1 Introduction
Shadowing is an important concept in the theory of dynamical systems. It was initially introduced in the context of hyperbolic differentiable dynamics [Reference Anosov3, Reference Bowen5] and generally refers to a situation in which coarse orbits, or pseudo-orbits, are approximated by true orbits. For background on the shadowing theory, we refer the reader to the monograph [Reference Aoki and Hiraide4]. In [Reference Morales15], by localizing global shadowing into pointwise shadowings, Morales introduced the notion of shadowable points. In [Reference Kawaguchi11], a shadowable point that is also an entropy point of a certain type is characterized by the structure of pseudo-orbits. In this article, we discuss the relationship among the shadowable points, sensitive points, and entropy points. For instance, one of the main results is that if the set of shadowable points is dense in the phase space, then any point located in the interior of the set of sensitive points is an entropy point (Theorem 1.2). Another result is that if the topological entropy is zero, then the denseness of the set of shadowable points is equivalent to almost chain continuity (Theorem 1.3). In contrast to [Reference Kawaguchi11], a particular focus of this article is on the terminal chain components. We also present a counter-example to a question raised by Ye and Zhang [Reference Ye and Zhang17] regarding entropy points.
We begin by defining shadowable points. Throughout, X denotes a compact metric space endowed with a metric d.
Definition 1.1 Let
$f\colon X\to X$
be a continuous map and let
$\xi =(x_i)_{i\ge 0}$
be a sequence of points in X. For
$\delta>0$
,
$\xi $
is called a
$\delta $
-pseudo orbit of f if
$d(f(x_i),x_{i+1})\le \delta $
for all
$i\ge 0$
. For
$\epsilon>0$
,
$\xi $
is said to be
$\epsilon $
-shadowed by
$x\in X$
if
$d(f^i(x),x_i)\leq \epsilon $
for all
$i\ge 0$
. We say that
$x\in X$
is a shadowable point for f if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon $
-shadowed by some
$y\in X$
. We denote by
$Sh(f),$
the set of shadowable points for f.
For a topological space Z, a subset S of Z is called a
$G_\delta $
-subset of Z if S is a countable intersection of open subsets of Z.
Remark 1.1
-
(1) Let
$f\colon X\to X$
be a continuous map. For any
$j,l\ge 1$
, let
$S_{j,l}$
denote the set of
$x\in X$
such that there is an open neighborhood U of x for which every
$\frac {1}{j}$
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0\in U$
is
$\frac {1}{l}$
-shadowed by some
$y\in X$
. We see that
$S_{j,l}$
is an open subset of X for all
$j,l\ge 1$
and therefore,
$$\begin{align*}Sh(f)=\bigcap_{l\ge1}\bigcup_{j\ge1}S_{j,l}; \end{align*}$$
$Sh(f)$
is a
$G_{\delta }$
-subset of X.
-
(2) A continuous map
$f\colon X\to X$
is said to have the shadowing property if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-pseudo orbit of f is
$\epsilon $
-shadowed by some point of X, which is equivalent to
$X=Sh(f)$
(see [Reference Kawaguchi8, Lemma 2.4]).
Sensitivity is a characteristic feature of chaotic dynamical systems. It is an element of some formal definitions of chaos and intuitively means that an arbitrarily small difference in initial conditions can be amplified to be a significant difference in later states (see, e.g., [Reference Glasner and Weiss7] for an in-depth look at the concept of sensitivity). The formal definition of sensitive points is as follows.
Definition 1.2 Given a continuous map
$f\colon X\to X$
and
$r>0$
,
$x\in X$
is called an r-sensitive point for f if for any
$\epsilon>0$
, there are
$y,z\in X$
and
$i\ge 0$
such that
$$\begin{align*}\max\{d(x,y),d(x,z)\}\le\epsilon \end{align*}$$
and
$d(f^i(y),f^i(z))>r$
. We denote by
$\mathrm{Sen}_r(f)$
the set of r-sensitive points for f. We also define the set
$\mathrm{Sen}(f)$
of sensitive points for f by
$$\begin{align*}\mathrm{Sen}(f)=\bigcup_{r>0}\mathrm{Sen}_r(f). \end{align*}$$
For a continuous map
$f\colon X\to X$
, a subset S of X is said to be f-invariant if
$f(S)\subset S$
.
Remark 1.2
-
(1) For any continuous map
$f\colon X\to X$
and
$r>0$
,
$\mathrm{Sen}_r(f)$
is a closed f-invariant subset of X. -
(2) A continuous map
$f\colon X\to X$
is said to be sensitive if
$X=\mathrm{Sen}_r(f)$
for some
$r>0$
.
Next, we recall the definition of entropy points. Note that the positive topological entropy is another characteristic feature of chaotic dynamical systems. The notion of entropy points is obtained by a concentration of positive topological entropy at a point [Reference Ye and Zhang17].
Let
$f\colon X\to X$
be a continuous map. For
$n\ge 1$
and
$r>0$
, a subset E of X is said to be
$(n,r)$
-separated if
$$\begin{align*}\max_{0\le i\le n-1}d(f^i(x),f^i(y))>r \end{align*}$$
for all
$x,y\in E$
with
$x\ne y$
. Let K be a subset of X. For
$n\ge 1$
and
$r>0$
, let
$s_n(f,K,r)$
denote the largest cardinality of an
$(n,r)$
-separated subset of K. We define
$h(f,K,r)$
and
$h(f,K)$
by
$$\begin{align*}h(f,K,r)=\limsup_{n\to\infty}\frac{1}{n}\log{s_n(f,K,r)} \end{align*}$$
and
$$\begin{align*}h(f,K)=\lim_{r\to0}h(f,K,r). \end{align*}$$
The topological entropy
$h_{\mathrm {top}}(f)$
of f is defined by
$h_{\mathrm {top}}(f)=h(f,X)$
.
Definition 1.3 Let
$f\colon X\to X$
be a continuous map. For
$x\in X$
, let
$\mathcal {K}(x)$
denote the set of closed neighborhoods of x.
-
(1)
$\mathrm{Ent}(f)$
is the set of
$x\in X$
such that
$h(f,K)>0$
for all
$K\in \mathcal {K}(x)$
. -
(2) For
$r>0$
,
$\mathrm{Ent}_r(f)$
is the set of
$x\in X$
such that
$h(f,K,r)>0$
for all
$K\in \mathcal {K}(x)$
. -
(3) For
$r,b>0$
,
$\mathrm{Ent}_{r,b}(f)$
is the set of
$x\in X$
such that
$h(f,K,r)\ge b$
for all
${K\in \mathcal {K}(x)}$
.
Remark 1.3 The following properties hold:
-
•
$\mathrm{Ent}(f)$
,
$\mathrm{Ent}_r(f)$
,
$r>0$
, and
$\mathrm{Ent}_{r,b}(f)$
,
$r,b>0$
, are closed f-invariant subsets of X, -
•
for all
$$\begin{align*}\mathrm{Ent}(f)\supset \mathrm{Ent}_r(f)\supset \mathrm{Ent}_{r,b}(f) \end{align*}$$
$r,b>0;$
-
• for any closed subset K of X and
$r>0$
, if
$h(f,K,r)>0$
, then
$K\cap \mathrm{Ent}_r(f)\ne \emptyset $
; -
• for any closed subset K of X and
$r,b>0$
, if
$h(f,K,r)\ge b$
, then
$K\cap \mathrm{Ent}_{r,b}(f)\ne \emptyset $
.
