Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes

TL;DR

This paper proves that for generic continuous maps on compact manifolds, physical-like measures match invariant measures on chain recurrent classes, with implications for typical points and entropy, and compares conservative and non-conservative systems.

Contribution

It establishes the equivalence of physical-like and invariant measures on chain recurrent classes for generic systems and explores their properties and differences.

Findings

Physical-like measures coincide with invariant measures on chain recurrent classes.

Every point's empirical measures are contained in the physical-like measure space.

Existence of a zero Lebesgue measure set with infinite topological entropy.

Abstract

For $C^0$ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical (that is, for any point in the base space, its empirical measures are contained in the space of physical-like measures) and (3) there is a subset of strongly regular set with Lebesgue zero measure but has infinite topological entropy. Moreover, some comparison between $C^0$ generic systems and $C^0$ conservative generic systems are discussed.

Equations

$${{\rm Y}}_n(x):=\frac{1}{n}\sum _{j=0}^{n-1}{\delta }_{f^j(x)},$$

$$p\omega_{f}(x):=\Big{\{}\mu\in{\mathcal{P}}:\ \ \exists\ n_{i}\rightarrow+\infty\mbox{ such that }{\lim_{i\rightarrow+\infty}}\ \Upsilon_{n_{i}}=\mu\Big{\}}.$$

$$G_{\mu }=\{x\in M:p{\omega }_f(x)=\{\mu \}\}$$

$$G_{\mu }(\epsilon )=\{x\in M:\text{mbox}{dist}^{\ast }(p{\omega }_f(x),\mu )<\epsilon \}$$

$$S{R}_f=\bigcup _{\mu \in {\mathcal{P}}_f,\mu ergodic}(G_{\mu }\cap S_{\mu }),$$

$$Leb(\Lambda )=0\text{ but }{h}_{top}(\Lambda )=h_{top}(f)=+\infty .$$

$$Leb(\Lambda)=1\text{ but }h_{top}(\Lambda)=0;\ \$$

$$Leb(I{R}_f)=0\text{ but }{h}_{top}(I{R}_f)=h_{top}(f)=+\infty .$$

$$Leb(\Lambda )=0\text{ but }{h}_{top}(\Lambda \cap {\Gamma }_f)=h_{top}(f)=+\infty ;$$

$$Leb(\Lambda )=0\text{ but }{h}_{top}(\Lambda \cap {\Gamma }_f)=h_{top}(f)=+\infty .$$

$$Leb(\Lambda )=0\text{ but }{h}_{top}(\Lambda )=h_{top}(f)=+\infty ?$$

$$\underset{\mu \in {\mathcal{O}}_f}{\sup }{h}_{\mu }(f)=0,$$

$$d(f(x_n),x_{n+1})<\delta \text{for any}n\ge 1.$$

$$d(f^n(y),x_n)<ϵ\text{for any}n\in \ge 1.$$

$$x_{i,j}^0=x\in U_i,x_{i,j}^{X_{i,j}}=y\in U_j.$$

$$M(ϵ):=ma{x}_{1\le i\ne j\le r_0}{X}_{i,j}.$$

$$x_{i_1,(i+1{)}_0}^0=f^{n_i}{x}_i\in U_{i_1},x_{i_1,(i+1{)}_0}^{X_{i_1,(i+1{)}_0}}=x_{i+1}\in U_{(i+1{)}_0}$$

$$\{f^t(x_1){\}}_{t=0}^{n_1}\cup {\Gamma }_{1_1,2_0}\cup \{f^t(x_2){\}}_{t=0}^{n_2}\cup \cdots \cup \{f^t(x_N){\}}_{t=0}^{n_N}\cup {\Gamma }_{N_1,(N+1{)}_0}.$$

$$d(f^{c_{i-1}+j}z,f^j{x}_i)<\epsilon ,j=0,1,\cdots ,n_i,i=1,2,\cdots ,N,$$

$$c_i=\{\begin{array}{ll}0,& \text{for }i=0\\ {\sum }_{j=1}^i[n_j+X_{j_1,(j+1{)}_0}],& \text{for}i=1,2,\cdots ,N.\end{array}$$

