omega_f(x)$. If $y\in I$ with
$\omega_f(y)\subset \omega_f(x)$, then for every $k\geq 1$ there
exists some $n_k\geq 0$ such that  $f^{n}(y)\in
J_{k,s_k}^{n-n_k(\text{mod }2^k)}$ for all $n\geq n_k$.
\end{lem}
\begin{proof}
Since $\omega_f(y)\subset \omega_f(x)$, then for every $k\geq 1$
there exists some $n_{k_0}\geq 0$ such that $f^n(y)\in
\bigcup_{i=0}^{2^k-1} J_{k,s_k}^i$ for all $n\geq n_{k_0}$.

{\bf Claim}: For every $k\geq 1$ there exists $n_{k_1}\geq n_{k_0}$, such that for every
$n\geq n_{k_1}$,  if $f^n(y)\in J_{k,s_k}^i$, then $f^{n+1}(y)\in
J_{k,s_k}^{i+1(\text{mod }2^k)}$.

{\bf Proof of the Claim}. If not, then there exists a sequence
$\{n_q\}$,  such that $f^{n_q}(y)\in J_{k,s_k}^{i_q}$ and
$f^{n_q+1}(y)\not \in J_{k,s_k}^{i_q+1(\text{mod }2^k)}$.
Without loss of generality, we assume $\lim_{q\to\infty}f^{n_q}(y)=a$,
$f^{n_q}(y)\in J_{k,s_k}^{i_0}$ and $f^{n_q+1}(y)\in
J_{k,s_k}^{i_1}$ with $i_1\neq i_0+1 (\text{mod }2^k)$. Then $a\in
J_k^{i_0}$ but $f(a)\in J_k^{i_1}$. This contradicts the fact that
$f(\omega_f(x)\cap J_k^i)=\omega_f(x)\cap J_k^{i+1(\text{mod
}2^k)}$. Then the proof of the Claim is complete.

Now, choose $n_k\geq n_{k_1}$ such that $f^{n_k}(y)\in J_{k,s_k}^0$,
then  $f^{n}(y)\in J_{k,s_k}^{n-n_k(\text{mod }2^k)}$ for all
$n\geq n_k$.
\end{proof}

\section{Chaos for zero entropy maps}
Throughout this section, if  without any other statements, we assume that $f\in C(I,I)$ with $h(f)=0$.
\subsection{Proximal relation}
First, we consider the proximal relation of $f$. If $\langle
x,y\rangle \in I^2$  is proximal, then $\omega_f(x)\cap
\omega_f(y)\neq \emptyset$.  Two maximal $\omega$-limit sets containing
$\omega_f(x)$ and $\omega_f(y)$ respectively are not disjoint, then
they coincide by Lemma \ref{interval-maximal-omega-set}.

We define a ``kneading sequence'' for one point according to the
periodic portion  of its maximal $\omega$-limit set, which can
characterize the proximal pair following the idea in  \cite{BL95}.
\begin{defn}\label{interval-defn-ckx}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. If $\omega_f(x)$ is
infinite,  then there exists a unique maximal $\omega$-limit set
$\omega_0$ which contains $\omega_f(x)$. Let $(J_{k, s_k}^i)_{k\geq
1, 0\leq i <2^k}$ be the periodic portion of $\omega_0$. By Lemma
\ref{interval-omega-lim-lem}, for every $k\geq 1$ we can define
$$c_x(k)=\min\left\{n_k\in \mathbb N:\ f^n(x)\in J_{k,s_k}^{n-n_k(\text{mod }2^k)}
\text{ for all } n\geq n_k\right\}$$
\end{defn}

It is easy to see that $c_x(k)=c_x(k+1)\ (\text{mod }2^k)$.

\begin{prop}\label{interval-prox-eq}
Let $f\in C(I,I)$ with $h(f)=0$ and  $ x,y \in I$. If $\omega_f(x)$
and $\omega_f(y)$ are  contained in the same maximal limit set
$\omega_0$ which is infinite, then the following conditions are
equivalent:
\begin{enumerate}
\item $\langle x,y \rangle $ is proximal;
\item $c_x(k)=c_y(k) (\text{mod }2^k)\text{ for all } k\geq 1$.
\end{enumerate}
\end{prop}
\begin{proof}
Since $\omega_0$ is infinite, $\omega_f(x)$ and $\omega_f(y)$ are
also infinite.  Let $(J_{k, s_k}^i)_{k\geq 1, 0\leq i <2^k}$ be the
periodic portion of $\omega_0$.

$(1)\Rightarrow (2)$. If not, there exists some $k\geq 1$ such that
$c_k(x)\neq c_k(y) (\text{mod }2^k)$. Let $\delta=\min_{0\leq
i<2^k-1}d(J_{k,s_k}^i, J_{k,s_k}^{i+1})>0$. Then it is easy to
verify that $|f^n(x)-f^n(y)|\geq \delta$ for all $n\geq
\max\{c_k(x), c_k(y)\}$. This is a contradiction,
since $\langle x,y \rangle$ is proximal.

$(2)\Rightarrow (1)$. For every $\varepsilon>0$, there exists an
appropriate $J_{k,s_k}^i$ such that $\text{diam}(J_{k,s_k}^i)\leq
\varepsilon$. Without loss of generality, we assume $c_k(x)\geq
c_k(y)$. Then $f^{c_k(x)}(x)$, $f^{c_k(x)}(y)\in J_{k,s_k}^0$ and
$f^{c_k(x)+i}(x)$, $f^{c_k(x)+i}(y)\in J_{k,s_k}^i$. Thus,
$|f^{c_k(x)+i}(x)- f^{c_k(x)+i}(y)|\leq \varepsilon$, which implies
$\langle x,y \rangle $ is proximal since $\varepsilon$ is arbitrary.
\end{proof}

Let $A$ be a subset of $\mathbb N$, we call $A$ has {\em Banach density $1$} if
$$\lim_{n\to \infty}\frac{\#(A\cap E_n)}{\#(E_n)}=1$$
for any sequence $\{E_n\}$ of intervals of positive integer,
where $E_n=\{a_n, a_n+1,\ldots, b_n\}$ and $\lim_{n\to\infty}\#(E_n)=\lim_{n\to\infty}(b_n-a_n+1)=\infty$.

Let $\mathcal F_{bd1}=\{A\subset \mathbb N:\ A$  has Banach density
$1\}$.  A pair $\langle x,y\rangle\in I^2$  is called {\em $\mathcal
F_{bd1}$-proximal} if $\{n\in \mathbb N:\ d(f^n(x),
f^n(y))<\frac{1}{p}\}\in \mathcal F_{bd1}$ for all $p\geq 1$.
Clearly, if  $\langle x,y\rangle \in I^2$ is asymptotic, then
$\langle x,y\rangle$ is $\mathcal F_{bd1}$-proximal.
\begin{prop}\label{interval-bd1-proximal}
Let $f\in C(I,I)$  with $h(f)=0$ and $x,y\in I$. If $\langle
x,y\rangle$ is  proximal, then $\langle x,y\rangle $ is  $\mathcal
F_{bd1}$-proximal.
\end{prop}
\begin{proof}
If $\omega_f(x)$ is finite, then it is easy to see that $\langle x,y\rangle$ is
asymptotic. Now assume that $\omega_f(x)$ is infinite, then so is
$\omega_f(y)$. Let $\omega_0$  be the maximal $\omega$-limit set
which contains both $\omega_f(x)$ and $\omega_f(y)$, then $\omega_0$
is also infinite.  Let $(J_{k, s_k}^i)_{k\geq 1, 0\leq i <2^k}$ be
the periodic portion of $\omega_0$.

Fix $p\geq 1$ and $k\geq 1$. Without loss of generality, assume
$c_k(x)\geq c_k(y)$.   Since the intervals $(J_{k,s_k}^i)_{0\leq i<2^k}$
are pairwise disjoint, there are at most $p$ distinct sets
$J_{k,s_k}^i$ with $\text{diam}(J_{k,s_k}^i)\geq 1/p $. Let
$\{E_n\}$ be a sequence of intervals of positive integer and
$\lim_{n\to\infty}\#(E_n)=\infty$. Then
\begin{align*}
\frac{\#\left\{i\in E_n: |f^i(x)- f^i(y)|< \frac{1}{p}\right\}}{\#(E_n)} &\geq
\frac{ \#\left\{i\in E_n: \text{diam}( J_{k,s_k}^{i(\text{mod }2^k)})< \frac{1}{p}\right\}-c_x(k)}{\#(E_n)}\\
& \geq 1-\frac{\left(\frac{\#(E_n)}{2^k}+2\right)p +c_x(k)}{\#(E_n)}\\
& \to 1-\frac{p}{2^k},\qquad \text{as }n\to \infty.
\end{align*}
Thus,
$$\lim_{n\to\infty}\frac{\#\left\{i\in E_n: |f^i(x)-f^i(y)|< \frac{1}{p}\right\}}{\#(E_n)}=1$$
since $k$ is arbitrary. Therefore, $\langle x,y\rangle$ is $\mathcal F_{bd1}$-proximal since $p$ and $\{E_n\}$ are  arbitrary.
\end{proof}

\begin{thm} \label{intevla-proximal-equivalence}
Let $f\in C(I,I)$ with $h(f)=0$. Then the proximal relation $Prox_f=\{\,\langle x, y\rangle \in I^2: \langle x,y \rangle \text{ is proximal}\,\}$ is an equivalence relation.
\end{thm}
\begin{proof}
The reflexivity and symmetry of $Prox_f$ is obviously. The
transitivity of $Prox_f$  is followed by Proposition
\ref{interval-bd1-proximal} and the property that the intersection
of two  Banach density $1$ sets also has Banach density $1$.
\end{proof}

For every $x\in I$, we define the
{\em proximal cell} of $x$ as $Prox_f(x)=\{y\in I: \langle
x,y\rangle$ is proximal$\}$. It is easy to see that $\{Prox_f(x): x\in
I\}$ is a partition of $I$.

\begin{prop}\label{interval-proximal-class}
Let $f\in C(I,I)$ with $h(f)=0$. If $Prox_f(x), Prox_f(y)$ are two
different  proximal cells, then there exists
$\delta>0$, such that
$$\liminf_{n\to \infty} d(f^n(u), f^n(v))\geq \delta$$
for all $u\in Prox_f(x), v\in Prox_f(y)$.
\end{prop}
\begin{proof}
Let $\omega_1$, $\omega_2$ be two maximal $\omega$-limit sets which contain $\omega_f(x)$ and $\omega_f(y)$ respectively.
If $\omega_1\neq \omega_2$, then $\omega_1\cap \omega_2=\emptyset$.
Let $\delta=d(\omega_1, \omega_2)>0$. Then $\liminf_{n\to \infty} d(f^n(u), f^n(v))\geq \delta$ for all $u\in Prox_f(x), v\in Prox_f(y)$.