Remark 1.4 Note that
$\mathrm{Ent}_r(f)\subset \mathrm{Sen}_r(f)$
for all
$r>0$
.
Chain components, which appear in the (so-called) fundamental theorem of dynamical systems by Conley, are essential objects for a global understanding of dynamical systems [Reference Conley6]. Let us recall the definition of chain components.
Definition 1.4 Given a continuous map
$f\colon X\to X$
and
$\delta>0$
, a finite sequence
$(x_i)_{i=0}^{k}$
of points in X, where
$k>0$
is a positive integer, is called a
$\delta $
-chain of f if
$d(f(x_i),x_{i+1})\le \delta $
for every
$0\le i\le k-1$
. For any
$x,y\in X$
, the notation
$x\rightarrow y$
means that for every
$\delta>0$
, there is a
$\delta $
-chain
$(x_i)_{i=0}^k$
of f with
$x_0=x$
and
$x_k=y$
. The chain recurrent set
$CR(f)$
for f is defined by
$$\begin{align*}CR(f)=\{x\in X\colon x\rightarrow x\}. \end{align*}$$
We define a relation
$\leftrightarrow $
in
$$\begin{align*}CR(f)^2=CR(f)\times CR(f) \end{align*}$$
by: for any
$x,y\in CR(f)$
,
$x\leftrightarrow y$
if and only if
$x\rightarrow y$
and
$y\rightarrow x$
. Note that
$\leftrightarrow $
is a closed equivalence relation in
$CR(f)^2$
and satisfies
$x\leftrightarrow f(x)$
for all
$x\in CR(f)$
. An equivalence class C of
$\leftrightarrow $
is called a chain component for f. We denote by
$\mathcal {C}(f)$
the set of chain components for f.
Remark 1.5 The following properties hold:
-
•
$CR(f)=\bigsqcup _{C\in \mathcal {C}(f)}C$
, a disjoint union; -
• every
$C\in \mathcal {C}(f)$
is a closed f-invariant subset of
$CR(f)$
; -
• for all
$C\in \mathcal {C}(f)$
,
$f|_C\colon C\to C$
is chain transitive, i.e., for any
$x,y\in C$
and
$\delta>0$
, there is a
$\delta $
-chain
$(x_i)_{i=0}^k$
of
$f|_C$
with
$x_0=x$
and
$x_k=y$
.
Following [Reference Akin, Hurley and Kennedy2], we define the terminal chain components as follows.
Definition 1.5 Given a continuous map
$f\colon X\to X$
, we say that a closed f-invariant subset S of X is chain stable if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-chain
$(x_i)_{i=0}^k$
of f with
$x_0\in S$
satisfies
$d(x_k,S)=\inf _{y\in S}d(x_k,y)\le \epsilon $
. We say that
$C\in \mathcal {C}(f)$
is terminal if C is chain stable. We denote by
$\mathcal {C}_{\mathrm {ter}}(f)$
the set of terminal chain components for f.
The following lemma is from [Reference Kawaguchi10].
Lemma 1.1 [Reference Kawaguchi10, Lemma 2.1]
Let
$f\colon X\to X$
be a continuous map. For any
$x\in X$
, there are
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$y\in C$
such that
$x\rightarrow y$
.
Given a continuous map
$f\colon X\to X$
and
$x\in X$
, the
$\omega $
-limit set
$\omega (x,f)$
of x for f is defined as the set of
$y\in X$
such that
$$\begin{align*}\lim_{j\to\infty}f^{i_j}(x)=y \end{align*}$$
for some sequence
$0\le i_1<i_2<\ldots $
. Note that
$\omega (x,f)$
is a closed f-invariant subset of X and satisfies
$$\begin{align*}\lim_{i\to\infty}d(f^i(x),\omega(x,f))=0. \end{align*}$$
Since, we have
$y\rightarrow z$
for all
$y,z\in \omega (x,f)$
, there is an unique
$C(x,f)\in \mathcal {C}(f)$
such that
$\omega (x,f)\subset C(x,f)$
and so
$$\begin{align*}\lim_{i\to\infty}d(f^i(x),C(x,f))=0. \end{align*}$$
The first result of this article is the following theorem. For a subset S of a topological space Z,
$\overline {S}$
and
$\mathrm {int}[S]$
denote the closure and the interior of
$S,$
respectively.
Theorem 1.1 Let
$f\colon X\to X$
be a continuous map. For any
$x\in Sh(f)$
and
$r>0$
,
-
(1) if
$C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$x\in \mathrm{Sen}_r(f)$
, then
$x\in \mathrm{Ent}_s(f)$
for all
$0<s<r$
; -
(2) if
$x\in \mathrm {int}[\mathrm{Sen}_r(f)]$
, then
$x\in \mathrm{Ent}_s(f)$
for all
$0<s<r$
.
The next lemma is a consequence of the Baire category theorem. The proof is left as an exercise for the reader.
Lemma 1.2 Let Z be a complete metric space. Every sequence
$A_j$
,
$j\ge 1$
, of closed subsets of Z satisfies
$\overline {\mathrm {int}[\bigcup _{j\ge 1}A_j]}=\overline {\bigcup _{j\ge 1}\mathrm {int}[A_j]}$
.
By Theorem 1.1 and Lemma 1.2, we obtain the following theorem.
Theorem 1.2 Given a continuous map
$f\colon X\to X$
, if
$X=\overline {Sh(f)}$
, then
$$\begin{align*}\overline{\mathrm{int}[\mathrm{Sen}(f)]}\subset \mathrm{Ent}(f). \end{align*}$$
Proof Letting
$(r_j)_{j\ge 1}$
be a sequence of numbers with
$0<r_1>r_2>\ldots $
and
$\lim _{j\to \infty }r_j=0$
, we have
$$\begin{align*}\mathrm{Sen}(f)=\bigcup_{j\ge1}\mathrm{Sen}_{r_j}(f). \end{align*}$$
We take a sequence
$(s_j)_{j\ge 1}$
of numbers such that
$0<s_j<r_j$
for all
$j\ge 1$
. By Theorem 1.1, we have
$$\begin{align*}\mathrm{int}[\mathrm{Sen}_{r_j}(f)]\cap Sh(f)\subset \mathrm{Ent}_{s_j}(f) \end{align*}$$
for all
$j\ge 1$
. Note that an open subset U of X and a subset B of X satisfy
$$\begin{align*}U\cap\overline{B}\subset\overline{U\cap B}. \end{align*}$$
By
$X=\overline {Sh(f)}$
, we obtain
$$\begin{align*}\mathrm{int}[\mathrm{Sen}_{r_j}(f)]=\mathrm{int}[\mathrm{Sen}_{r_j}(f)]\cap\overline{Sh(f)}\subset\overline{\mathrm{int}[\mathrm{Sen}_{r_j}(f)]\cap Sh(f)}\subset \overline{\mathrm{Ent}_{s_j}(f)}=\mathrm{Ent}_{s_j}(f) \end{align*}$$
for all
$j\ge 1$
. With the aid of Lemma 1.2, we obtain
$$\begin{align*}\overline{\mathrm{int}[\mathrm{Sen}(f)]}=\overline{\mathrm{int}[\bigcup\nolimits_{j\ge1}\mathrm{Sen}_{r_j}(f)]}=\overline{\bigcup\nolimits_{j\ge1}\mathrm{int}[\mathrm{Sen}_{r_j}(f)]}\subset\overline{\bigcup\nolimits_{j\ge1}\mathrm{Ent}_{s_j}(f)}\subset \mathrm{Ent}(f); \end{align*}$$
thus, the theorem has been proved.