$$dist(\mu ,\nu ):=\sum _{n=1}^{\infty }\frac{|\int g_{n}d\mu -\int g_{n}d\nu |}{2^{n+1}\Vert g_n\Vert }.$$

$$\sum _{n=L+1}^{\infty }\frac{2}{2^{n+1}}<\frac{ϵ}{5}.$$

$$|\int gd\mu -\sum _{i=1}^k{\theta }_i\int gd{\mu }_i|=|\int gd\mu -\int gd(\sum _{i=1}^k{\theta }_i{\mu }_i)|<\frac{ϵ}{5}.$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{j=1}^{n-1}g(f^j{x}_i)=\int gd{\mu }_i,i=1,2,\cdots ,k,g\in F.$$

$$|\frac{1}{N}\sum _{j=1}^{N-1}g(f^j{x}_i)-\int gd{\mu }_i|<\frac{ϵ}{5},i=1,2,\cdots ,k,g\in F.$$

$$|\int gd\mu -\frac{1}{mN}\sum _{i=1}^{mN}g(y_i)|<\frac{2ϵ}{5}.$$

$$|\frac{1}{p}\sum _{i=1}^{p}g(f^{i}z)-\frac{1}{mN}\sum _{i=1}^{mN}g(y_i)|\le \frac{2M(\delta )}{N}+\frac{ϵ}{5}<\frac{2ϵ}{5}.$$

$$|\int gd\mu -\frac{1}{p}\sum _{i=1}^{p}g(f^{i}z)|<\frac{4ϵ}{5}.$$

$${\mathcal{O}}_f\subseteq \{\mu |S_{\mu }\subseteq \Lambda \text{ where }\Lambda \in \overline{{\omega }_f}\}\cap \overline{\{\mu |\mu \in p{w}_f(x),x\in M\}}.$$

$${\mathcal{O}}_f\subseteq \{\mu |S_{\mu }\text{ is contained in a chain recurrent class}\}\subseteq \overline{{\mathcal{P}}_p(f)}.$$

Full text

1. Introduction

Let $f:M\rightarrow M$ be a continuous map on a compact manifold $M$, which does not necessarily preserve any smooth measures with respect to the Lebesgue measure. Let $\mathcal{P}$ denote the space of all the probability measures endowed with the weak∗ topology, and ${\mathcal{P}}_{f}\subset{\mathcal{P}}$ denote the space of $f$-invariant probability measures.

For any point $x\in M$ and for any integer number $n\geq 1$, the empirical probability or time-average measure $\Upsilon_{n}(x)$ of the $f$-orbit of $x$ up to time $n$, is defined by

[TABLE]

where $\delta_{y}$ is the Dirac probability measure supported at $y\in M$. Consider the sequence \big{\{}\Upsilon_{n}\big{\}}_{n\in\mathbb{N}^{+}} of empiric probabilities in the space ${\mathcal{P}}$, and define the p-omega-limit set $p\omega_{f}(x)\subset{\mathcal{P}}$ as follows:

[TABLE]

It is standard to check that $p\omega_{f}(x)\subset{\mathcal{P}}_{f}$. From [23] we know that $p\omega_{f}(x)$ is always nonempty, weak∗-compact and connected. Among the most useful concepts in the ergodic theory, the physical probability measures play an important role.

Definition 1.1*.*

We call a measure $\mu\in\mathcal{P}$ physical or SRB (Sinai-Ruelle-Bowen), if the set

[TABLE]

has positive Lebesgue measure. The set $G_{\mu}$ is called basin of statistical attraction of $\mu$, or in brief, basin of $\mu$ (even if $\mu$ is not physical).

Remark 1.2*.*

The above definition of physical or SRB measures is not adopted by all the authors. Some mathematicians require the measure $\mu$ to be ergodic to call it physical. Besides, some mathematicians when studying $C^{1+\alpha}$ systems do not define SRB as a synonym of physical measure, but take into account the property of absolute continuity on the unstable foliation. But, in the scenario of continuous systems, and even for $C^{1}$ systems, the unstable conditional measures can not be defined because the unstable foliation may not exist (see [38]).