If $\omega_1=\omega_2$, then there are two cases.

Case 1. $\omega_1$ is a periodic orbit. Then there exist two
periodic points $p_1$ and  $ p_2$, such that
$\lim_{n\to\infty}|f^n(x)-f^n(p_1)|=0$ and
$\lim_{n\to\infty}|f^n(y)-f^n(p_2)|=0$. Since $ \langle x,y\rangle $
is not proximal, one has $p_1\neq p_2$. Let $\delta=\min_{n\geq
0}|f^n(p_1)-f^n(p_2)|>0$. Then $\liminf_{n\to \infty} d(f^n(u),
f^n(v))\geq \delta$ for all $u\in Prox_f(x), v\in Prox_f(y)$.

Case 2. $\omega_1$ is infinite. Let $(J_{k, s_k}^i)_{k\geq 1, 0\leq
i <2^k}$ be the periodic  portion of $\omega_1$. Since $\langle
x,y\rangle$ is not proximal, there exists some $k\geq 1$ such that
$c_x(k) \neq c_y(k)\ (\text{mod }2^k)$. Let
$\delta=\frac{1}{2}\min\{d(J_{k,s_k}^i, J_{k,s_k}^{i+1}):\ 0\leq
i<2^k-1\}>0$. Then $\liminf_{n\to \infty} d(f^n(u), f^n(v))\allowbreak\geq
\delta$ for all $u\in Prox_f(x), v\in Prox_f(y)$.
\end{proof}

\begin{rem} (1). It has been shown that the proximal relation is an equivalence relation for
a zero entropy interval map, but it may not be closed in $I\times I$. For example, let $f(x)=x^2$,
then $\langle 0, x\rangle $ is proximal for all $x\in [0,1)$, but $\langle 0, 1\rangle$ is not proximal.

(2). Recall that a pair $\langle x, y\rangle\in I^2$ is called {\em regionally proximal}, if for every
$\varepsilon>0$, there exist $u, v$ and $n\geq 1$ such that $|x-u|<\varepsilon, |y-v|<\varepsilon$
and $|f^n(u)-f^n(v)|<\varepsilon$. Obviously, every proximal pair is regionally proximal.
Let $Q_f$ denote the set of regionally proximal pairs. It is not hard to verify that
$$Q_f=\bigcap_{k=1}^{\infty}\overline{\bigcup_{n=k}^{\infty}(f\times f)^{-n}V_k}, $$
where $V_k=\{\langle x,y\rangle \in I^2:\ |x-y|<1/k\}$. Then $Q_f$ is a closed subset
of $I^2$, but it may not be an equivalence relation. For example, let $f(x)=2x^2$ on $[0,1/2]$
and $2(x-1/2)^2+1/2$ on $[1/2,1]$, then $\langle 0, 1/2 \rangle$ and $\langle 1/2, 1\rangle $
are regionally proximal, but $\langle 0, 1\rangle$ is not regionally proximal.
\end{rem}

\subsection{$f$-nonseparable pair}
\begin{defn}
Let $f\in C(I,I)$ with $h(f)=0$. A proper pair $\langle u,v\rangle \in I^2$ is called
{\em $f$-nonseparable}, if there exists some $x\in I$ with infinite $\omega_f(x)$ and let
$(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic portion of $\omega_f(x)$, such that
there exists a nested sequence $J_1^{i_1}\supset J_2^{i_2}\supset \cdots$ satisfying
$\bigcap_{k=1}^\infty J_k^{i_k}=[u,v]\text{ or }[v,u]$.
\end{defn}

The definition of $f$-nonseparability first appeared in \cite{S86}. Our definition is
slightly different from the origin one, but it is easy to see that they are equivalent.

\begin{lem}
Let $f\in C(I,I)$ with $h(f)=0$ and $u<v\in I$. If $\langle u,v\rangle$ is $f$-nonseparable, then
\begin{enumerate}
  \item $[u,v]$ is wandering, i.e. $f^n([u,v])\cap [u,v]=\emptyset$ for all $n\geq 1$,
  \item $u,v$ are not the endpoints of $I$, i.e. \@$u,v\in (0,1)$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) By the structure of $\omega$-limit set (see Theorem \ref{interval-structure-of-limit-set}),
it is easy to see that  $[u,v]$ is wandering.

(2) Let $u,v \in \omega_f(x)$. Then $f^n([u,v])\cap Per(f)=\emptyset$ for all $n\geq 0$ since
$[u,v]\in C(x)$. If $u=0$, then $[u,v)$ is open in $I$. By Lemma \ref{interval-Ruette-lemma}
there exists some $m\geq 0$ such that  $f^m([u,v))\cap Per(f)\neq \emptyset$. This is a contradiction.
Thus, $u>0$ and one can show $v<1$ similarly.
\end{proof}

A point $x\in I$ is called an {\em eventually periodic point},
if there exists some $n\geq 0$ such that $f^n(x)$ is
a periodic point. Denote $EP(f)$ by the set of eventually periodic points. Then
$\overline{EP(f)}\supset \bigcup_{x\in I}\omega_f(x)$. Moreover,
if $u<v\in I$ and  $\langle u,v\rangle$ is $f$-nonseparable, then
$[u,v]\cap EP(f)=\emptyset$. But $\{u,v\}\subset
\overline{EP(f)}$, i.e. $u$ is a limit point of $EP(f)$ from the
left, $v$ is a limit point of $EP(f)$ from the right. Therefore,
for every $x\in I$ there is at most one point such that they can
form a $f$-nonseparable pair.

\begin{prop}
Let $f\in C(I,I)$ with $h(f)=0$ and $x\in I$. Suppose that $\omega_f(x)$ is infinite and  $u, v \in \omega_f(x)$.
\begin{enumerate}
\item If $\langle u,v\rangle$ is $f$-nonseparable, then so is $\langle f(u), f(v)\rangle $.
\item There exists a unique pair $\langle y,z \rangle \in \omega_f(x)\times \omega_f(x)$
such that $f(y)=u, f(z)=v$. Moreover, if $\langle u,v\rangle$ is $f$-nonseparable, then so is $\langle y,z\rangle$.
\end{enumerate}
\end{prop}
\begin{proof}
By Theorem \ref{interval-structure-of-limit-set}, Theorem \ref{interval-adding-machine} and the definition of $f$-nonseparability.
\end{proof}

\begin{prop}\label{interval-f-nonsep-eq}
Let $f\in C(I,I)$ with $h(f)=0$ and $u<v\in I$. Then the following conditions are equivalent:
\begin{enumerate}
\item $\langle u,v\rangle$ is $f$-nonseparable;
\item there exists an $\omega$-limit set $\omega_0$ which is infinite such that $\{u,v\}\in \omega_0$ and $(u,v)\cap Per(f)=\emptyset$.
\end{enumerate}
\end{prop}
\begin{proof}  By the definition of $f$-nonseparability and Theorem \ref{interval-structure-of-limit-set},
(1)$\Rightarrow$(2) is obvious. It remains to show
(2)$\Rightarrow$(1). Let $(J_k^i)_{k\geq 1, 0\leq i <2^k}$ be the
periodic portion of $\omega_0$. If $\langle u, v\rangle$ is not
$f$-nonseparable, then there exist some $k\geq 1$ and two different
intervals $J_k^{i_1}$ and $J_k^{i_2}$ such that $u\in J_k^{i_1},
v\in J_k^{i_2}$. Then by Lemma \ref{interval-key-lemma} there exists
a periodic point $p$ between $J_k^{i_1}$ and $J_k^{i_2}$. Therefore,
$p\in (u,v)$. This is a contradiction.
\end{proof}

\begin{prop}
Let $f\in C(I,I)$ with $h(f)=0$. Then the set of $f$-nonseparable pairs is either empty or countable.
\end{prop}
\begin{proof}
Suppose that the set of $f$-nonseparable pairs is not empty. Let
$A=\{x\in I: \omega_f(x)$ is a maximal $\omega$-limit set and there
exist $u,v\in \omega_f(x)$  such that $\langle u,v\rangle$ is
$f$-nonseparable$\}$. Then $A$ is at most countable since for every
$x\in A$, $C(x)$ contains a non-degenerate interval and any two
different $C(x)$ and $C(y)$ are pairwise disjoint.
Then by Theorem \ref{interval-structure-of-limit-set}, Theorem
\ref{interval-adding-machine} and Proposition \ref{interval-f-nonsep-eq},
for every $x\in A$, $\omega_f(x)\times \omega_f(x)$ contains exactly countable
$f$-nonseparable pairs .
\end{proof}

\subsection{Chaos in the sense of Li-Yorke}

\begin{lem}\label{intervl-nonseparable-lemma-new}
Let $f\in C(I,I)$ with $h(f)=0$ and $u<v\in I$. Suppose $\langle
u,v \rangle$ is $f$-nonseparable  with respect to $\omega_f(x)$,
$(J_k^i)_{k\geq 1, 0\leq i <2^k}$ and $\bigcap_{k=1}^\infty
J_k^{i_k}=[u,v]$. If $A_1$ and $A_2$ are neighborhoods of $u$ and
$v$ respectively, then there exists some $n\geq 1$ such that $[u,v]$ is
in the interior of $f^{2^n}(A_1)\cap f^{2^n}(A_2)$.
\end{lem}

\begin{proof}
By Lemma \ref{interval-Ruette-lemma}, there exists some $n_j\geq 0$ such
that $f^{n_j}(A_j)$  contains periodic points $y_j$  for $j=1,2$. The
periods of $y_1, y_2$ are some powers of 2. Let $2^p$ be a common
multiple of their periods and let $q>p$ such that $2^q>\max\{n_1,
n_2\}$. Let $z_j=f^{2^{q-n_j}}(y_j)$ for $j=1,2$. Then
$f^{2^p}(z_j)=z_j\in f^{2^{q-n_j}}(f^{n_j}(A_j))=f^{2^q}(A_j)$ for
$j=1,2$. Moreover, $z_j\not \in J_q^{i_q}$ since the period of $z_j$
is less than $2^q$. Suppose for instance that $z_1$ is in the left
of $J_q^{i_q}$, the case in the right being symmetric.

Let $g=f^{2^q}$. There exists $i_{q_0}\geq 0$ such that
$J_{q+1}^{i_{q+1}}\cup J_{q+1}^{i_{q_0}} \subset J_q^{i_q}$,
$g(J_{q+1}^{i_{q+1}})\supset J_{q+1}^{i_{q_0}}$ and
$g(J_{q+1}^{i_{q_0}})\supset J_{q+1}^{i_{q+1}}$,  and a fixed point
$c$ for $g$ between $J_{q+1}^{i_{q+1}}$ and $J_{q+1}^{i_{q_0}}$.