Given a continuous map
$f\colon X\to X$
, we say that
$x\in X$
is
-
• an equicontinuity point for f if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$y\in X$
with
$d(x,y)\le \delta $
satisfies
$$\begin{align*}\sup_{i\ge0}d(f^i(x),f^i(y))\le\epsilon, \end{align*}$$
-
• a chain continuity point for f [Reference Akin1] if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon $
-shadowed by x, i.e., satisfies
$$\begin{align*}\sup_{i\ge0}d(f^i(x),x_i)\le\epsilon. \end{align*}$$
We denote by
$EC(f)$
(resp.
$CC(f)$
) the set of equicontinuity (resp. chain continuity) points for f. By
$$\begin{align*}EC(f)=X\setminus \mathrm{Sen}(f)=\bigcap_{j\ge1}[X\setminus \mathrm{Sen}_{j^{-1}}(f)], \end{align*}$$
we see that
$EC(f)$
is a
$G_\delta $
-subset of X. It is easy to see that
$$\begin{align*}CC(f)=Sh(f)\cap EC(f) \end{align*}$$
and so
$CC(f)$
is a
$G_\delta $
-subset of X. We say that f is almost chain continuous if
$CC(f)$
is a dense (
$G_\delta $
-)subset of X.
Theorem 1.3 For a continuous map
$f\colon X\to X$
, if
$h_{\mathrm {top}}(f)=0$
, then the following conditions are equivalent:
-
(A)
$X=\overline {Sh(f)}$
; -
(B) f is almost chain continuous.
Proof The implication (A)
$\implies $
(B): Since
$Sh(f)$
is a
$G_\delta $
-subset of X, if
$X=\overline {Sh(f)}$
, then
$Sh(f)$
is a dense
$G_\delta $
-subset of X. Due to Theorem 1.2, if
$X=\overline {Sh(f)}$
and
$$\begin{align*}\mathrm{int}[\mathrm{Sen}(f)]\ne\emptyset, \end{align*}$$
then
$\mathrm{Ent}(f)\ne \emptyset $
and so
$h_{\mathrm {top}}(f)>0$
. By
$X=\overline {Sh(f)}$
and
$h_{\mathrm {top}}(f)=0$
, we obtain
$$\begin{align*}\mathrm{int}[\mathrm{Sen}(f)]=\emptyset; \end{align*}$$
therefore,
$EC(f)=X\setminus \mathrm{Sen}(f)$
is a dense
$G_\delta $
-subset of X. By the Baire category theorem, we conclude that
$$\begin{align*}CC(f)=Sh(f)\cap EC(f) \end{align*}$$
is a dense
$G_\delta $
-subset of X.
The implication (B)
$\implies $
(A) is a direct consequence of the inclusion
$CC(f)\subset Sh(f)$
.
Remark 1.6 By Theorem 1.3, if a continuous map
$f\colon X\to X$
satisfies
$h_{\mathrm {top}}(f)=0$
and
$X=Sh(f)$
, i.e., the shadowing property, then f is almost chain continuous. An alternative proof of this can be given as follows.
Proof By Theorem A.1, for a continuous map
$f\colon X\to X$
and
$x\in X$
,
$x\in CC(f)$
holds exactly if
$C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$CC(f|_{C(x,f)})=C(x,f)$
. Every continuous map
$f\colon X\to X$
satisfies
$CR(f|_{CR(f)})=CR(f)$
and
$$\begin{align*}\mathcal{C}(f)=\mathcal{C}(f|_{CR(f)})=\mathcal{C}_{\mathrm{ter}}(f|_{CR(f)}). \end{align*}$$
Assume that
$h_{\mathrm {top}}(f)=0$
and
$X=Sh(f)$
. By Theorem B.1,
$X=Sh(f)$
implies
$CR(f)=Sh(f|_{CR(f)})$
. If
$\mathrm{Sen}(f|_{CR(f)})\ne \emptyset $
, by taking
$x\in \mathrm{Sen}(f|_{CR(f)})$
and
${C\in \mathcal {C}(f)}$
with
$x\in C$
, we obtain
$C(x,f|_{CR(f)})=C$
and
$x\in \mathrm{Sen}_r(f|_{CR(f)})$
for some
$r>0$
. By
$C\in \mathcal {C}_{\mathrm {ter}}(f|_{CR(f)})$
and Theorem 1.1, we obtain
$x\in \mathrm{Ent}_s(f|_{CR(f)})$
for all
$0<s<r$
and so
$h_{\mathrm {top}}(f)>0$
, a contradiction. It follows that
$\mathrm{Sen}(f|_{CR(f)})=\emptyset $
; thus,
$EC(f|_{CR(f)})=CR(f)$
. We obtain
$$\begin{align*}CC(f|_{CR(f)})=Sh(f|_{CR(f)})\cap EC(f|_{CR(f)})=CR(f) \end{align*}$$
(cf. [Reference Moothathu14, Corollary 6]). This implies that
$CC(f|_C)=C$
for all
$C\in \mathcal {C}(f)$
. As a consequence, if
$h_{\mathrm {top}}(f)=0$
and
$X=Sh(f)$
, then
$$\begin{align*}CC(f)=\{x\in X\colon C(x,f)\in\mathcal{C}_{\mathrm{ter}}(f)\}. \end{align*}$$
On the other hand, by Theorem 1.1 of [Reference Kawaguchi12],
$X=Sh(f)$
implies that
$$\begin{align*}\{x\in X\colon C(x,f)\in\mathcal{C}_{\mathrm{ter}}(f)\} \end{align*}$$
is a dense
$G_{\delta }$
-subset X. By these conditions, we conclude that
$CC(f)$
is a dense
$G_{\delta }$
-subset X, i.e., f is almost chain continuous.
Let
$f\colon X\to X$
be a continuous map. For
$\delta ,r>0$
and
$n\ge 1$
, we say that two
$\delta $
-chains
$(x_i)_{i=0}^n$
,
$(y_i)_{i=0}^n$
of f is
$(n,r)$
-separated if
$d(x_i,y_i)>r$
for some
$0\le i\le n$
. Let
$$\begin{align*}s_n(f,X,r,\delta) \end{align*}$$
denote the largest cardinality of a set of pairwise
$(n,r)$
-separated
$\delta $
-chains of f. The next lemma is from [Reference Misiurewicz13].