Not any system, principally in the $C^{1}$ context, possesses physical measures. This problem can be easily dodged by substituting the definition of physical measure by a weaker concept: *physical-like *measure, also called SRB-like or observable measure (see Definition 1.3). In [17] it was proved that for any $C^{0}$ system $f$, there exists a nonempty set composed by all the observable or physical-like measures.

Definition 1.3*.*

Choose any metric $\operatorname{dist}^{*}$ that induces the weak∗ topology on the space ${\mathcal{P}}$ of probability measures. A probability measure $\mu\in\mathcal{P}$ is called physical-like (or SRB-like or observable) if for any $\varepsilon>0$ the set

[TABLE]

has positive Lebesgue measure. The set $G_{\mu}(\varepsilon)$ is called *basin of $\varepsilon$-partial statistical attraction *of $\mu$, or in brief, $\varepsilon$-basin of $\mu$.

We denote by $\mathcal{O}_{f}$ the set of physical-like measures for $f$. It is standard to check that every physical-like measure is $f$-invariant and that ${\mathcal{O}}_{f}$ does not depend on the choice of the metric in ${\mathcal{P}}$. There is a basic fact for ${\mathcal{O}}_{f}$.

Theorem 1.4**.**

(Characterization of physical-like measures [17])**

Let $f:M\rightarrow M$ be a continuous map on a compact manifold $M$. Then, the set $\mathcal{O}_{f}$ of physical-like measures is nonempty, weak∗ compact, and contains the limits of the convergent subsequences of the empiric probabilities for Lebesgue almost all the initial states $x\in M$. Besides, no proper subset of ${\mathcal{O}}_{f}$ has the latter three properties simultaneously.

A point $x\in M$ is *irregular *(see [34, 35, 7, 4, 5]) if the sequence of time-averages along its orbit is not convergent, that is, $\#pw_{f}(x)>1$. It is also called points with historic behaviour [40, 43]. Otherwise, $x$ is called *regular *(called quasi-regular in [33, 23]). Let $IR_{f}$ be the set of irregular points and $R_{f}$ be the set of regular points. In [7] it is proved that $IR_{f}$ carries full topological entropy for hyperbolic systems. This result is generalized to systems with specification-like properties [45, 46]. The set of strongly regular points (called regular in [33]), denoted by $SR_{f},$ which means that

[TABLE]

where $S_{\mu}$ denotes the support of $\mu.$ This set has full measure for any invariant measure by Birkhoff ergodic theorem and ergodic decomposition theorem.

A $\delta$-pseudo-orbit (or $\delta$-chain) of $f$ from $x$ to $y$ is a sequence $\{x_{i}\}^{n}_{i=0}$ with $x_{0}=x,x_{n}=y$ and $d(f(x_{k}),x_{k+1})<\delta$ for $k=0,1,...,n-1.$ A point $x\in X$ is called chain recurrent if there is an $\delta$-chain from $x$ to itself for any positive $\delta$. We can define an equivalence relation on the set of chain recurrent points in such a way that two points x and y are said to be equivalent if for every $\delta>0$ there exist an $\delta$-chain from $x$ to $y$ and an $\delta$-chain from $y$ to $x$. The equivalence classes of this relation are called chain recurrent classes (or chain components). These are compact invariant sets and cannot be decomposed into two disjoint compact invariant sets, hence serve as building blocks of the dynamics. The topology of chain recurrent classes and the set of chain recurrent points have been always in particular interest [3, 39, 41, 49].

A point $x\in M$ is called physically-typical, if $pw_{f}(x)\subseteq\mathcal{O}_{f}.$ Let $h_{top}(A)$ denote the topological entropy of $A\subseteq M$ defined by Bowen [16] and let $Leb(A)$ denotes the Lebesgue measure of $A$. Let $\mathcal{H}(M),\ \mathcal{C}(M)$ denote the space of all homeomorphisms and continuous maps respectively. Now we are ready to state our main result.