Case 1. $J_{q+1}^{i_{q+1}}$ is in the left of
$J_{q+1}^{i_{q_0}}$, then by  connectedness $g(A_1)\supset
J_{q+1}^{i_{q+1}}$ since $g(u)\in J_{q+1}^{i_{q_0}}$ and $z_1\in
g(A_1)$. Hence, $g^2(A_1)\supset J_{q+1}^{i_{q_0}}\cup \{z_1, c\}$
and by the connectedness $g^2(A_1)\supset [z_1, c]\supset
J_{q+1}^{i_{q+1}}$.

Case 2. $J_{q+1}^{i_{q+1}}$ is in the right of
$J_{q+1}^{i_{q_0}}$, then by the connectedness $g^2(A_1)\supset
J_{q+1}^{i_{q_0}}\cup \{z_1, c\}$ since $z_1\in g^2(A_1)$ and
$g^2(u)\in J_{q+1}^{i_{q+1}}$. Then $g^3(A_1)\supset
J_{q+1}^{i_{q+1}}\cup\{z_1,c\}$ and by the connectedness $g^3(A_1)\supset
[z_1, c]$. Therefore, $[u,v]$ are in the interior of $g^4(A_1)$,
since there exists a point $h\in J_{q+1}^{i_{q_0}}\cap \omega_f(x)$
such that $g(h)=v$ and a neighborhood $D$ of $h$ with $g(D)\subset
A_2$, then $D$ contains an eventually periodic point $e$ and $g(e)$
must be in the right of $v$.

As a conclusion, we get $[u,v]$ is in the interior of $g^4(A_1)$.
Similarly,  we have $[u,v]$ is in the interior of $g^4(A_2)$. Let
$n=q+2$, then $[u,v]$ is in the interior of $f^{2^n}(A_1)\cap
f^{2^n}(A_2)$.
\end{proof}

By Lemma \ref{intervl-nonseparable-lemma-new} and induction, we have the following result.
\begin{prop}\label{interval-covering-map}
Let $f\in C(I,I)$ with $h(f)=0$ and $u,v\in I$. If $\langle u,v
\rangle$ is  $f$-nonseparable,  then there exist two sequences of
closed intervals $\{U_n\}$ and $\{V_n\}$  which are neighborhoods of
$u$ and $v$ respectively, and a sequence of positive number
$\{k_n\}$ such that
\begin{enumerate}
\item  $U_n\supset U_{n+1}$, $\lim_{n\to\infty}\text{diam}(U_n)= 0$,
\item  $V_n\supset V_{n+1}$, $\lim_{n\to\infty}\text{diam}(V_n)=0$,
\item $f^{2^{k_n}}(U_n)\cap f^{2^{k_n}}(V_n) \supset U_{n+1}\cup V_{n+1}$.
\end{enumerate}
\end{prop}

\begin{thm}\label{interval-LY-chaos}
Let $f\in C(I,I)$ with $h(f)=0$, then the following conditions are equivalent:
\begin{enumerate}
\item $f$ is Li-Yorke chaotic;
\item there exists a scrambled pair;
\item there exists a $\delta$-scrambled Cantor set for some $\delta>0$;
\item there exists an $f$-nonseparable pair.
\end{enumerate}
\end{thm}
\begin{proof}
(3)$\Rightarrow$(1)$\Rightarrow$(2) is trivial.

(2)$\Rightarrow$(4). Let $\langle x, y\rangle \in I^2$ be a
scrambled pair. Then $\omega_f(x)$ and  $\omega_f(y)$ are contained
in the same maximal $\omega$-limit set $\omega_0$ which is infinite.
Let $(J_{k,s_k}^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic portion
of $\omega_0$. Since $\limsup_{n\to\infty}|f^n(x)-f^n(y)|>0$, there
exists a sequence $\{n_i\}$ such that
$\lim_{i\to\infty}f^{n_i}(x)=a$ and $\lim_{i\to\infty}f^{n_i}(y)=b$
for $a\neq b$. Then $\langle a,b \rangle$ is $f$-nonseparable. In
fact, if not, then there exist $k\geq 1$ and $i_0\neq i_1$ such that
$a\in J_k^{i_0}$ and  $b\in J_k^{i_1}$. Thus, there are infinitely
many $n_i$ such that $f^{n_i}(x)\in J_{k,s_k}^{i_0}$ and
$f^{n_i}(y)\in J_{k,s_k}^{i_1}$. This is a contradiction since
$\langle x, y\rangle$ is proximal.

(4)$\Rightarrow$(3). Let $\langle u,v\rangle\in I^2$ be an $f$-nonseparable pair and $\delta=|v-u|$.
Let $\{U_n\}$, $\{V_n\}$ and $\{k_n\}$ as in Proposition \ref{interval-covering-map}. Let $t_0=0$ and $t_m=\sum_{n=1}^m 2^{k_n}$ for $m\geq 1$.

First we build a family of closed subintervals $\{E_{a_0a_1,\ldots,a_m}: m\geq 0, a_i\in\{0,1\}\}$ satisfying
 the following properties:
\begin{enumerate}
\item[(a)] $E_{a_0a_1,\ldots,a_ma_{m+1}}\subset E_{a_0a_1,\ldots,a_m}$,
\item[(b)] $E_{a_0a_1,\ldots,a_m}\cap E_{b_0b_1,\ldots,b_m}=\emptyset$ if $a_0a_1,\ldots,a_m\neq b_0b_1,\ldots,b_m$,
\item[(c)] for $\alpha=a_0a_1,\ldots,a_m$, $f^{t_i}(E_\alpha) \subset W_i$ for $i=0,1,\ldots,m-1$
and $f^{t_m}(E_\alpha)=W_m$ where $W_i= U_i$ if $a_i=0$ and $W_i= V_{i}$ if $a_i=1$.
\end{enumerate}

Let $E_0=U_1$, $E_1=V_1$. Suppose that $E_{a_0a_1,\ldots,a_m},
a_i\in\{0,1\}$ are already defined. For $a_0a_2,\ldots, a_ma_{m+1}$,
we have $E_{a_0a_1,\ldots,a_m} \stackrel{f^{t_m}}{\longrightarrow}
W_m \stackrel{f^{2^{k_{m+1}}}}{\longrightarrow} W_{m+1}$, where notation
$A\stackrel{g}{\longrightarrow}B$ means $g(A) \supset B$. Let $F$ be
a subinterval of $W_m$ of minimal length such that
$f^{2^{k_{m+1}}}(F)=W_{m+1}$ and $E_{a_0a_1,\ldots,a_ma_{m+1}}$ be a
subinterval of $E_{a_0a_1,\ldots,a_m}$ of minimal length such that
$f^{t_m}(E_{a_0a_1,\ldots,a_ma_{m+1}})=F$.
Then it is easy to verify that $E_{a_0a_1,\ldots,a_ma_{m+1}}$ satisfies the requirement.

With $\mathbb Z_+=\{0,1,2,\ldots\}$ let $\Sigma$ be the Cantor space
$\{0,1\}^{\mathbb Z_+}$  regard as the set of infinite words. For
every $\alpha=(a_0 a_1, a_2, \ldots)\in \Sigma$, let
$E_\alpha=\bigcap_{m=0}^{\infty}E_{a_0a_1\ldots a_m}$. Then
$E_\alpha$ is either a nonempty compact interval or a single point.
Moreover $E_\alpha\cap E_\beta=\emptyset$ if $\alpha\neq \beta\in \Sigma$.

Let $\Lambda=\{\alpha\in\Sigma: E_\alpha$ is not reduced to a single
point$\}$. The set  $\Lambda$ is at most countable because the sets
$(E_\alpha)_{\alpha\in\Lambda}$ are pairwise disjoint and nondegenerate
intervals. Let
$$X=\bigcup_{\alpha\in \Sigma} E_\alpha\setminus \bigcup_{\beta\in \Lambda} int(E_\beta).$$
It is easy to see that $X$ is a totally disconnected compact set.
Define $\phi:X \to \Sigma$  by $\phi(x)=\alpha$ if $x\in E_\alpha$.
Clearly, the map $\phi$ is well defined, continuous and onto.

Fix $\gamma=(c_0c_1\ldots)=(010101\ldots)$. Define $\psi: \Sigma\to\Sigma$ by
\[\psi((a_n)_{n\in\mathbb Z_+})=(a_0c_0a_0a_1c_0c_1\ldots a_0a_1\ldots a_n c_0c_1\ldots c_n\ldots).\]
The map $\psi$ is clearly continuous, thus $\psi(X)$ is
compact.

For every $\alpha \in \Sigma$, choose $x_\alpha\in X$ such that
$\phi(x_\alpha)=\psi(\alpha)$  and let $S=\{x_\alpha\in X:
\alpha\in\Sigma\}$. If $\psi(\alpha)\not \in \Lambda$ then there is
a unique choice for $x_\alpha$ and if  $\psi(\alpha) \in \Lambda$
then there are two possible choices. Consequently, $S$ is equal to
$\phi^{-1}(\psi(X))$ deprived of a countable set. Let $\alpha,
\beta$ be two distinct elements of $\Sigma$. By the definition of
$\psi$, for every $N\geq 0$ there exists some $m\geq N$ such that the
$m$-th coordinates of $\psi(\alpha)$ and $\psi(\beta)$ are distinct.
Then either $f^{t_m}(x_\alpha)\in U_{\psi(\alpha)_m}$ and $f^{t_m}(x_\beta)\in
V_{\psi(\alpha)_m}$, or the converse. Thus,
\[\limsup_{n\to\infty} |f^n(x_\alpha)-f^n(x_\beta)|\geq \delta.\]

According to the choice of $\gamma$ and the definition of $\psi$,
for every $N\geq 0$  there exists some $m\geq N$ such that for every
$\alpha\in\Sigma$ the $m$-th coordinate of $\psi(\alpha)$ is 0,
which implies $f^{t_m}(x_\alpha)\in U_{\psi(\alpha)_m}$. Thus,
$$\liminf_{n\to\infty}|f^n(x)-f^n(y)|=0$$
for all $x,y\in S$ since $\lim_{n\to\infty}diam(U_n)=\lim_{n\to\infty}diam(V_n)=0$.
By the choice of $\gamma$, it is also easy to see that $\{u,v\}\in\omega_f(x)$ for all $x\in S$.

Since $X$ is totally disconnected and $\phi^{-1}(\psi(X))$ is an
uncountable closed subset  of $X$, then $S$ contains some Cantor subset
$K$. Therefore, $K$ is a $\delta$-scrambled Cantor set.
\end{proof}

\begin{rem} It should be noticed that in the above Theorem {\rm (3)$\Leftrightarrow$(4)}
was proved in \cite{S86} and {\rm (2)$\Leftrightarrow$(3)} was proved in \cite{KS89}, but here we give a new proof.
\end{rem}

\begin{cor}\label{interval-f-non-sep-it}
Let $f\in C(I,I)$ with $h(f)=0$. If $\langle u,v \rangle$ is $f$-nonseparable, then $\langle u, v\rangle$ is an IT-pair.
\end{cor}

\begin{proof}
Let $\{U_n\}$, $\{V_n\}$ and $\{k_n\}$ as in Proposition
\ref{interval-covering-map}.  Let $t_n=\sum_{j=1}^n k_n$. If $U, V$
are neighborhood of $u, v$ respectively, then there exists some $N\geq 1$
such that $U_n\subset U$ and $V_n\subset V$ for all $n\geq N$.