Lemma 1.3 (Misiurewicz)
$$\begin{align*}h_{\mathrm{top}}(f)=\lim_{r\to0}\lim_{\delta\to0}\limsup_{n\to\infty}\frac{1}{n}\log{s_n(f,X,r,\delta)}. \end{align*}$$
Let
$f\colon X\to X$
be a continuous map. Following [Reference Ye and Zhang17], we define the set
$\mathrm{Ent}_{up}(f)$
of uniform entropy points for f by
$$\begin{align*}\mathrm{Ent}_{up}(f)=\bigcup_{r,b>0}\mathrm{Ent}_{r,b}(f). \end{align*}$$
Note that
$$\begin{align*}\mathrm{Ent}(f)=\overline{\mathrm{Ent}_{up}(f)}. \end{align*}$$
The following lemma can be proved by Lemma 1.3 and a similar argument as in the proof of Theorem 1.2 in [Reference Kawaguchi11]. Note that the “only if” part is rather trivial.
Lemma 1.4 For a continuous map
$f\colon X\to X$
and
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
,
$h_{\mathrm {top}}(f|_C)>0$
holds if and only if
$C\cap \mathrm{Ent}_{up}(f)\ne \emptyset $
.
The following lemma is proved in [Reference Kawaguchi11] (see [Reference Kawaguchi11, Lemmas 1.2 and 3.1]).
Lemma 1.5 Let
$f\colon X\to X$
be a continuous map. Let
$x,y\in X$
and
$r,b>0$
. If
$x\rightarrow y$
, then:
-
(1)
$x\in Sh(f)$
implies
$y\in Sh(f)$
; -
(2)
$x\in Sh(f)$
and
$y\in \mathrm{Ent}_r(f)$
imply
$x\in \mathrm{Ent}_s(f)$
for all
$0<s<r$
; -
(3)
$x\in Sh(f)$
and
$y\in \mathrm{Ent}_{r,b}(f)$
imply
$x\in \mathrm{Ent}_{s,b}(f)$
for all
$0<s<r$
.
By using Lemmas 1.1, 1.4, and 1.5, we obtain the following theorem.
Theorem 1.4 For a continuous map
$f\colon X\to X$
, if
$X=Sh(f)$
, then the following conditions are equivalent:
-
(A)
$X=\mathrm{Ent}_{up}(f)$
; -
(B)
$h_{\mathrm {top}}(f|_C)>0$
for all
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
.
Proof The implication (A)
$\implies $
(B) is a direct consequence of Lemma 1.4.
The implication (B)
$\implies $
(A): Due to Lemma 1.1, for any
$x\in X$
, there are
${C\in \mathcal {C}_{\mathrm {ter}}(f)}$
and
$y\in C$
such that
$x\rightarrow y$
. By
$h_{\mathrm {top}}(f|_C)>0$
, we obtain
$z\in \mathrm{Ent}_{r,b}(f)$
for some
$z\in C$
and
$r,b>0$
. Since
$y,z\in C$
, we have
$y\rightarrow z$
. This with
$x\rightarrow y$
implies
$x\rightarrow z$
. By
$x\in Sh(f)$
and Lemma 1.5, we obtain
$x\in \mathrm{Ent}_{s,b}(f)$
for all
$0<s<r$
; thus,
$x\in \mathrm{Ent}_{up}(f)$
. Since
$x\in X$
is arbitrary, we conclude that
$X=\mathrm{Ent}_{up}(f)$
.
We recall the definition of C-entropy points from [Reference Ye and Zhang17]. For a continuous map
$f\colon X\to X$
and an open cover
$\mathcal {U}$
of X, let
$h(f,\mathcal {U})$
denote the entropy of f relative to
$\mathcal {U}$
(see [Reference Walters16] for details). We say that
$(x,y)\in X^2$
with
$x\ne y$
is an entropy pair for f if for any closed neighborhoods A of x and B of y,
$$\begin{align*}h(f,\{X\setminus A,X\setminus B\})>0 \end{align*}$$
whenever
$A\cap B=\emptyset $
. We denote by
$E_2(f)$
the set of entropy pairs for f. We say that
$x\in X$
is a C-entropy point for f if
$(x,y)\in E_2(f)$
for some
$y\in X$
with
$x\ne y$
. Let
$E_1(f)$
denote the set of C-entropy points for f. By Theorem 3.4 of [Reference Ye and Zhang17], we know that
$E_1(f)\subset \mathrm{Ent}(f)$
. Ye and Zhang [Reference Ye and Zhang17] raised the following question.
Question [Reference Ye and Zhang17, Question 6.11]
Does
$E_1(f)\subset \mathrm{Ent}_{up}(f)$
hold for any homeomorphism
$f\colon X\to X$
?
In Section 3, we present an example of a homeomorphism
$f\colon X\to X$
with the following properties:
-
(1)
$X=Sh(f)$
; -
(2) there exists
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
such that C is an infinite set and satisfies
$C\subset E_1(f)\setminus \mathrm{Ent}_{up}(f)$
.
This article consists of three sections and three appendices. In the next section, we prove Theorem 1.1. In Section 3, we present the example just mentioned above. In Appendix A, we prove Theorem A.1. In Appendices B and C, Theorems B.1 and C.1 are proved.
2 Proof of Theorem 1.1
In this section, we prove Theorem 1.1. For the proof, we need a few lemmas.
Let
$f\colon X\to X$
be a continuous map. For a closed f-invariant subset S of X and
$r>0$
, we denote by
$\mathrm{Sen}_r^\ast (f|_S)$
the set of
$x\in S$
such that for any
$\delta>0$
, there are
$\delta $
-chains
$(x_i)_{i=0}^k$
,
$(y_i)_{i=0}^k$
of
$f|_S$
with
$x_0=y_0=x$
and
$d(x_k,y_k)>r$
. For
$x\in X$
and
$\epsilon>0$
, let
$B_\epsilon (x)$
denote the closed
$\epsilon $
-ball centered at x:
$B_\epsilon (x)=\{y\in X\colon d(x,y)\le \epsilon \}$
.
Lemma 2.1 Let
$f\colon X\to X$
be a continuous map and let
$C\in \mathcal {C}(f)$
. For any
$x\in C$
and
$r>0$
, if
$$\begin{align*}x\in Sh(f)\cap \mathrm{Sen}_r^\ast(f|_C), \end{align*}$$
then
$x\in \mathrm{Ent}_s(f)$
for all
$0<s<r$
.