Theorem A**.**

*There is a dense $G_{\delta}$ subset $\mathcal{R}\subseteq\mathcal{H}(M)\ (or\ \mathcal{C}(M))$ in $C^{0}$ topology such that for any $f\in\mathcal{R},$

(1) $\mathcal{O}_{f}=\{\mu|\ S_{\mu}\text{ is contained in a chain recurrent class}\}.$

(2) for any $x\in M$, $x$ is physically-typical.

(3) there is a subset $\Lambda\subseteq SR_{f}$ such that*

[TABLE]

Here we also list a recent related result [21] that there is a dense $G_{\delta}$ subset $\mathcal{R}\subseteq\mathcal{H}(M)\ (or\ \mathcal{C}(M))$ such that for any $f\in\mathcal{R},$

(4) there is a subset $\Lambda\subseteq R_{f}$ such that

[TABLE]

(5)

[TABLE]

Here item (3) of Theorem A gives us more information on Lebesgue measure and topological entropy in $SR_{f}$ which still has full measure for any invariant measure.

Remark 1.5*.*

By [20] for a $C^{0}$ generic $f,$ periodic measures are dense in $\mathcal{O}_{f}$ so that $\inf_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)=0.$ For $C^{1}$ partially hyperbolic diffeomorphisms (which form an open set in the space of diffemorphisms), it is proved in [18] that $\inf_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)>0,$ where $h_{\mu}(f)$ denotes the metric entropy of $\mu.$ Thus $\{f:\inf_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)=0\}$ is not $C^{r}$ generic for $r\geq 1.$ For $C^{1}$ partially hyperbolic diffeomorphisms, note that $\inf_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)>0$ implies that any periodic measure (if exists) is not in $\mathcal{O}_{f}$ and any periodic point (if exists) is not physically-typical. Combining with classical Kupka-Smale’s theorem that periodic measures and points exist and must be hyperbolic in $C^{1}$ generic diffeomorphisms, items (1) and (2) of Theorem A both are false in $C^{1}$ generic diffeomorphisms.

Remark 1.6*.*

A point $x\in M$ is called without physical-like behavior if $pw_{f}(x)\cap\mathcal{O}_{f}=\emptyset.$ Denote $\Gamma_{f}$ be the set of such points. By Theorem 1.4, $\Gamma$ always has zero Lebesgue measure. It is proved in [19] that such points form a nonempty set with full topological entropy in all transitive Anosov diffeomorphisms for which such systems form an open subset in the space of diffeomorphisms by structural stability. This implies that $\{f|\ \Gamma_{f}=\emptyset\}$ is not $C^{r}$ generic for any $r\geq 1$, different from item (2) of Theorem A which implies that $\{f|\ \Gamma_{f}=\emptyset\}$ is $C^{0}$ generic.

Remark 1.7*.*

Takens’ last problem said that: Whether there are persistent classes of smooth dynamical systems such that the set of initial states which give rise to orbits with historic behaviour has positive Lebesgue measure? S. Kiriki, T. Soma [28] proved that any Newhouse open set in the space of $C^{r}$-diffeomorphisms ($r\geq 2$) on a closed surface is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior so that Takens’ last problem holds. This implies that the parts for Lebesgue measure in item (4) and item (5) are false for $C^{r}$ generic diffeomorphisms ($r\geq 2$).

We point out some similar results as Theorem A and [21] for $C^{0}$ generic conservative systems. Let $\mathcal{H}_{Leb}(M),\ \mathcal{C}_{Leb}(M)$ denote the space of all conservative homeomorphisms and continuous maps respectively.

Theorem 1.8**.**

There is a dense $G_{\delta}$ subset $\mathcal{R}\subseteq\mathcal{H}_{Leb}(M)\ (or\ \mathcal{C}_{Leb}(M))$ in $C^{0}$ topology such that for any $f\in\mathcal{R},$ above item (3), (4) and (5) are true.

On the other hand, item (2) of Theorem A implies that for $C^{0}$ generic systems, $\Gamma_{f}$ is empty (carrying zero topological entropy) and $M\setminus\Gamma_{f}=M$ (carrying full topological entropy), but here we point out a following different result for $C^{0}$ generic conservative systems.