For every $s\in \{0,1\}^{\{0,1,\ldots, k\}}$, by the proof of
Theorem \ref{interval-LY-chaos}  there exists a $w_s$ such that for
$0\leq i \leq k$, $f^{t_{N+i}}(w_s)\in U_{N+i+1}$ if $s(i)=0$ and
$f^{t_{N+i}}(w_s)\in V_{N+i+1}$ if $s(i)=1$, then
$$w_s\in \bigcap_{i=0}^{k}f^{-t_{N+i}}A_{s(i)}$$
where $A_0=U$, $A_1=V$.

Thus, $\{t_N, t_{N+1}, \ldots\}$ is an infinite independence set of $(U, V)$.
Therefore,  $\langle u, v\rangle$ is an IT-pair and $(X,T)$ is not
tame since $\langle u, v\rangle$ is not in the diagonal.
\end{proof}

\begin{defn}
Let $(X,T)$ be a TDS and $S, R\subset X$.
\begin{enumerate}
\item $S$ and $R$ are called {\em equivalent} if there exists a bijection $\phi: S\to R$ such that
$$\lim_{n\to \infty}d(f^n(\phi(x)), f^n(x))=0\quad \text{for any }x\in S.$$
\item $S$ and $R$ are called {\em separable} if there exists $\delta>0$ such that
$$\liminf_{n\to \infty}d(f^n(x), f^n(y))\geq \delta\quad \text{for any } x\in S \text{ and }y\in R.$$
\end{enumerate}
\end{defn}

\begin{prop}\label{interval-scr-eq-sep}
Let $f\in C(I,I)$ with $h(f)=0$. Then every two maximal scrambled sets are either equivalent or separable.
\end{prop}
\begin{proof}
Let $S$ and $R$ be two maximal scrambled sets. If $S$ and $R$ are
contained in different proximal cells, then by
Proposition \ref{interval-proximal-class} there exists $\delta>0$
such that
$$\liminf_{n\to \infty}|f^n(x)-f^n(y)|\geq \delta\quad \text{for any } x\in S \text{ and }y\in R.$$
Now assume that $S$ and $R$ are contained in the same proximal
cell $Prox_f(z)$.  We define a relationship $\sim$ in
$Prox_f(z)$. Let $a,b\in Prox_f(z)$, $a \sim b $ iff $\langle a,b
\rangle$ is asymptotic. Then $\sim$ is a equivalence relation on
$Prox_f(z)$. It is easy to see that in $Prox_f(z)$ every maximal
scrambled set contains exactly one representative point for every
one of those $\sim$ equivalent classes. Thus, there exists a
bijection $\phi: S\to R$ such that
\[\lim_{n\to \infty}|f^n(x)- f^n(\phi(x))|=0\qquad \text{for any }x\in S. \qedhere\]
\end{proof}

\begin{rem}
The Proposition \ref{interval-scr-eq-sep} was proved by Balibrea and Lopez in \cite{BL95}.
It seems that they used some lemmas which are only available for piecewise monotone maps.
\end{rem}
{\bf Quesition}: If $h(f)=0$, is every maximal scrambled set uncountable?

\begin{defn}\cite{X05} Let $(X,T)$ be a TDS and $n\geq 2$.  A tuple $\langle x_1, x_2,\ldots,
x_n\rangle \in X^n$ is called {\em $n$-scrambled} if
$$ \liminf_{k\to\infty}\max_{1\leq i<j\leq n}d\left(f^k(x_i), f^k(x_j)\right)
=0,\quad\limsup_{k\to\infty}\min_{1\leq i<j\leq n}d\left(f^k(x_i), f^k(x_j)\right)>0.$$

The system $(X,T)$ (or the map T) is called {\em $n$-chaotic in the sense
of Li-Yorke} if there exists an uncountable subset $S\subset X$ such
that every essential tuple $\langle x_1, x_2,\ldots,\allowbreak x_n \rangle\in S^n$ is $n$-scrambled.
\end{defn}

\begin{thm}\label{interval-3-srca-tuple}
Let $f\in C(I,I)$ with $h(f)=0$. Then there is no $3$-scrambled tuple.
\end{thm}
\begin{proof}
If $\langle x_1, x_2, x_3\rangle\in I^3 $ is a $3$-scrambled tuple,
then  there exists a sequence $\{n_q\}$ in $\mathbb N$, such that $\lim_{q\to\infty}f^{n_q}(x_i)= a_i$
for $i=1,2,3$ and $a_1$,$ a_2$ and $a_3$ are pairwise distinct. Let
$\omega_0$ be the maximal $\omega$-limit set contains
$\omega_f(x_i)$ for $i=1,2,3$, then $\omega_0$ is infinite. Let
$(J_{k,s_k}^i)_{k\geq 1, 0\leq i <2^k}$ be the periodic portion of
$\omega_0$.  Since $\langle x_1, x_2\rangle $ and $\langle
x_1, x_3\rangle$ are proximal, by Proposition \ref{interval-prox-eq},
 for every $k\geq 1$,
$a_1$, $a_2$ and $a_3$ must be in the same $J_k^{r_k}$. Thus,
$\{a_1,a_2,a_3\}\subset \bigcap_{k=1}^{\infty}J_k^{r_k}$ , which
implies $\#(\bigcap_{k=1}^{\infty}J_k^{r_k} \cap \omega_0)\geq 3$.
This contradicts the  structure of $\omega$-limit set (see
Theorem \ref{interval-structure-of-limit-set}).
\end{proof}

\begin{cor}
There exists a TDS which is $2$-chaotic in the sense of Li-Yorke but
does not  have any $3$-scrambled tuple.
\end{cor}
\begin{proof}
Any interval map, which is chaotic in the sense of Li-Yorke and has
zero entropy,  satisfies the requirement.
\end{proof}

\subsection{Sequence entropy pair}
In \cite{FS91}, Franzova and Smital showed that positive sequence topological entropy can characterize Li-Yorke chaos:
\begin{thm}\label{interval-null-vs-chaos}
Let $f\in C(I,I)$. Then $f$ is Li-Yorke chaotic iff it is not null.
\end{thm}

\begin{thm}\label{interval-null-eq-tame}\cite{GY09}
Let $f\in C(I,I)$. Then $f$ is null iff it is tame.
\end{thm}

The structure of the set of IN-pairs (or sequence entropy pairs) was studied by Tan, Ye and Zhang in \cite{TYZ09}.

\begin{thm}\label{interval-3-seq-ent-pair}\cite{TYZ09}
Let $f\in C(I,I)$ with $h(f)=0$. If $f$ is not null, then there exist exactly countable IN-pairs, but no essential $3$-IN-tuple.
\end{thm}

\begin{thm}\label{interval-seq-ent-pair-str}\cite{TYZ09}
Let $f\in C(I,I)$ with $h(f)=0$. If $\langle x, y \rangle$ is an IN-pair and $x<y$, then
\begin{enumerate}
\item both $\omega_f(x)$ and $\omega_f(y)$ are infinite,
\item $[x,y]$ is wandering, i.e. $f^n([x,y])\cap [x,y]=\emptyset$ for all $n\geq 1$,
\item $\langle x, y\rangle $ is asymptotic,
\item $[x, y]\cap EPer(f)=\emptyset$, but $\{x, y\}\cap \overline{Per(f)}\neq \emptyset $ and $\{x,y\}\subset \overline{EPer(f)}$.
\end{enumerate}
\end{thm}

\begin{thm}\label{interval-IT-IN-eq}
Let $f\in C(I,I)$ with $h(f)=0$ and $x<y \in I$. Then the following conditions are equivalent:
\begin{enumerate}
\item $\langle x , y \rangle$ is $f$-nonseparable;
\item $\langle x , y \rangle$ is an IT-pair;
\item $\langle x , y \rangle$ is an IN-pair.
\end{enumerate}
\end{thm}

\begin{proof}
(1)$\Rightarrow$(2) is proved in Corollary \ref{interval-f-non-sep-it} and (2)$\Rightarrow$(3) is trivial.

It remains to show (3)$\Rightarrow$(1). Assume that $\langle x , y
\rangle$ is an IN-pair.  Without loss of generality, assume that $x$ is a
limit point of $Per(f)$ from the left. Let $U_1$ and $U_2$ be two
disjoint connected neighborhoods of $x$ and $y$ respectively. Then
there are periodic points $p$, $q$ and $n\geq 1$ such that $p\in
U_1$ and $q\in f^n(U_2)$. Without loss of generality, assume that $p$ and
$q$ are fixed points and $n=1$, since the periods of $p, q$ are the
powers of $2$ and $\langle x,y \rangle$ is also an IN-pair for $f^n$
for every $n\geq 1$.

{\bf Claim: } There exists $n\geq 1$ such that the subinterval $[x,y]$ is in the interior of $f^{n}(U_1)\cap f^{n}(U_2)$.

{\bf Proof of the Claim: }Clearly, we have $p<x$, but there are two cases about the position of $q$.

Case 1, $q>y$. (a) $f(x)>y, f(y)>y$. Since $q$ is a fixed point
and $f(x)>y$, then by the connectedness $[x,y]$ is in the interior of
$f^k(U_1)$ for all $k>1$. Since $\langle x,y\rangle$ is an IN-pair,
there exist $z\in U_2$ and $1<n_1<n_2$ such that $f^{n_1}(z)\in U_1$
and $f^{n_2}(z)\in U_1$. Then  one of $f^{n_1}(z) , f^{n_2}(z)$ must
be on the left side of $x$ since $[x,y]$ is wandering. Since $q>y$, $q$
is a fixed point and $q\in f(U_2)$, by the connectedness we have $[x,y]$
is in the interior of $f^{n_1}(U_2)$ or $f^{n_2}(U_2)$. (b)
$f(x)<x, f(y)<x$. This is the symmetric case of (a).