Proof We fix
$s,\epsilon>0$
such that
$s+2\epsilon <r$
. By
$x\in Sh(f)$
, we obtain
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon $
-shadowed by some
$y\in X$
. Since
$x\in \mathrm{Sen}_r^\ast (f|_C)$
, there is a pair
$$\begin{align*}(\alpha_0,\alpha_1)=((x_i^{(0)})_{i=0}^k,(x_i^{(1)})_{i=0}^k) \end{align*}$$
of
$\delta $
-chains of
$f|_C$
such that
$x_0^{(0)}=x_0^{(1)}=x$
and
$d(x_k^{(0)},x_k^{(1)})>r$
. Since
$x,x_k^{(0)}, x_k^{(1)}\in C$
, we have
$x_k^{(0)}\rightarrow x$
and
$x_k^{(1)}\rightarrow x$
; therefore, there is a pair
$$\begin{align*}(\beta_0,\beta_1)=((y_i^{(0)})_{i=0}^l,(y_i^{(1)})_{i=0}^m) \end{align*}$$
of
$\delta $
-chains of f such that
$y_0^{(0)}=x_k^{(0)}$
,
$y_0^{(1)}=x_k^{(1)}$
, and
$y_l^{(0)}{\kern-1pt}={\kern-1pt}y_m^{(1)}{\kern-1pt}={\kern-1pt}x$
. Let
${n{\kern-1pt}={\kern-1pt}2k{\kern-1pt}+{\kern-1pt}l{\kern-1pt}+{\kern-1pt}m}$
and let
$$\begin{align*}(\gamma_0,\gamma_1)=(\alpha_0\beta_0\alpha_1\beta_1,\alpha_1\beta_1\alpha_0\beta_0)=((z_i^{(0)})_{i=0}^n,(z_i^{(1)})_{i=0}^n), \end{align*}$$
a pair of
$\delta $
-chains of f. Note that
$z_0^{(0)}=z_0^{(1)}=z_n^{(0)}=z_n^{(1)}=x$
and
$$\begin{align*}d(z_k^{(0)},z_k^{(1)})=d(x_k^{(0)},x_k^{(1)})>r. \end{align*}$$
For any
$N\ge 1$
and
$u=(u_j)_{j=1}^N\in \{0,1\}^N$
, let
$$\begin{align*}\gamma_u=(w_i^{(u)})_{i=0}^{nN}=\gamma_{u_1}\gamma_{u_2}\ldots\gamma_{u_N}, \end{align*}$$
a
$\delta $
-chain of f. Since
$w_0^{(u)}=x$
, by the choice of
$\delta $
, we have
$x_u\in X$
such that
$$\begin{align*}d(f^i(x_u),w_i^{(u)})\le\epsilon \end{align*}$$
for all
$0\le i\le nN$
. Note that
$$\begin{align*}d(x,x_u)=d(w_0^{(u)},x_u)\le\epsilon \end{align*}$$
and so
$x_u\in B_\epsilon (x)$
for all
$u\in \{0,1\}^N$
. For any
$u,v\in \{0,1\}^N$
,
$u\ne v$
implies
$u_j\ne v_j$
and so
$$ \begin{align*} d(f^{k+(j-1)n}(x_u),f^{k+(j-1)n}(x_v))&\ge d(w_{k+(j-1)n}^{(u)},w_{k+(j-1)n}^{(v)})-2\epsilon \\ &=d(z_k^{(0)},z_k^{(1)})-2\epsilon>r-2\epsilon>s \end{align*} $$
for some
$1\le j\le N$
. It follows that
$\{x_u\colon u\in \{0,1\}^N\}$
is an
$(nN,s)$
-separated subset of
$B_\epsilon (x)$
with
$$\begin{align*}|\{x_u\colon u\in\{0,1\}^N\}|=2^N. \end{align*}$$
Since
$N\ge 1$
is arbitrary, we obtain
$$ \begin{align*} h(f,B_\epsilon(x),s)&\ge\limsup_{N\to\infty}\frac{1}{nN}\log{s_{nN}(f,B_\epsilon(x),s)}\\ &\ge\limsup_{N\to\infty}\frac{1}{nN}\log{2^N}=\frac{1}{n}\log{2}>0. \end{align*} $$
Since
$\epsilon>0$
with
$s+2\epsilon <r$
is arbitrary, we conclude that
$x\in \mathrm{Ent}_s(f)$
, completing the proof of the lemma.
Lemma 2.2 Let
$f\colon X\to X$
be a continuous map and let
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
. For any
$x\in C$
and
$r>0$
,
$x\in \mathrm{Sen}_r(f)$
implies
$x\in \mathrm{Sen}_s^\ast (f|_C)$
for all
$0<s<r$
.
Proof Since
$x\in C$
and
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
, for any
$\epsilon>0$
, there is
$\gamma>0$
such that every
$\gamma $
-chain
$(x_i)_{i=0}^k$
of f with
$x_0=x$
satisfies
$$\begin{align*}d(x_i,C)\le\epsilon \end{align*}$$
for all
$0\le i\le k$
. By
$x\in \mathrm{Sen}_r(f)$
, we obtain a pair
$$\begin{align*}((x_i^{(0)})_{i=0}^k,(x_i^{(1)})_{i=0}^k) \end{align*}$$
of
$\gamma $
-chains of f with
$x_0^{(0)}=x_0^{(1)}=x$
and
$d(x_k^{(0)},x_k^{(1)})>r$
. For each
$j\in \{0,1\}$
, we take a sequence
$y_i^{(j)}\in C$
,
$0\le i\le k$
, with
$y_0^{(j)}=x_0^{(j)}=x$
and
$$\begin{align*}d(x_i^{(j)},y_i^{(j)})=d(x_i^{(0)},C)\le\epsilon \end{align*}$$
for all
$0<i\le k$
. For every
$\delta>0$
, if
$\epsilon ,\gamma>0$
are sufficiently small, then
$$\begin{align*}((y_i^{(0)})_{i=0}^k,(y_i^{(1)})_{i=0}^k) \end{align*}$$
is a pair of
$\delta $
-chains of
$f|_C$
such that
$y_0^{(0)}=y_0^{(1)}=x$
and
$d(y_k^{(0)},y_k^{(1)})>s$
. This implies
$x\in \mathrm{Sen}_s^\ast (f|_C)$
; therefore, the lemma has been proved.
By using these lemmas, we prove Theorem 1.1.
Proof of Theorem 1.1
(1): We take
$\epsilon>0$
such that
$s+2\epsilon <r$
. Let
$y\in \omega (x,f)$
. Since
$\mathrm{Sen}_r(f)$
is a closed f-invariant subset of X,
$x\in \mathrm{Sen}_r(f)$
implies
$\omega (x,f)\subset \mathrm{Sen}_r(f)$
. Note that
$$\begin{align*}\omega(x,f)\subset C(x,f). \end{align*}$$
By
$C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$
and Lemma 2.2, we obtain
$$\begin{align*}y\in \mathrm{Sen}_r(f)\cap C(x,f)\subset \mathrm{Sen}_{s+2\epsilon}^\ast(f|_{C(x,f)}). \end{align*}$$
Note that
$y\in \omega (x,f)$
implies
$x\rightarrow y$
. By
$x\in Sh(f)$
and Lemma 1.5, we obtain
$y\in Sh(f)$
and so
$$\begin{align*}y\in Sh(f)\cap \mathrm{Sen}_{s+2\epsilon}^\ast(f|_{C(x,f)}). \end{align*}$$
This with Lemma 2.1 implies
$y\in \mathrm{Ent}_{s+\epsilon }(f)$
. By
$x\rightarrow y$
,
$x\in Sh(f)$
, and Lemma 1.5, we conclude that
$x\in \mathrm{Ent}_s(f)$
.