Theorem 1.9**.**

*There is a dense $G_{\delta}$ subset $\mathcal{R}\subseteq\mathcal{H}_{Leb}(M)\ (or\ \mathcal{C}_{Leb}(M))$ such that for any $f\in\mathcal{R},$ above item (1) and (2) of Theorem A are false; and moreover

(6) $\Gamma_{f}$ is not only nonempty but also carries full infinite topological entropy and $M\setminus\Gamma_{f}\subsetneq M$ carries zero topological entropy;

(7) $Leb(IR_{f})=0$ and $h_{top}(IR\cap\Gamma_{f})=h_{top}(f)=+\infty;$

(8) there is a subset $\Lambda\subseteq SR_{f}$ such that*

[TABLE]

(9) there is a subset $\Lambda\subseteq R_{f}\setminus SR_{f}$ such that

[TABLE]

Remark 1.10*.*

Note that items (6), (7), (8) and (9) are false for $C^{0}$ generic systems, since by item (2) of Theorem A one has $\Gamma_{f}=\emptyset.$ Note that item (3) is weaker than item (8) and item (5) is weaker than item (7). Similarly one can ask a weaker version of item (9) for a $C^{0}$ generic system, whether there is a subset $\Lambda\subseteq R_{f}\setminus SR_{f}$ such that

[TABLE]

Up to now this is still unknown.

Remark 1.11*.*

By [20] for $C^{0}$ generic systems, ergodic measures are dense in $\mathcal{O}_{f}$ so that by variational principle $\sup_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)=h_{top}(f)=+\infty.$ However, for $C^{0}$ generic conservative systems,

[TABLE]

since in this case Lebesgue measure is ergodic with zero metric entropy [25] and by Theorem 1.4 $\mathcal{O}_{f}$ is just composed by the Lebesgue measure.

The proofs of these results are not only based on some classical results but also based on some very recent results.

2. Periodic Approximation

Let $(X,f)$ be a topological dynamical system, meaning that $f:X\rightarrow X$ is a continuous map on a compact metric space $X$.

2.1. Various periodic orbit tracing properties

In this section we recall some orbit tracing properties. For more details and results, one may refer to  [23, 42]. For any $\delta>0$, a sequence $\{x_{n}\}_{n=0}^{+\infty}$ is called a $\delta$-pseudo-orbit if

[TABLE]

A $\delta$-pseudo-orbit $\{x_{n}\}_{n=0}^{+\infty}$ is called a periodic $\delta$-pseudo-orbit, if there is $T\in\mathbb{N}$ such that $x_{i+T}=x_{i},\forall i\geq 0,$ where the smallest positive integer $T$ satisfying this is called its period. A pseudo-orbit $\{x_{n}\}_{n=0}^{+\infty}$ is $\epsilon$-shadowed by the orbit of a point $y\in X$, if

[TABLE]

Definition 2.1*.*

We say that $(X,f)$ satisfies the (periodic) shadowing property, if for any $\epsilon>0$, there exists $\delta>0$ such that any (periodic) $\delta$-pseudo-orbit is $\epsilon$-shadowed by the orbit of a (periodic) point in $X$.

We introduce a notion of gluing property on a set for which the tracing point may be not in this set.

Definition 2.2*.*

Let $\Lambda\subseteq X$ be an invariant and closed subset. We say that $\Lambda$ has the periodic gluing orbit property, if for any $\varepsilon>0$ there exists $M(\varepsilon)\in\mathbb{N}$ such that for any points $x_{1},\cdots,x_{k}\in X$ and any integers $n_{1},\cdots,n_{k}\geq 1,$ there exist $m_{1},\cdots,m_{k}\leq M(\varepsilon)$ and a periodic point $x\in X$ with period $\sum_{j=1}^{k}(m_{j}+n_{j})$ satisfying $d(f^{l+\sum_{j=0}^{i-1}(m_{j}+n_{j})}(x),f^{l}(x_{i}))<\varepsilon$ for all $0\leq l\leq n_{i}-1$ and $1\leq i\leq k,$ where $m_{0}=n_{0}=0.$

Lemma 2.3**.**

Suppose $(X,f)$ has periodic shadowing property and $\Lambda$ is a chain recurrent class. Then $\Lambda$ has periodic gluing property.