Case 2, $q<x$. (a) There exists some $k\geq 1$ such that $f^k(x)>y,
f^k(y)>y$. Without loss of generality, we can assume $k=1$. Since
$q$ is a fixed point and $f(x)>y$, then by the connectedness $[x,y]$ is
in the interior of $f(U_1)$. Since $p\in f(U_2)$ and $f(y)>y$, then
by the connectedness $[x,y]$ is in the interior of $f(U_2)$. (b) We
have $f^k(x)<x, f^k(y)<x$ for all $k\geq 1$. Then $f^k([x,y])$ is in
the left of $[x,y]$  for all $k\geq 1$. Thus if $z\in U_2$ and
$f^n(z)\in U_2$ for some $n\geq 1$, then $z>y$. Since $\langle
x,y\rangle$ is an IN-pair, there exist $u\in U_1, v\in U_2$ and
$1<n_1<n_2$ such that $f^{n_1}(u)\in U_2$,  $f^{n_2}(u)\in U_2$ and
$f^{n_1}(v)\in U_2$,  $f^{n_2}(v)\in U_2$. Then $f^{n_1}(u)>y$ and
$f^{n_1}(v)>y$. Thus $[x,y]$ is in the interior of $f^{n_1}(U_1)$
and $f^{n_1}(U_2)$. This completes the proof of the Claim.

Similarly to the usage of Lemma \ref{intervl-nonseparable-lemma-new}
to prove Theorem  \ref{interval-LY-chaos}, we can get that there
exists some $z\in I$ such that $x, y\in \omega_f(z)$. Thus, by
Proposition \ref{interval-f-nonsep-eq} $\langle x, y\rangle$ is
$f$-nonseparable.
\end{proof}

\section{Positive entropy Maps}

\begin{thm} \label{interval-postive-entropy-eq1}
Let $f\in C(I,I)$. Then the following conditions are equivalent:
\begin{enumerate}
\item $h(f)>0$;
\item there exists a subsystem which is Devaney chaotic \cite{L93};
\item there exists $n\geq 1$ such that $f^n$ has a strongly mixing subsystem \cite{XY92};
\item $f$ is distributionally chaotic \cite{SS94}.
\end{enumerate}
\end{thm}

\begin{defn}
\cite{X05} Let $(X,T)$ be a TDS and $n\geq 2$. $(X,T)$ is said to be
{\em $n$-sensitive},  if there exists some $\delta>0$  such that for every
nonempty open subset $U\subset X$, there exist $n$ distinct points
$x_1, x_2, \ldots, x_n \in U$ and $k\geq 1$ satisfying $\min_{1\leq
i< j\leq n}\{d(f^k(x_i), f^k(x_j))\}\geq \delta$.
\end{defn}

\begin{thm} \label{Devaney-chaos-thm}
Let $(X,T)$ be a TDS. If $(X,T)$ is Devaney chaotic, then
\begin{enumerate}
\item $(X,T)$ is infinite sensitive (i.e. $n$-sensitive for all $n\geq 2$) \cite{YZ08},
\item $(X,T)$ is infinite chaotic in the sense of Li-Yorke (i.e. there is an uncountable subset $S$ of X
which is $n$-scrambled for all $n\geq 2$) \cite{X05}.
\end{enumerate}
\end{thm}

\begin{lem} \label{tran-sys-lemma}
Let $(X,T)$ be a TDS. If $(X,T)$ is transitive and has two periodic points, then
\begin{enumerate}
\item $Prox_T$ is not an equivalence relation,
\item there are two maximal scrambled sets which are neither equivalent nor separable.
\end{enumerate}
\end{lem}
\begin{proof}
(1) Without loss of generality, we assume that there are two fixed points
$p_1$ and $p_2$,  since for every $n\geq 1$ $\langle x_1,x_2\rangle\in X^2$ is
proximal for $T$ iff so is for $T^n$. Then it is easy to see that
for every transitive point $x\in X$, $\langle x,p_1\rangle$ and $\langle
x, p_2\rangle$ are proximal, but $\langle p_1, p_2\rangle$ can not
be proximal.

(2)  Let $S$ and $R$  be the maximal scrambled sets which contain
$\{x, p_1\}$ and $\{x, p_2\}$  respectively. Clearly, $S$ and $R$
are not separable. Next, we show that $S$ and $R$ also are not
equivalent. If there exists a bijection $\phi: S\to R$ such that
$$\lim_{n\to \infty}d(f^n(y)- f^n(\phi(y)))=0\quad \text{for any }y\in S, $$
then $\langle p_1, \phi(p_1)\rangle$ is asymptotic. Thus, $\langle
p_1, p_2 \rangle$ is  proximal since $\langle p_2, \phi(p_1)
\rangle$ is proximal. This is a contradiction.
\end{proof}

Now we state the main result of this paper: there are various
equivalent conditions of  positive entropy which may relate to the
number ``$3$".

\begin{thm}
Let  $f\in C(I,I)$. Then the following conditions are equivalent:
\begin{enumerate}
\item $h(f)>0$;
\item $Prox_f$ is not an equivalence relation;
\item there exist two maximal scrambled sets which are neither equivalent nor separable;
\item there exists some $3$-scrambled tuple;
\item there exists some $3$-sensitive transitive subsystem;
\item there exists some essential $3$-IN-tuple.
\end{enumerate}
\end{thm}
\begin{proof}
(1)$\Rightarrow$(2) By Theorem \ref{interval-postive-entropy-eq1}(2) and Lemma \ref{tran-sys-lemma}(1).

(2)$\Rightarrow$(1) By Theorem \ref{intevla-proximal-equivalence}.

(1)$\Rightarrow$(3) By Theorem \ref{interval-postive-entropy-eq1}(2) and Lemma \ref{tran-sys-lemma}(2).

(3)$\Rightarrow$(1) By Theorem \ref{interval-scr-eq-sep}.

(1)$\Rightarrow$(4) By Theorem \ref{interval-postive-entropy-eq1}(2) and Theorem \ref{Devaney-chaos-thm}(2).

(4)$\Rightarrow$(1) By Theorem \ref{interval-3-srca-tuple}.

(1)$\Rightarrow$(5) By Theorem \ref{interval-postive-entropy-eq1}(2) and Theorem \ref{Devaney-chaos-thm}(1).

(5)$\Rightarrow$(1) If $h(f)=0$, by Theorem
\ref{interval-adding-machine}, any infinite  transitive subsystem is
at most $2$ to $1$ extension of the adding machine, then it is not
$3$-sensitive \cite{SYZ08}.

(1)$\Rightarrow$ (6) In \cite{HY06}, it shows that if $h(f)>0$, then there exists some essential $3$-IN-tuple.

(6)$\Rightarrow$(1) By Theorem \ref{interval-3-seq-ent-pair}.
\end{proof}

\begin{thm}\cite{Block1992}
Let $f\in C(I,I)$. If $f$ is transitive, then exactly one of the following alternatives holds:
\begin{enumerate}
\item $f$ is strongly mixing,
\item there exists a fixed point $c\in (0,1)$ such that $f([0,c])=[c,1]$, $f([c,1])=[0,c]$
and both $f^2|_{[0,c]}$ and $f^2|_{[c,1]}$ are strongly mixing.
\end{enumerate}
\end{thm}
\begin{thm}\cite{Block1992}\label{interval-strong-mixing}
Let $f\in C(I,I)$. Then the following conditions are equivalent:
\begin{enumerate}
\item $f^2$ is transitive;
\item $f$ is weakly mixing;
\item $f$ is strongly mixing;
\item for any $\varepsilon>0$ and non degenerate subinterval $J\subset I$, there exists a $N>0$ such that
$$f^n(J)\supset [\varepsilon, 1-\varepsilon] \text{ for all } n\geq N.$$
\end{enumerate}
\end{thm}

\begin{thm}\label{interval-stro-mix-eq1}
Let $f\in C(I,I)$. Then the following conditions are equivalent:
\begin{enumerate}
\item $f$ is strongly mixing;
\item $\mathcal C(\mathcal U)>2$ for every open cover $\mathcal U$ which consists of
two non dense open sets, where $\mathcal C(\mathcal U)=\lim_{n\to \infty}N(\bigvee_{i=1}^{n-1}f^{-i}(\mathcal U))$;
\item $f$ is uniformly positive entropy;
\item $f$ is topological K system.
\end{enumerate}
\end{thm}
\begin{proof}

(4)$\Rightarrow$(3) and (3)$\Rightarrow$(2) is trivial.

(2)$\Rightarrow$(1). We first prove a claim which implies that $f$ is transitive.

{\bf Claim}: Let $(X,T)$ be a TDS. If $\mathcal C(\mathcal U)>2$ for
every open cover  $\mathcal U$ of $X$ which consists of two non
dense open sets, then $T$ is transitive.

{\bf Proof of the Claim: } We follow the idea in \cite{BHM00}. If
not, there exist two nonempty open  subsets $U,V$ of $X$ such that $T^nU\cap
V=\emptyset$ for all $n\geq 1$.

Case 1.  If there is some $n\geq 1$ such that $T^nV \cap U \neq
\emptyset$, let $m$ be the minimum of such values. Let $E=V\cap
T^{-m}U$, then $E\cap T^{-n}E=\emptyset$ for all $n\geq 1$. Choose
two closed  subsets $U_1$ and $V_1$  of $E$ and $T^{-1}E$
respectively which have non-empty interior. Let
$\mathcal{U}=\{U_1^c, V_1^c\}$, it is easy to verify that
$\mathcal{C}(\mathcal{U})=2$. This is a contraction.

Case 2. $T^n V\cap U=\emptyset$ for all $n\geq 1$. Choose two closed
subsets $U_1$ and $V_1$   of $U$ and $V$ respectively which have
non-empty interior. Let $\mathcal{U}=\{U_1^c, V_1^c\}$, it is easy
to verify that $\mathcal{C}(\mathcal{U})=2$. This also is a
contraction.

Thus, the proof of the Claim is complete.

So $f$ is transitive. Now assume that $f$ is not strongly mixing, then
there exists a fixed  point $c\in (0,1)$ such that $f([0,c])=[c,1]$,
$f([c,1])=[0,c]$. There exists $\varepsilon>0$ such that
$c+\varepsilon, c-\varepsilon\in (0,1)$. Let $\mathcal U=\{[0,
c+\varepsilon), (c-\varepsilon, 1]\}$. It is easy to verify that
$\mathcal C(\mathcal U)=2$. This also is a contradiction.

(1)$\Rightarrow$(4) To show that $f$ is topological K , it is
sufficient to show that every $k$-tuple of non-empty open  subsets
$(U_1, \ldots, U_k)$ has an independence set of positive density.