(2): Due to Lemma 1.1, there are
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$y\in C$
such that
$x\rightarrow y$
. Let
$(\epsilon _j)_{j\ge 1}$
be a sequence of numbers with
$0<\epsilon _1>\epsilon _2 >\ldots $
and
$\lim _{j\to \infty }\epsilon _j=0$
. By
$x\in Sh(f)$
, for each
$j\ge 1$
, we have
$\delta _j>0$
such that every
$\delta _j$
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon _j$
-shadowed by some
$z\in X$
. Since
$x\rightarrow y$
, for each
$j\ge 1$
, there is a
$\delta _j$
-chain
$(x_i^{(j)})_{i=0}^{k_j}$
of f with
$x_0^{(j)}=x$
and
$x_{k_j}^{(j)}=y$
. We obtain a sequence
$x_j\in X$
,
$j\ge 1$
, such that
$$\begin{align*}d(f^i(x_j),x_i^{(j)})\le\epsilon_j \end{align*}$$
for all
$j\ge 1$
and
$0\le i\le k_j$
. Note that
$$\begin{align*}d(x_j,x)=d(x_j,x_0^{(j)})\le\epsilon_j \end{align*}$$
and
$$\begin{align*}d(f^{k_j}(x_j),y)=d(f^{k_j}(x_j),x_{k_j}^{(j)})\le\epsilon_j \end{align*}$$
for all
$j\ge 1$
. Since
$x\in \mathrm {int}[\mathrm{Sen}_r(f)]$
, we have
$x_j\in \mathrm{Sen}_r(f)$
for all j large enough. Since
$\mathrm{Sen}_r(f)$
is a closed f-invariant subset of X, it follows that
$f^{k_j}(x_j)\in \mathrm{Sen}_r(f)$
for all j large enough; thus, by
$\lim _{j\to \infty }f^{k_j}(x_j)=y$
, we obtain
$$\begin{align*}y\in \mathrm{Sen}_r(f)\cap C. \end{align*}$$
Note that
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
,
$x\rightarrow y$
, and
$x\in Sh(f)$
. The rest of the proof is similar to that of (1).
3 Example
In this section, we present an example of a homeomorphism
$f\colon X\to X$
with the following properties:
-
(1)
$X=Sh(f)$
; -
(2) there exists
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
such that C is an infinite set and satisfies
$C\subset E_1(f)\setminus \mathrm{Ent}_{up}(f)$
.
The following example is taken from [Reference Kawaguchi10]. We modify Example 4.3 of [Reference Kawaguchi10] to obtain the homeomorphism.
Example 3.1 Let
$\sigma \colon [-1,1]^{\mathbb {Z}}\to [-1,1]^{\mathbb {Z}}$
be the shift map, i.e.,
$$\begin{align*}\sigma(x)_n=x_{n+1} \end{align*}$$
for all
$x=(x_n)_{n\in \mathbb {Z}}\in [-1,1]^\mathbb{Z}$
and
$n\in \mathbb {Z}$
. Let d be the metric on
$[-1,1]^{\mathbb {Z}}$
defined by
$$\begin{align*}d(x,y)=\sup_{n\in\mathbb{Z}}2^{-|n|}|x_n-y_n| \end{align*}$$
for all
$x=(x_n)_{n\in \mathbb {Z}},y=(y_n)_{n\in \mathbb {Z}}\in [-1,1]^{\mathbb {Z}}$
. We fix a sequence
$s=(s_k)_{k\ge 1}$
of numbers with
$0<s_1<s_2<\ldots $
and
$\lim _{k\to \infty }s_k=1$
. Put
$$\begin{align*}S=\{-1,1\}\cup\{-s_k\colon k\ge1\}\cup\{s_k\colon k\ge1\}, \end{align*}$$
a closed subset of
$[-1,1]$
.
We define a closed
$\sigma $
-invariant subset X of
$S^{\mathbb {Z}}$
by for any
$x=(x_n)_{n\in \mathbb {Z}}\in S^{\mathbb {Z}}$
,
$x\in X$
if and only if the following conditions are satisfied:
-
•
$|x_n|\le |x_{n+1}|$
for all
$n\in \mathbb {Z}$
; -
• for any
$n\in \mathbb {Z}$
and
$k\ge 1$
, if
$x_n=s_k$
, then
$x_{n+j}=-s_k$
for all
$1\le j\le k$
; -
• for every
$n\in \mathbb {Z}$
, if
$x_n=1$
, then
$x_{n+j}=-1$
for all
$j\ge 1$
.
We define
$z=(z_n)_{n\in \mathbb {Z}}$
and
$z^{(m)}=(z_n^{(m)})_{n\in \mathbb {Z}}$
,
$m\in \mathbb {Z}$
, as follows:
-
•
$z_n=-1$
for all
$n\in \mathbb {Z}$
; -
•
$z_m^{(m)}=1$
for all
$m\in \mathbb {Z}$
; -
•
$z_n^{(m)}=-1$
for all
$m,n\in \mathbb {Z}$
with
$m\ne n$
.
Let
$f=\sigma |_X\colon X\to X$
,
$X_k=X\cap \{-s_k,s_k\}^{\mathbb {Z}}$
,
$k\ge 1$
, and let
$$\begin{align*}X_\infty=X\cap\{-1,1\}^{\mathbb{Z}}=\{z\}\cup\{z^{(m)}\colon m\in\mathbb{Z}\}. \end{align*}$$
A similar argument as in Example 4.3 of [Reference Kawaguchi10] shows that
$$\begin{align*}CR(f)=\{x=(x_n)_{n\in\mathbb{Z}}\in X\colon|x_n|=|x_{n+1}|\:\:\text{for all }n\in\mathbb{Z}\}=X_\infty\cup\bigcup_{k\ge1}X_k \end{align*}$$
and
$\mathcal {C}(f)=\{X_\infty \}\cup \{X_k\colon k\ge 1\}$
. Note that
$\mathcal {C}_{\mathrm {ter}}(f)=\{X_\infty \}$
.
As in Example 4.3 of [Reference Kawaguchi10], by taking a sequence
$s=(s_k)_{k\ge 1}$
that rapidly converges to
$1$
, we can show that
$X=Sh(f)$
, i.e., f has the shadowing property. Since
$X_\infty \in \mathcal {C}_{\mathrm {ter}}(f)$
and
$h_{\mathrm {top}}(f|_{X_\infty })=0$
, by Lemma 1.4, we obtain
$$\begin{align*}X_\infty\cap \mathrm{Ent}_{up}(f)=\emptyset. \end{align*}$$
Note that the sequence
$X_k$
,
$k\ge 1$
, satisfies
$$\begin{align*}\lim_{k\to\infty}X_k=X_\infty \end{align*}$$
with respect to the Hausdorff distance. For every
$k\ge 1$
, since
$X_k$
is a mixing subshift of finite type,
$f|_{X_k}$
is mixing and satisfies the shadowing property. This implies that for every
$k\ge 1$
,
$f|_{X_k}$
has uniformly positive entropy (upe) and so satisfies
$$\begin{align*}(x,y)\in E_2(f) \end{align*}$$
for all
$x,y\in X_k$
with
$x\ne y$
(see, e.g., [Reference Kawaguchi9] for details). Given any
$u,v\in X_\infty $
with
$u\ne v$
, by taking two sequences
$(x_k)_{k\ge 1}$
,
$(y_k)_{k\ge 1}$
such that
-
•
$x_k,y_k\in X_k$
and
$x_k\ne y_k$
for all
$k\ge 1$
; -
•
$\lim _{k\to \infty }x_k=u$
and
$\lim _{k\to \infty }y_k=v$
,
we obtain
$(u,v)\in E_2(f)$
. It follows that
$X_\infty \subset E_1(f)$
; thus, this example satisfies the desired properties.