**Proof. ** For any $\varepsilon>0,$ by periodic shadowing there exists $\delta>0$ such that any periodic $\delta$-pseudo orbit in $\Lambda$ can be $\varepsilon$ shadowed by a periodic orbit in $M$.

Take and fix for $\Lambda$ a finite cover $\alpha=\{U_{1},U_{2},\cdots,U_{r_{0}}\}$ by nonempty open balls $U_{i}$ in $\Lambda$ satisfying $diam(U_{i})<\delta$, $i=1,2,\cdots,r_{0}$. Since $\Lambda$ is chain transitive, for $x\in U_{i},y\in U_{j},$ and any $i,j=1,2,\cdots,r_{0},$ there exist a positive integer $X_{i,j}$ and $\delta$-chain $\Gamma_{i,j}:=\{x^{0}_{i,j},x^{1}_{i,j},\cdots,x^{X_{i,j}}_{i,j}\}$ such that

[TABLE]

Let

[TABLE]

Now let us consider a given sequence of points $x_{1},x_{2},\cdots,x_{N}\in\,\Lambda,$ and a sequence of positive numbers ${n_{1},n_{2},\cdots,n_{N}}$. Take and fix $U_{i_{0}},U_{i_{1}}\in\alpha$ so that $x_{i}\in U_{i_{0}},f^{n_{i}}x_{i}\in U_{i_{1}},i=1,2,\cdots,N.$ Take $\delta$-chain $\Gamma_{i_{1},{(i+1)_{0}}}:=\{x^{0}_{i_{1},(i+1)_{0}},x^{1}_{i_{1},(i+1)_{0}},\cdots,x^{X_{i_{1},(i+1)_{0}}}_{i_{1},(i+1)_{0}}\}$ such that

[TABLE]

for $i=1,2,\cdots,N$ where $x_{N+1}=x_{1},U_{(N+1)_{0}}=U_{1_{0}}$ Thus we get a periodic $\delta$-pseudoorbit:

[TABLE]

Hence there exists a periodic point $z\in M$ $\varepsilon$-shadowing the above sequence with period $p=\sum_{i=1}^{N}[n_{i}+X_{i_{1},(i+1)_{0}}].$ Clearly $p\in[\sum_{i=1}^{N}n_{i},\,\,\sum_{i=1}^{N}n_{i}+NM(\epsilon)].$ More precisely,

[TABLE]

where $c_{i}$ is defined as follows:

[TABLE]

∎

2.2. Approximation by periodic measures

Let $\mathcal{P}_{p}(f)$ denotes the space of all periodic measures.

Lemma 2.4**.**

Suppose $\Lambda$ is a compact invariant set with periodic gluing property. Then $\mathcal{P}_{f}(\Lambda)\subseteq\overline{\mathcal{P}_{p}(f)}.$

**Proof. ** Take a sequence of non-zero functions $\{f_{n}\}$ dense in the space $C^{0}(X,\mathbb{R})$ of all continuous functions. Then the weak∗ topology can be defined as following: for any $\mu,\nu\in\mathcal{P}(X),$

[TABLE]

Fix $\epsilon>0$ and $L>0$ such that

[TABLE]

Let $\mu\in\mathcal{P}_{f}(\Lambda)$ and $F=\{\frac{g_{i}}{\|g_{i}\|}\}_{i=1}^{L}$. Then one has to show that there is $\mu_{p}\in\mathcal{P}_{p}(X)$ such that for any $g\in F,$ $|\int gd\mu-\int gd\mu_{p}|<\frac{4\epsilon}{5}.$

Choose $\delta>0$ such that for all $g\in F$ one has $|g(x)-g(y)|<\frac{\epsilon}{5}$ whenever $d(x,y)<\delta,\ x,y\in X.$ Let $M(\delta)$ be the number given in the gluing property.