Since $f$ is strongly mixing, by Theorem \ref{interval-strong-mixing}(4),
there exist some $n\geq 1$ and nonempty
open subset $V_i\subset U_i$ for $1\leq i\leq k $ such that
$\bigcap_{i=1}^n f^n V_i \supset \bigcup_{i=1}^k V_i$. Then
$n\mathbb{N}=\{n,2n,3n,\ldots\}$ is an independence set for $(U_1,
\ldots, U_k)$ since
$$f^{-n} V_{s(1)}\cap f^{-2n}V_{s(2)}\cap\cdots \cap f^{-mn}V_{s(m)}\neq \emptyset$$
holds for all $m\geq 1$ and all $s\in\{1,2,\ldots k\}^m$.
\end{proof}

\begin{rem}
Recall a TDS $(X,T)$ is called to be of {\em completely positive entropy}
if each of its non-trivial factors has positive entropy.
In \cite{B92}, it showed that there exists a TDS which is of completely
positive entropy but not of uniformly positive entropy.  There are
also some examples for interval maps. For example, let $f(x)= 1/2 +
2x$ on $[0, 1/4]$, $3/2-2x$ on $[1/4, 1/2]$ and $1-x$ on $[1/2, 1]$.
It is easy to check that $f$ is of  completely positive entropy but
not of uniformly positive entropy. Moreover, $f^2$ is also of
completely positive entropy but not transitive.
\end{rem}
\section{Topological null system}

In \cite{HY09}, Huang and Ye introduced the notion of maximal
pattern entropy. For a TDS $(X,T)$, $n \in \mathbb{N}$ and a
finite open cover $\mathcal{U}$, let
$$p^*_{X,\mathcal{U}}(n) = \max_{(t_1<t_2<\cdots<t_n)\in\mathbb{Z}_+^n }N \left(\bigvee^n_{i=1} T^{-t_i}\mathcal{U}\right).$$

The maximal pattern entropy of $T$ with respect to $\mathcal{U}$ is
defined by
$$h^*_{top}(T,\mathcal{U}) = \lim_{n\to\infty} \frac{1}{n}\log p^*_{X,\mathcal{U}}(n).$$

The {\em maximal pattern entropy} of $(X,T)$ is
$$h^*_{top}(T)= \sup_{\mathcal{U}\in \mathcal{C}_X^o} h^*_{top}(T,\mathcal{U}).$$
Then a TDS $(X,T)$ is null iff $h^*_{top}(T)=0$, since $h^*_{top}(T)=\sup_A h_A(T)$.

In \cite{HY09}, Huang and Ye proved that for a null TDS
defined on a zero dimensional space, $p^*_{X,\mathcal{U}}(n)$ is of
polynomial order for each open cove $\mathcal{U}$ of $X$. They also

{\bf Conjecture:} If a TDS $(X,T)$ is null, then it is true that
$p^*_{X,\mathcal{U}}(n)$ is of polynomial order for each open cover
$\mathcal{U}$ of $X$.

In the following section, we prove that the conjecture holds
for interval maps. Before doing this, we need some lemmas. Let $\omega(f)=\bigcup\{\omega_f(x):x\in I\}$.

\begin{lem} \cite{FS91} \label{Smital-lemma} Let $f\in C(I,I)$. If $f$ is null, then for every
$\varepsilon > 0$ there are points $x_1, x_2, \ldots, x_k \in
\omega(f)$ and an open set $U \supset \omega(f)$ with the following property: if
$$f^j(x)\in U \quad for  \quad 0\leq j \leq r ,$$
then there exists some $i\in\{1,2,\ldots, k\}$ such that for any $j$ with $0\leq j \leq r$,
$$|f^j(x)-f^j(x_i)|<\varepsilon.$$
\end{lem}

\begin{lem}\label{Sarkovskii-lemma} {\rm (\cite{Sh67} or \cite[Corollary IV.13]{Block1992})}
Let $f\in C(I,I)$. Then for any neighborhood $U$ of
$\omega(f)$ there is an integer $q>0$ such that the number of points
of an arbitrary trajectory lying outside $U$ is less than $q$.
\end{lem}

\begin{thm} \label{interval-poloynomial-order}
Let $f\in C(I,I)$. If $f$ is null, then $p^*_{I,\mathcal{U}}(n)$ is of polynomial order for each open
cover $\mathcal{U}$ of $I$.
\end{thm}

\begin{proof}
We follow the idea in \cite{FS91}. Let $\mathcal{U}$ be an open cover of $I$ with
Lebesgue number $\delta$ and $n \in \mathbb{N}$. For any
$\vec{t}=(t_1<t_2<\cdots<t_n)\in\mathbb{Z}_+^n$, it is
well known that
$$N\left(\bigvee^n_{i=1}T^{-t_i}\mathcal{U}\right)\leq S\left(\vec{t},f,\frac{\delta}{2}\right),$$
where $S\left(\vec{t},f,\frac{\delta}{2}\right)$ is the minimal
cardinality of $\left(\vec{t},f,\frac{\delta}{2}\right)$-spanning sets.
Recall that a set $E \subset I$ is called a $\left(\vec{t},f,\varepsilon\right)$-spanning set,
if for any $x \in I$, there exists some $y \in E$ such that
$|f^{t_i}(x)-f^{t_i}(y)|<\varepsilon$ for $1 \leq i \leq n$. Let
$\varepsilon=\frac{\delta}{4}$ and $U$ and $x_1,\ldots,x_k$ be as in
Lemma \ref{Smital-lemma}. Let $\{K_i\}_{i=1}^s$ be pairwise disjoint
set with diam$(K_i)<\varepsilon$ for any $i$, and $K_1\cup\cdots\cup
K_s=I\backslash U$. Assign to any $x\in I$ an itinerary
$\alpha_{\vec{t}}(x)=\{\alpha_{t_i}(x)\}_{i=1}^n$ such
that $\alpha_{t_i}(x)=K_j$ if $f^{t_i}(x)\in K_j$. If $f^{t_i}(x)\in
U$, let $M(t_i)$ be the maximal subinterval of the set of nonnegative
integers such that $t_i\in M(t_i)$ and $f^k(x)\in U $, for all $k
\in M(t_i)$. Then by Lemma \ref{Smital-lemma}, there exists some $r\in\{1,\ldots,k\}$
such that $|f^k(x)-f^k(x_r)|<\varepsilon<\frac{\delta}{2}$ for any
$k \in M(t_i)$. Put $\alpha_{t_j}(x)=r$ for any $t_j\in M(t_i)$.

It is easy to see that for any $x,y \in I$,
$\alpha_{\vec{t}}\,(x)=\alpha_{\vec{t}}\,(y)$
implies $|f^{t_i}(x)-f^{t_i}(y)|<2\varepsilon=\frac{\delta}{2}$ for
all $1\leq i \leq n$. So we get
$$S\left(\vec{t},f,\frac{\delta}{2}\right)\leq C\left(\vec{t}\,\right),$$
where $C\left(\vec{t}\,\right)$ is the number of all the possible
codes $\alpha_{\vec{t}}\,(x)$. By Lemma \ref{Sarkovskii-lemma},
there exists an integer $q>0$ such that the number of points of  an
arbitrary trajectory lying outside $U$ is less than $q$.
Consequently, every code $\alpha_{\vec{t}}\,(x)$ consists of at most
$2q+1$ blocks, and each block is formed by only one of the symbols
$1,\ldots,k,K_1,\ldots,K_s$ (with possible repetitions). Therefore,
$$N \left(\bigvee^n_{i=1} T^{-t_i}\mathcal{U}\right)\leq C\left(\vec{t}\,\right)\leq (k+s)^{2q+1}n^q.$$
We remark that the choice of $k,s$ and $q$ depend only on $\mathcal{U}$ but not $\vec{t}$. Thus,
$$p^*_{I,\mathcal{U}}(n) \leq (k+s)^{2q+1}n^q$$
for all $n\geq 1$
\end{proof}

\begin{rem}
Recall that a space $X$ is called a {\em tree} if it is a connected space
that is a union of finite  number of intervals, but does not contain
a subset homeomorphic to a circle. If one is acquainted  with the
dynamical properties of the tree maps, it is not hard to see that
Theorem  \ref{interval-stro-mix-eq1} and  Theorem
\ref{interval-poloynomial-order} hold for tree maps, but in
Theorem \ref{interval-stro-mix-eq1}(2), the number ``$\,2$" should
be replaced by a sufficient large integer which is associated with
the number of endpoints of the tree.
\end{rem}
\subsection*{Acknowledgement}
The author would like to thank Prof. Xiangdong Ye
for the careful reading and helpful suggestions.
This work was supported in part by the National Natural Science Foundation of China (Nos.\@ 11001071, 11071231).

redpajama_set_name: RedPajamaArXiv


text:
Jeff Clark & Marc Tharp
Presented by Davis Arts Council at Kenley Amphitheater, Layton UT
A Kenley Amp favorite! Don't miss Jeff Clark & Marc Tharp perform some of your favorite hits.
Jeff Clark and Marc Tharp return to the Kenley Amphitheater to present another offering of their acoustic blend of piano, vocals, and guitar. Join us for a laid back summer evening performance of recognizable songs from many genres, including folk, jazz, pop, adult contemporary, and more.
Jeff Clark is a singer, songwriter, and recording artist who has been entertaining audiences for years from backyards to concert halls. He is currently the frontman for Toast, a tribute to the band Bread. Jeff
Jeff Clark is a singer, songwriter, and recording artist who has been entertaining audiences for years from backyards to concert halls. He is currently the frontman for Toast, a tribute to the band Bread. Jeff and his beautiful wife Trudi and 3 kids reside in Syracuse, Utah.
Marc Tharp is a long-time spectator, first time performer at Kenley. He has played piano in the Dixie College and University of Utah jazz bands. He currently plays piano as a volunteer at the Joseph Smith Memorial Building in Salt Lake City. Marc resides with his wife Stacy and three daughters in Syracuse. Marc is hoping that his music will one day allow him to work his true passion as a fixed asset accountant.
Tickets for this event are free but required. Tickets will be available on Monday, September 14 at 10 a.m. tickets.davisarts.org
Kenley Amphitheater
All parking options are free! The closest places are the Layton City Building/Surf 'n Swim parking lot, along the road on Wasatch Drive, or at Layton High School.
If your event is being affected, canceled, or postponed due to the response to the Coronavirus (COVID-19) crisis, please provide the details below.:
Face masks are required in and around The Kenley Amphitheater. Hand sanitizing stations will be located throughout the venue. Social distancing measures will be in place. For more info, please visit our website: www.davisarts.org/covid/
MORE FROM Davis Arts Council
VIRTUAL Chamber Music Mini Series
Presented by Davis Arts Council at Online/Virtual Space
Jan 8 - Mar 5, 2021
CONNECT WITH Davis Arts Council

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text:
Home » Ostrobothnian Chamber Orchestra » Ostrobothnian Chamber Orchestra
Ostrobothnian Chamber Orchestra
The Ostrobothnian Chamber Orchestra (OCO), founded by its honorary conductor Juha Kangas in 1972, has been a professional orchestra since 1989. The homogeneous sound and dynamic impact of the orchestra are a result of years of playing together. Sakari Oramo served as artistic director of the orchestra between 2013 and 2019. His successor in this position is Malin Broman.
The orchestra's repertoire covers periods in the history from the baroque to the present day. The OCO has made an outstanding contribution to the promotion of contemporary Finnish music. Many Finnish composers have written for and dedicated works to the Orchestra. The OCO has also close contacts with composers both in Scandinavia and the Baltic countries and has premièred more than 160 works to date.
The OCO was awarded the Nordic Council Music Prize in 1993 and has since been the recipient of many other distinctions. The OCO performs regularly with top international soloists and has released more than 60 recordings. Its foreign tours have taken it to several European countries, Japan and USA.
For further information, please visit the Ostrobothnian Chamber Orchestra's website.
Russian Masquerade
BIS-2365 SACD
Arrangements old and new for string orchestra
Mozart - Complete Works for Flute and Orchestra
Nordgren - Transe-Choral
BIS-1356 CD
Shostakovich - Suite on Finnish Themes
Finntastic - Musical Souvenirs from Finland
Nordgren - Towards Equilibrium
BIS-826 CD
Mozart - Complete Music for Solo Flute and Orchestra


In this Q&A, Matt McLarty, team lead at the API Academy, discusses microservices, the API economy and how the ongoing effort to make it easier for developers to create software might even lead to the emergence of "software factories".
Matt McLarty is an experienced software architect who leads the API Academy at CA Technologies. He helps organizations with their strategy and architecture for APIs, microservices and enterprise integration. Matt recently co-authored the book Microservice Architecture for O'Reilly, with his API Academy colleagues.