Appendix A: A theorem on chain continuity points
In this appendix, we prove the following theorem.
Theorem A.1 Let
$f\colon X\to X$
be a continuous map. Let
$x\in X$
and
$C=C(x,f)$
. The following conditions are equivalent:
-
(A)
$x\in CC(f)$
; -
(B)
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$C\subset CC(f)$
; -
(C)
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$CC(f|_C)=C$
; -
(D)
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$CC(f|_C)\ne \emptyset $
.
For the proof, we need several lemmas.
Lemma A.1 For a continuous map
$f\colon X\to X$
and
$x,y\in X$
, if
$x\in CC(f)$
and
$x\rightarrow y$
, then
$y\in CC(f)$
.
Proof Let
$\epsilon>0$
. By
$x\in CC(f)$
, we have
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon $
-shadowed by x. Let
$\xi =(y_i)_{i\ge 0}$
be a
$\delta $
-pseudo orbit of f with
$y_0=y$
. We fix a sequence
$(\delta _j)_{j\ge 1}$
of numbers with
$\delta \ge \delta _1>\delta _2>\ldots $
and
$\lim _{j\to \infty }\delta _j=0$
. Since
$x\rightarrow y$
, for every
$j\ge 1$
, there is a
$\delta _j$
-chain
$\alpha _j=(x_i^{(j)})_{i=0}^{k_j}$
of f with
$x_0^{(j)}=x$
and
$x_{k_j}^{(j)}=y$
. By
$x\in CC(f)$
and
$\lim _{j\to \infty }\delta _j=0$
, we obtain
${\lim _{j\to \infty }f^{k_j}(x)=y}$
. For each
$j\ge 1$
, let
$$\begin{align*}\xi_j=(y_i^{(j)})_{i\ge0}=\alpha_j\xi, \end{align*}$$
a
$\delta $
-pseudo orbit of f. Given any
$j\ge 1$
, since
$y_0^{(j)}=x_0^{(j)}=x$
, we have
$$\begin{align*}\sup_{i\ge0}d(f^i(x),y_i^{(j)})\le\epsilon \end{align*}$$
which implies
$$\begin{align*}\sup_{i\ge0}d(f^i(f^{k_j}(x)),y_i)=\sup_{i\ge0}d(f^{i+k_j}(x),y_{i+k_j}^{(j)})\le\epsilon. \end{align*}$$
Letting
$j\to \infty $
, we obtain
$$\begin{align*}\sup_{i\ge0}d(f^i(y),y_i)\le\epsilon. \end{align*}$$
Since
$\xi $
is arbitrary, we conclude that
$y\in CC(f)$
, proving the lemma.
Lemma A.2 For a continuous map
$f\colon X\to X$
and
$C\in \mathcal {C}(f)$
, if
$C\cap CC(f)\ne \emptyset $
, then
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
.
Proof Let
$x\in C\cap CC(f)$
. Let
$y\in C$
,
$z\in X$
and
$y\rightarrow z$
. By
$x,y\in C$
, we have
$x\rightarrow y$
. By
$x\rightarrow y$
and
$y\rightarrow z$
, we obtain
$x\rightarrow z$
. Since
$x\in CC(f)$
, this implies
$z\in C$
; thus, we conclude that
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
.
Lemma A.3 Let
$f\colon X\to X$
be a continuous map and let
$C\in \mathcal {C}(f)$
. If
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
, then
$CC(f|_C)\subset CC(f)$
.
Proof Let
$x\in CC(f|_C)$
and
$\epsilon>0$
. By
$x\in CC(f|_C)$
, we have
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of
$f|_C$
with
$x_0=x$
is
$\epsilon $
-shadowed by x. Let
$0<\beta \le \epsilon $
. Since
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$x\in C$
, there is
$\gamma>0$
such that every
$\gamma $
-pseudo orbit
$\xi =(y_i)_{i\ge 0}$
of f with
$y_0=x$
satisfies
$$\begin{align*}\sup_{i\ge0}d(y_i,C)\le\beta. \end{align*}$$
If
$\beta ,\gamma>0$
are sufficiently small, then by letting
$x_0=x$
and taking
$x_i\in C$
,
$i>0$
, such that
$$\begin{align*}d(y_i,x_i)=d(y_i,C)\le\beta \end{align*}$$
for all
$i>0$
, we obtain a
$\delta $
-pseudo orbit
$\xi '=(x_i)_{i\ge 0}$
of
$f|_C$
with
$x_0=x$
. It follows that
$$\begin{align*}\sup_{i\ge0}d(f^i(x),x_i)\le\epsilon \end{align*}$$
and so
$$\begin{align*}\sup_{i\ge0}d(f^i(x),y_i)\le\epsilon+\beta\le2\epsilon. \end{align*}$$
Since
$\xi $
is arbitrary, we obtain
$x\in CC(f)$
and thus
$CC(f|_C)\subset CC(f)$
, proving the lemma.
Lemma A.4 Let
$f\colon X\to X$
be a continuous map. Let
$x\in X$
and let
$C=C(x,f)$
. If
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and
$CC(f|_C)\ne \emptyset $
, then
$x\in CC(f)$
.
Proof Let
$y\in CC(f|_C)$
and
$z\in \omega (x,f)$
. By
$C\in \mathcal {C}_{\mathrm {ter}}(f)$
and Lemma A.3, we obtain
$y\in CC(f)$
. Since
$\omega (x,f)\subset C$
, we have
$y,z\in C$
and so
$y\rightarrow z$
. By Lemma A.1, we obtain
$z\in CC(f)$
. Then, for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=z$
is
$\epsilon $
-shadowed by z. For
$0<\beta \le 2\epsilon $
, we fix
$k>0$
with
$d(z,f^k(x))\le \beta $
. By continuity of f, we have
$\gamma>0$
such that every
$\gamma $
-pseudo orbit
$\xi =(y_i)_{i\ge 0}$
of f with
$y_0=x$
satisfies
$$\begin{align*}\sup_{0\le i\le k}d(f^i(x),y_i)\le\beta \end{align*}$$
and so
$$\begin{align*}d(z,y_k)\le d(z,f^k(x))+d(f^k(x),y_k)\le2\beta. \end{align*}$$
If
$\beta ,\gamma>0$
are sufficiently small, then
$$\begin{align*}(z,f^{k+1}(x),f^{k+2}(x),\dots) \end{align*}$$
and
$$\begin{align*}(z,y_{k+1},y_{k+2},\dots) \end{align*}$$
are
$\delta $
-pseudo orbits of f and so
$\epsilon $
-shadowed by z, which implies
$$\begin{align*}\sup_{i\ge k+1}d(f^i(x),y_i)\le2\epsilon. \end{align*}$$
Note that
$$\begin{align*}\sup_{0\le i\le k}d(f^i(x),y_i)\le\beta\le2\epsilon. \end{align*}$$
Since
$\xi $
is arbitrary, we conclude that
$x\in CC(f)$
, completing the proof.
By these lemmas, we prove Theorem A.1.
Appendix B: A theorem on chain recurrent shadowable points
The aim of this appendix is to prove the following theorem.