By ergodic decomposition theorem for the subsystem $f|_{\Lambda}$, there exist finite ergodic measures $\mu_{i}\in\mathcal{P}(\Lambda)$ ($i=1,2\cdots,k$) and positive numbers $\theta_{i}$ with $\sum_{i=1}^{k}\theta=1$ such that for any $g\in F,$

[TABLE]

We may assume $\theta_{i}$ are all rational numbers. Then there exist natural numbers $m_{i}$ such that $\theta_{i}=\frac{m_{i}}{m}$ where $m=\sum_{i=1}^{k}m_{i}.$ By Birkhoff ergodic theorem for the subsystem $f|_{\Lambda}$, we can choose $x_{i}\in\Lambda$ such that

[TABLE]

Take $N$ large enough such that $\frac{2M(\delta)}{N}<\frac{\epsilon}{5}$ and

[TABLE]

Let $\{y_{n}|1\leq n\leq mN\}\subseteq\Lambda$ be a sequence of points defined by letting $y_{n}$ run $m_{1}$ times through $\{x_{1},fx_{1},\cdots,f^{N-1}x_{1}\},$ then $m_{2}$ times through $\{x_{2},fx_{2},\cdots,f^{N-1}x_{2}\},$ etc., finally $m_{k}$ times through $\{x_{k},fx_{k},\cdots,f^{N-1}x_{k}\}.$ Then for any $g\in F,$

[TABLE]

By gluing property, there exist $0\leq q_{i}\leq M(\delta),i=1,2,\cdots,m,$ and a periodic point $z\in X$ with period $p=mN+\sum_{j=1}^{m}q_{j}$ satisfying $d(f^{t}(z),f^{t}(y_{i}))<\varepsilon$ for all $a_{t}\leq t\leq b_{t},\ t=0,1\cdots,m-1$ where $a_{t}=tN+\sum_{i=1}^{t}q_{i},\ b_{t}=tN+\sum_{i=1}^{t}q_{i}+N.$

Note that for any $g\in F,$

[TABLE]

Thus for any $g\in F,$

[TABLE]

Now we complete the proof. ∎

2.3. Basic facts

Let $A\subseteq X$ be a nonempty compact invariant set. We call $A$ internally chain transitive if for any $a,b\in A$ and any $\epsilon>0$, there is an $\epsilon$-chain $\{x_{i}\}^{n}_{i=0}$ contained in $A$ with connecting $a$ and $b$. Let $\omega_{f}=\{A\subseteq X:\exists x\in X~{}\text{with}~{}\omega_{f}(x)=A\}$ and denote the collection of internally chain transitive sets by $ICT.$ It has been shown [27] that every $\omega$-limit set is internally chain transitive, and the converse has also been shown in a variety of contexts, including Axiom A diffeomorphisms [13], shifts of finite type [8], topologically hyperbolic maps [9], and in certain Julia sets [11, 10], systems with orbit limit shadowing [24]. From [32] for systems with shadowing property, $\overline{\omega_{f}}=ICT.$

Proposition 2.5**.**

*For any dynamical system $(X,f),$

(1) $\{\mu|\ S_{\mu}\subseteq\Lambda\text{ where }\Lambda\in{\omega_{f}}\}\subseteq\{\mu|\ S_{\mu}\subseteq\Lambda\text{ where }\Lambda\in\overline{\omega_{f}}\}\subseteq\{\mu|\ S_{\mu}\subseteq\Lambda\text{ where }\Lambda\in ICT\}\subseteq\{\mu|\ S_{\mu}\subseteq\Lambda\text{ where }\Lambda\text{ is a chain recurrent class}\}.$

(2) for any $x\in M,$ $pw_{f}(x)\subseteq\{\mu|\ S_{\mu}\subseteq\Lambda\text{ where }\Lambda\in{\omega_{f}}\}.$

(3) If $X$ is a compact manifold, then*

[TABLE]

*(4) If $X$ is a compact manifold and $f$ has periodic shadowing, then $\mathcal{O}_{f}\subseteq\overline{\mathcal{P}_{p}(f)}.$

Proof. (1) From [32] we know that $ICT$ is closed in Hausdorff distance. By [27] ${\omega_{f}}\subseteq ICT$ and thus $\overline{\omega_{f}}\subseteq ICT.$ Note that any internally chain transitive set should be contained in one chain recurrent class. Thus one has item (1).