Ellsworth Community College provides a variety of academic support services to help students be successful. Most of these services are housed in The Hub, located in the lower level of Kruse Main Hall (entrance via ramp on the south side of the building). The goal of The Hub is to provide a setting in which students can study/relax/collaborate as well as find all the academic support services they need. The Hub offers group and individual study areas, study pods, a small conference room for groups, and a large conference room. We've listed several resources below that will be helpful to you. If you aren't sure which of these departments or services to use, email eccinfo@iavalley.edu and we'll put you in touch with the right person!
Academic Advising – Each ECC student is assigned an academic advisor, who assists in choosing courses and seeing that the appropriate coursework is taken to assure graduation, transfer to another college or university, or completion of a career program. Students should maintain regular contact with their academic advisor throughout the year and are invited to contact members of the Hub staff whenever needed. If you're unsure who your academic advisor is, ask a Hub staff member.
Accommodations for Students with Disabilities – ECC provides individualized assistance to students with disabilities who identify themselves and request help.
Athletic Study Tables – Coaches will let their student-athletes know when their mandatory study groups will meet during the academic year.
Confidential Resources – ECC partners with Crisis Intervention Service for sexual assault advocacy services. We are also able to provide a selection of off-campus counseling resources.
TRIO Student Support Services – TRIO SSS services help inspire and support ECC students in their pursuit of a college degree through one-on-one advising, financial literacy workshops, scholarship opportunities, and more.
Tutoring – During fall and spring terms, free peer tutoring provides assistance for students enrolled in specific ECC courses; our tutors are students who have successfully completed the course. Normal tutoring includes two 50-minute sessions per week, arranged one-on-one during regular Hub hours; drop-in tutoring is also available. Students are encouraged to seek assistance as soon as they experience difficulty in a course, as it may take up to two weeks to arrange a tutoring schedule. To be assigned a peer tutor, students should contact a faculty member, complete the online form, or visit with Student Success Specialist Wanda Koehrsen. Students interested in becoming a tutor should contact the course instructor and complete a tutor application form in the Hub.
Writing Center – The Writing Center provides students with one-on-one assistance with writing assignments for all classes. Students can get help at any stage of the writing process, and drop-in assistance is available. To schedule an appointment, visit the Hub, or contact Student Success Specialist Wanda Koehrsen.
Library – A variety of resources, including computer stations and online databases, are available in ECC's Osgood Library in Kruse Main Hall (main floor, above the Hub). Stop in or check the Library web page for details.
The Hub is zoned for social activity in the south area, a collaborative study area in the center, and a quiet study zone at the east end of the facility.
Use of electronic devices is allowed as long as it's not disruptive to others.
Food and beverages are allowed in the social zone.
Students will use their student ID card to sign in each time they visit.


Here are a few 'large' (more then two employees) companies (with offices) which are using kickstarter to get the money to run their company (with expenses), while building a game project. This, to me, is a terrible idea bound for failure. But I can't quite put my finger on exactly why. This sets a bad example for anyone just starting out. Kickstarter works best when it's used to fund projects, not companies. But then on the other hand, maybe funding is funding.
Employee compensation is the largest single cost of a company. Between salary and healthcare. All those people in the picture? That's a lot of expense. Although it's quite possible that some of the people in this picture were let go with the cancellation of Ghost Recon Social game.
Again, a ton of concept art. Someone has spent a lot of time (and money), perhaps personal, perhaps company, to create great looking concept art. Although some of it is reused from a previous title, they also indicate having a large team. And therefore large expenses.
The obvious counter example. They did really, really well without even naming a project at all. Their kickstarter sums it up really well "The world of video game design is a mysterious one." There are surprise successes, and terrible failures. Do not expect the same results, even if you put in the same effort. Even if you could make witty videos, you won't be able to get the same results. One thing I notice in watching the videos, Tim Shafer understands that this is funding for a project.
Being but one person there is no way to find all the successful and unsuccessful kickstarter projects, nor determine which ones were funded at what stage. But what I can say to the beginner, the person like myself looking to complete their first or second project, understand that a Game Idea != Game, and that Project != a company. Err on the side of caution, and fund a project with a kickstarter. If the project does well, use the revenue to then start a company.
Xcom: Sometimes the Redshirts die.


"The form we take to exist arises from framing nothing" (Spencer-Brown, 1969, p. 105).
Quehenberger's artwork presented here connects the ancient theory of everything with 20th century physics by following the truly Platonic aim for visualizing the ideal world by means of digital arts. Plato's triangles cradled by the wet-nurse-of- becoming becomes alive in the higher dimensional realms of quantum physics and Spencer-Brown's circles gain a new relevance.
1. See her thesis, On the Hermeneutics of the Penrose Patterns, where she hypothesizes that the geometry of 5-dimensional space is the ontologic basis that existence precedes.


text:
Padirac (en francès Padirac) és un municipi francès, situat al departament de l'Òlt i a la regió d'Occitània.

Demografia

Administració

Referències

Municipis de l'Òlt

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When using XtraReports for LightSwitch based on a filtered query, the default behavior for the XtraReports preview screen is to show a textbox in which you can type the value of the parameter. With Date values the XtraReports has a calendar dropdown, but for numeric or string data like identifiers or codes that refer to primary keys in lookup tables, we have to provide some code to build a combobox list to facilitate life for the user.
This article is a follow-up on my article on how to create a 3 column report.
The report preview screen uses a numeric up-down control to enter a category Id. Here the user would appreciate an combobox with a list of the names and not the id's.
The solution I provide is based on the XtraReports tutorial Provide Custom Editors for Report Parameters. Some lines of coding are needed to replace the standard up-down box by a DevExpress ComboBoxEdit control. This control is installed together with the XtraReports controls.
The code in the example under step 3. will illustrate this.
We need to add code in the preview screen to generate a local list object with values from the categories table that can be used by the ComboBoxEdit control.
In the solution explorer right click on the report preview screen and select View Screen Code.
define a customize parameters event handler, in here the ComboBoxEdit will be created.
In the Activated event populate the list with the categories entities.
When we open the preview screen, click on the ComboBoxEdit control the list with 4 Categories opens.
The XtraReports preview screen can be based on a filtered query, the parameters of which are presented automatically in a text box. With a little coding and a ComboBoxEdit control we can then make the selection of the parameter values user-friendlier.