Theorem B.1 Every continuous map
$f\colon X{\kern-1pt}\to{\kern-1pt} X$
satisfies
$Sh(f){\kern-1pt}\cap{\kern-1pt} CR(f){\kern-1pt}\subset{\kern-1pt} Sh(f|_{CR(f)})$
.
Let
$f\colon X\to X$
be a continuous map. For
$x,y\in X$
and
$\delta>0$
, the notation
$x\rightarrow _\delta y$
means that there is a
$\delta $
-chain
$(x_i)_{i=0}^k$
of f with
$x_0=x$
and
$x_k=y$
. We define an equivalence relation
$\leftrightarrow _\delta $
in
$$\begin{align*}CR(f)^2=CR(f)\times CR(f) \end{align*}$$
by: for any
$x,y\in CR(f)$
,
$x\leftrightarrow _\delta y$
if and only if
$x\rightarrow _\delta y$
and
$y\rightarrow _\delta x$
. It is easy to see that
$x\leftrightarrow _\delta f(x)$
for all
$x\in CR(f)$
. It is also not difficult to see that
$x\leftrightarrow _\delta y$
for all
$x,y\in CR(f)$
with
$d(x,y)\le \delta $
.
Proof of Theorem B.1
Let
$x\in Sh(f)\cap CR(f)$
. Then, for any
$\epsilon>0$
, we have
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_0=x$
is
$\epsilon $
-shadowed by some
${y\in X}$
. Let
$\xi =(x_i)_{i\ge 0}$
be a
$\delta $
-pseudo orbit of
$f|_{CR(f)}$
with
$x_0=x$
. For every
$i\ge 0$
, since
$d(f(x_i),x_{i+1})\le \delta $
, we have
$x_i\leftrightarrow _\delta f(x_i)$
and
$f(x_i)\leftrightarrow _\delta x_{i+1}$
; therefore,
$x_i\leftrightarrow _\delta x_{i+1}$
. This implies that
$x_0\leftrightarrow _\delta x_k$
for all
$k>0$
. Then, for any
$k>0$
, there is a
$\delta $
-chain
$(y_i)_{i=0}^l$
of f with
$y_0=x_k$
and
$y_l=x_0$
. Since
$$\begin{align*}\xi_k=(x_0,x_1,\dots,x_{k-1},y_0,y_1,\dots,y_{l-1},x_0,x_1,\dots,x_{k-1},y_0,y_1,\dots,y_{l-1},\dots) \end{align*}$$
is a
$\delta $
-pseudo orbit of f with
$x_0=x$
, letting
$$\begin{align*}X_k=\{x\in X\colon \xi_k\text{ is }\epsilon\text{-shadowed by }x\}, \end{align*}$$
we have
$X_k\ne \emptyset $
. Note that
$X_k$
is a closed
$f^{k+l}$
-invariant subset of X. By taking
$u\in X_k$
and
$v\in \omega (u,f^{k+l})$
, we obtain
$v\in X_k\cap CR(f^{k+l})\subset X_k\cap CR(f)$
and thus
$$\begin{align*}CR(f)\cap\bigcap_{i=0}^{k-1}f^{-i}(B_\epsilon(x_i))\ne\emptyset. \end{align*}$$
It follows that:
$$\begin{align*}CR(f)\cap\bigcap_{i=0}^\infty f^{-i}(B_\epsilon(x_i))=\bigcap_{k>0}\left(CR(f)\cap\bigcap_{i=0}^{k-1}f^{-i}(B_\epsilon(x_i))\right)\ne\emptyset, \end{align*}$$
i.e.,
$\xi $
is
$\epsilon $
-shadowed by some
$y\in CR(f)$
. Since
$\xi $
is arbitrary, we obtain
$x\in Sh(f|_{CR(f)})$
, proving the theorem.
Appendix C: A theorem on shadowable points
In this appendix, we prove the following theorem (Theorem C.1) that is not used elsewhere in this article but illustrates how “shadowable points” localize the concept of shadowing. Given a continuous map
$f\colon X\to X$
and a subset S of X, we say that f has the shadowing on S if for any
$\epsilon>0$
, there is
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_i\in S$
for all
$i\ge 0$
is
$\epsilon $
-shadowed by some
$x\in X$
.
Theorem C.1 Let
$f\colon X\to X$
be a continuous map. A closed subset A of X satisfies
$A\subset Sh(f)$
if and only if there is a closed f-invariant subset S of X such that
-
(1)
$A\subset S$
; -
(2) S is chain stable;
-
(3) f has the shadowing on S.
Proof First, we prove the “if” part. Let
$x\in S$
and
$0<\beta \le \epsilon $
. By (3), we have
$\delta>0$
such that every
$\delta $
-pseudo orbit
$(x_i)_{i\ge 0}$
of f with
$x_i\in S$
for all
$i\ge 0$
is
$\epsilon $
-shadowed by some
$y\in X$
. By (2), there is
$\gamma>0$
such that every
$\gamma $
-pseudo orbit
$\xi =(y_i)_{i\ge 0}$
of f with
$y_0=x$
satisfies
$$\begin{align*}\sup_{i\ge0}d(y_i,S)\le\beta. \end{align*}$$
If
$\beta ,\gamma>0$
are sufficiently small, then by taking
$x_i\in S$
,
$i\ge 0$
, such that
$$\begin{align*}d(y_i,x_i)=d(y_i,S)\le\beta \end{align*}$$
for every
$i\ge 0$
, we obtain a
$\delta $
-pseudo orbit
$\xi '=(x_i)_{i\ge 0}$
of f with
$x_i\in S$
for all
$i\ge 0$
. It follows that:
$$\begin{align*}\sup_{i\ge0}d(f^i(y),x_i)\le\epsilon \end{align*}$$
and so
$$\begin{align*}\sup_{i\ge0}d(f^i(y),y_i)\le\epsilon+\beta\le2\epsilon \end{align*}$$
for some
$y\in X$
. Since
$\xi $
is arbitrary, it follows that
$x\in Sh(f)$
and thus
$S\subset Sh(f)$
. By (1), we obtain
$A\subset Sh(f)$
.
Next we prove the “only if” part. Let
$$\begin{align*}S=A\cup\{y\in X\colon x\rightarrow y\:\:\text{for some }x\in A\}. \end{align*}$$
Since A is closed in X and the relation
$\rightarrow $
is closed in
$X^2$
, S is a closed subset of X. Note that
$z\rightarrow f(z)$
for all
$z\in X$
. For any
$x\in A$
, we have
$x\rightarrow f(x)$
and so
$f(x)\in S$
. For any
$x\in A$
,
$y\in X$
with
$x\rightarrow y$
, we have
$y\rightarrow f(y)$
and so
$x\rightarrow f(y)$
; therefore,
$f(y)\in S$
. It follows that S is f-invariant. Since
$y\in S$
for all
$x\in S$
,
$y\in X$
with
$x\rightarrow y$
, it also follows that S is chain stable. By
$A\subset Sh(f)$
and Lemma 1.5, we have
$S\subset Sh(f)$
. By using Lemma 2.4 in [Reference Kawaguchi8], we conclude that f has the shadowing on S, completing the proof.
Acknowledgements
The author would like to thank the reviewer for a careful reading of the manuscript and helpful comments.