text:
Addiction Angel Organization Empowers Those Struggling With Addiction
The Addiction Angel Organization knows that the heroin epidemic has made its mark all over the country, and one place the mounting rates of death as a result of the heroin epidemic has made plenty of headlines is the state of New York. At one point reports showed that New York experienced more deaths due to heroin overdose than from homicide, while across the country some statics have shown that the overdose death rate has quadrupled in recent years. Staten Island in New York is among one of the hardest hit communities statewide, with record numbers of overdose deaths resulting from the heroin epidemic. State officials and law enforcement agencies everywhere are beginning to revolutionize the methods they use to effectively address the heroin epidemic in their areas, with more compassionate programs being initiated with the intention of supporting those who struggle most. As heroin and opiate addiction has become [...]
Addiction Angel	2016-10-21T13:29:46+00:00	October 21st, 2016|
Staten Island Addiction Treatment
Staten Island Addiction and addiction all over the country has been increasing steadily from the 1990s. The largest killers were the narcotic drugs, prescription narcotic pain killers, cocaine and heroin. These drugs accounted for nearly half of the deaths in this particular study. Poisoning deaths, of which drug poisoning accounts for about 95%, now run behind only auto accidents as the leading cause of accidental injury death in the US. The demographics are also interesting. Drug overdose deaths are twice as common in men as in women meaning about 2/3 of these deaths. The highest rates are in the age group between 35 and 54 years of age, accounting for about 60% of all the poisoning deaths in the year 2004. The study also found the rates among white people rising much faster than in blacks while rates among Hispanics went up much less than among non-Hispanics. Among American Indian Alaska [...]
Addiction Angel	2016-10-17T10:29:12+00:00	October 17th, 2016|
Battling Addiction
We care about people battling addiction because of the feeling of hopelessness, and despair they undergo. All of us have commitments to fulfill in life, and we should all strive in this excellence by helping others, including those battling addictions. I interviewed about 8 people who were on their road to recovery from addiction. I also used discussion forums to get some of these ideas. From my findings, I came to realize that most people in the society partly blame people with addiction problems and others do not think it as a problem that should be shared by all societal members. Individuals in the addition network requires all the help they can get, when it comes from members of the society it carries more weight given that some of our actions contribute to their addiction problems. You must change the way you think to change your life, this paves way [...]
Addict Misconceptions
Addicts are Lazy People. Addicts may appear lazy but they aren't always lazy people. They are normally very encouraged to satisfy their cravings. It actually takes a large quantity of energy to plan and ensure there suffices of the drug through the remainder of the day and for tomorrow and the day after that. Whether it is making certain you have computer system access for pornography or that you have enough alcohol allocated out for the next day. Planning to get and allocate your drug to make sure you have enough of it, while concealing it from those around you, takes a lot of effort. When an addict becomes sober, they are quite surprised to discover a lot of leisure time and do not know ways to utilize it due to the fact that it has actually always been taken in by addicting behaviors. This is why it is very [...]
Addiction Angel	2016-10-07T11:36:08+00:00	October 7th, 2016|
Staten Island Substance Abuse
Staten Island substance abuse is a growing menace in the New York area. Anyone can be a victim of it, disregarding age, gender and ethnicity. Substance abuse can be broadly defined as the situation in which the use of a substance causes a detrimental effect on the physical and mental condition of an individual. Further, substance abuse can also have devastating effect on the financial, social and even legal standing in the society of a person. The 'substance' in substance abuse can be almost anything. It can be either legal or illegal. Even the non prescribed use of prescriptive medicines has and can be termed as substance abuse. In cases of substance abuse, injections, inhaling and oral usage of the substance is common. However, sniffing and smoking of the said substance is also seen in some cases. Drug Addiction in Staten Island Maybe the most [...]
Addiction Angel	2016-09-30T09:35:01+00:00	September 30th, 2016|
Staten Island Heroin Abuse
Addiction to pain killers is running rampant in the state of New York and has resulted in the recent explosion of Staten Island heroin abuse. Heroin importation has dramatically increased over the past decade. Heroin is cheaper than prescription opioids and has similar physical effects on the user. Many patients who are no longer able to obtain prescription pain killers from a legitimate medical practitioner will simply turn to heroin to avoid withdrawal symptoms and relieve pain. Heroin In Staten Island, NY Heroin is a drug that is both extremely illegal and very very addictive. It comes from opium from the poppy plant before it is refined to morphine, then further changed chemically to become heroin. Despite its extremely bad reputation for an easy overdose, heroin continues to be a commonly abused drug in Staten Island, NY. Heroin is sold and used in a number of forms including white or brown powder, a [...]
Addiction Angel	2016-09-29T19:11:52+00:00	September 29th, 2016|Tags: Heroin Abuse Staten Island, Heroin In Staten Island, Heroin Staten Island NY, Staten Island Heroin Abuse|
Family Involvement is Important in Substance Abuse Treatment
For family and friends of drug- or alcohol-addicted individuals, addressing the addiction is one of the most difficult aspects of helping the addicted person seek treatment. Often, over time, daily family involvement has only managed to enable the addict. Family members frequently do not know how to bring up the issue of addiction therapy, and opt to ignore the problem for fear of pushing their loved one away during a confrontation or intervention. These are legitimate concerns, and while families should understand that approaching their loved one should be a gentle and supportive process, they also need to understand that most patients seek substance abuse treatment because of positive family involvement and intervention. Prior to Substance Abuse Treatment and Intervention Each family is different, and the best way to approach family involvement with addiction therapy will differ with every person. There are counselors in your area who are trained to [...]
7 Tips for Mothers of Adult Addicts
Suzanne Degges-White, Ph.D. Before a child is born, most parents are already carrying a heavy burden. They recognize that a great deal of responsibility comes with bringing a child into this world and typically believe that every choice they make from the moment of conception onwards is going to play a role in how their child turns out. For the most part, they may be right. Some choices made during pregnancy can definitely influence a child's physiology and future health. Consuming alcohol, using drugs and some medications, eating nutritiously, among others, can all influence the health of an unborn child. However, as of the moment of conception, some unique personalitycharacteristics and physiological potentials are already pretty much fixed, regardless of pre- and post-birth parenting choices that are made. If you are the mother or father of an adult child who is not making the choices that are necessary for [...]
4 Life-Changing Ways to Release the Shame of Addiction
When we are in shame, we don't see the big picture; we don't accurately think about our strengths and limitations. We just feel alone, exposed and deeply flawed. ~ Brené Brown Do you tend to feel more guilt or shame when you feel you have done something wrong? Brené Brown's book, I Thought It Was Just Me talks about shame and how it affects our lives. I found her section about addiction particularly interesting. We all feel shame at some point in our lives. As individuals we have a tendency to respond with either shame or guilt in any given situation. It makes us feel humiliated and uncomfortable because of something we did wrong. We lose our self-respect. We may feel mortified and embarrassed and we may want to pull back and just be alone. Addiction and shame go hand in hand. It is hard to understand where one starts and the [...]


text:
Home US Army Combined Arms Center (USACAC) Repository Social intelligence and emotional intelligence: a literature review of their similarities and...
Social intelligence and emotional intelligence: a literature review of their similarities and differences.
7.pdf [1.14 MB]
Collection USACAC Repository
Title Social intelligence and emotional intelligence: a literature review of their similarities and differences.
Author Rodman, Joseph
Abstract This is the second of three papers that the Human Dimension Capabilities Development Task Force (HDCDTF) will be producing on social intelligence (SI) in 2016. The first paper, published in February 2016, discussed the historical and theoretical development of SI. It sketched an outline of existing research that has attempted to define SI. The paper provided readers with the appropriate context--including important findings, challenges, and limitations that others have revealed and encountered in the scientific and popular literature--to consider as the Army, and particularly its Human Dimension efforts, develops and executes its interest in the concept. The second paper, presented here, examines how scholars and practitioners from a variety of fields have attempted to distinguish SI from the seemingly similar concept of emotional intelligence (EI). Both concepts have received significant attention from a broad audience, including the Army; however, common conceptualizations of each are often conflated. In order for the Army to approach its interest in SI and EI deliberately and effectively, it will be important to discuss how the relationship between SI and EI has been described in the scientific and popular literature. This paper addresses these concerns and provides a waypoint from which the Army can expand its interest and understanding of SI and EI and their potential relationship to each other. The third paper, to be published in August 2016, will investigate methods that the Army might consider implementing or learning from that attempt to (1) assess social intelligence in individuals and (2) develop or improve social intelligence (or components of social intelligence) in individuals. Supporting Objective #1.2 ("Social Intelligence") in The Army Human Dimension Strategy, which charges the Army to, "Develop trusted Army Professional as effective team members who thrive in complex social environments, adapt to diverse cultures, communicate effectively, and build relationships" provides the catalyst for the series and serves as the ultimate goal of its research. Given the success of traditional measures of cognitive intelligence (IQ, most notably) in predicting certain outcomes of human performance, it is logical and valuable for the Army to pursue interest in a similar mediating construct for leadership and performance that might help it better manage its personnel and prepare for the changing operational environment of the future. SI and EI are two such constructs that have received significant attention in the scientific community, private industry, popular literature, and the Army. Both represent increased appreciation of the importance of what is variously called "people skills" or "soft skills" in organizational effectiveness and individual performance. At the heart of this concern is identifying and developing skills or personal characteristics that enhance the effectiveness of individuals' interaction with others in order to improve organizational, leader, and team outcomes. A key underlying issue--and the main subject of this paper--is the conceptual similarity between SI and EI. This is largely a product of the ambiguity of both constructs and a lack of scientific consensus of what they each consist of. Naturally, this creates considerable confusion in delineating where SI ends and EI begins. Such an understanding is critical if the Army is to move forward in a meaningful way with its interest in either or both concepts. In order to properly contextualize how SI and EI may interrelate, this paper first provides an abbreviated examination of EI similar to that presented on SI in the previous white paper.
Series Human Dimension Capabilities Development Task Force (HDCTF), White Paper
Publisher Fort Leavenworth, KS : Mission Command - Capabilities Development and Integration Directorate (CDID),
Date, Original 2016-05
Format PDF: Adobe Reader required; 49 p.; 1.14 MB.
Release statement Approved for public release; Distribution is unlimited
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text:
Home / Case Studies / Axxon PSIM face recognition provided security for the Tunisian presidential elections
Axxon PSIM face recognition provided security for the Tunisian presidential elections
Presidential elections were held in the Republic of Tunisia in November of 2014. Axxon PSIM face recognition was one of the most important security components for polling stations.
In June 2014, the electoral commission of the Republic of Tunisia announced a tender for creating a security system for polling stations during the presidential elections. The elections followed the Arab Spring, which involved military action and revolutionary events in the Middle East, including Tunisia. Due to the country's instability and the threat of terrorist infiltration at the polling stations, security was a major priority during the elections.
To prevent a terrorist threat, AxxonSoft's partner Synergie developed a unique solution for Tunisia based on the Axxon PSIM facial recognition system. This solution is designed for use in public security systems and at sites with elevated access control requirements. The system analyzes faces in images from video surveillance cameras and compares them with faces in a database (for example, a database of wanted persons). Depending on the degree of similarity, it follows a specified script to warn the operator, alert emergency services, and so on. This proposal was submitted to the tender and accepted by the electoral commission of the Republic of Tunisia.
The Tunisian presidential elections were held on November 23, 2014. Synergie deployed the security system at 150 polling stations throughout the country. The system included 269 cameras transmitting live images that were analyzed by Axxon PSIM face recognition in real time. A database was set up that contained about 1000 persons wanted on suspicion of involvement in terrorist organizations. As a result, the electoral commission's primary goal was accomplished, and the presidential elections were held without any security violations.
Altre Informazioni Importanti
"Il Sistema è così intuitivo che permette di operare con facilità anche a chi ha poca familiarità con i computer. Questo è uno dei vantaggi chiave perché ci permette di equipaggiare anche le filiali regionali con lo stesso sistema"
Oleg Pronko, Chief Specialist della sezione tecnica di sicurezza del dipartimento di sicurezza Raiffeisen
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Well, I saw this project and thought I need to do that and share it with my followers.
It's super cute, right? This project used a lot of supplies which I usually don't use a my recorded projects but I had to share it. I changed this project up a bit with the embossing paste and I love the texture it creates plus these colors together!
I hope you enjoyed today's project and give it a try!
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The price is right with Lantana. It's one of our top picks for those who have a little more budget, thanks to its excellent value, great location across the river from the old town and an exceptionally pretty design.
The rooms at Lantana are almost a carbon copy of La Residencia, right down the pattern floor tiles, patina copper fixtures and basement pool/gym. It's so similar that we asked if they were the same owner (apparently not). Opened in 2015, Lantana has a slight edge. It is just a bit spiffier and luxurious, giving guests a feeling that this is even more of a steal.
Lantana is located on the other side of the river, on the northern edge of An Hoi Islet looking back at the mainland. The pseudo French colonial exterior with shades of antebellum is twee, but who cares about the exterior because the rooms are tasteful and elegant, balancing decor with comfort. Enter a room and it feels like a calming exhale. The room we saw had light grey walls, white linens, dark wood furniture, blue quilted throw pillows and bed runner. Watercolour paintings adorned the wall and the bathroom had everything a traveller needed: contemporary fixtures, a blue tiled shower stall with dispensers of soaps and lotions. Lux amenities included large wall mounted flatscreen TV, desk, minibar, safety box, air-con and WiFi. If you can afford it, the deluxe river view room is worth it as you get your own private balcony.
Though they rate themselves a four-star, it's really an excellent three-star. Our only concern is how well everything stands the test of time, whether materials were chosen because they were good quality or because they were pretty. The room rate includes a buffet breakfast, an L-shaped underground swimming pool, well-equipped gym, a three times a day shuttle to An Bang and a "private" section of the beach with chairs. The terrace at the front of the house seals this deal. No bicycles are provided -- it's an easy seven-minute walk to the old town. Head east, cross the pedestrian bridge and you'll be right at the landmark Japanese Bridge.
Check rates at Lantana Boutique Hotel on Booking.com.