(2) It is easy to check for any $x\in X$, $\cup_{\mu\in pw_{f}(x)}S_{\mu}\subseteq\omega_{f}(x)$ and thus for any $\mu\in pw_{f}(x)$, $\mu(\omega_{f}(x))=1.$ Thus one has item (2).

(3) Given $\mu\in\mathcal{O}_{f},$ by definition there exists $x_{n}\in M$ such that $dist^{*}(pw_{f}(x_{n}),\mu)<\frac{1}{n}.$ By the choice of $x_{n},$ there exists $\mu_{n}\in pw_{f}(x_{n})$ such that $d(\mu_{n},\mu)<\frac{1}{n}$ and $\mu_{n}(\omega_{f}(x_{n}))=1.$ Let $\Lambda_{n}=\omega_{f}(x_{n})$. Take a convergent subsequence $\{\Lambda_{n_{k}}\}$ in Hausdorff distance for which the limit set is a closed set, denoted by $\Lambda.$ Then $\Lambda\in\overline{\omega_{f}}.$ For $\epsilon>0,$ let $\Lambda_{\epsilon}:=\{x|\ d(x,\Lambda)\leq\epsilon\}$. Then $\mu(\Lambda_{\epsilon})\geq\lim_{n\rightarrow\infty}\mu_{n_{k}}(\Lambda_{\epsilon})=1$ and thus $\mu(\Lambda)=1.$ Now one ends the proof of item (3).

(4) By Proposition 2.5 (1), (3), Lemma 2.3 and Lemma 2.4,

[TABLE]

∎

3. Proofs

3.1. Proof of Theorem A.

(1) It is known (see [31] and [29, 30], respectively) that periodic shadowing is a generic property in $\mathcal{C}(M)$ and in $\mathcal{H}(M)$. By Proposition 2.5 (1), (3), Lemma 2.3 and Lemma 2.4,

[TABLE]

By [20] $\mathcal{O}_{f}=\overline{\mathcal{P}_{p}(f)}$ is a generic property in $\mathcal{C}(M)$ and in $\mathcal{H}(M)$. Thus one ends the proof of item (1) of Theorem A.

(2) By Proposition 2.5 (1), (2) and Theorem A (1), for any $x\in M,$ $pw_{f}(x)\subseteq\mathcal{O}_{f}.$

(3) Fix $f\in\mathcal{H}(M)\ (or\ \mathcal{C}(M))$. By classical variational principle, take $\mu_{n}$ to be a sequence of ergodic measures with $\lim_{n\rightarrow\infty}h_{\mu_{n}}(f)=h_{top}(f).$ Let $\Lambda=\cup_{i=1}^{\infty}G_{\mu_{i}}\cap S_{\mu_{i}}$. Then $\Lambda\subseteq SR_{f}.$ Note that $\mu_{i}(G_{\mu_{i}}\cap S_{\mu_{i}})=1$ and thus $\mu_{i}(\Lambda)=1.$ From [16, Theorem 1] one knows that for any invariant measure $\omega$ and any set $Y\subseteq M$ with $\omega(Y)=1$, then $h_{\omega}(f)\leq h_{top}(Y).$ Thus $h_{top}(\Lambda)\geq\lim_{n\rightarrow\infty}h_{\mu_{n}}(f)=h_{top}(f).$

From [20, 2] there is a dense $G_{\delta}$ subset $\mathcal{R}_{1}\subseteq\mathcal{H}(M)\ (or\ \mathcal{C}(M))$ such that for any $f\in\mathcal{R}_{1},$ there is no physical measure. Thus $Leb(G_{\mu_{i}})=0$ so that $Leb(\Lambda)=0.$ From [50, 30] there is a dense $G_{\delta}$ subset $\mathcal{R}_{2}\subseteq\mathcal{H}(M)\ (or\ \mathcal{C}(M))$ such that for any $f\in\mathcal{R}_{2},$ $h_{top}(f)=+\infty.$ Let $\mathcal{R}=\mathcal{R}_{1}\cap\mathcal{R}_{2}$ and then one ends the proof. ∎

3.2. Proof of Theorem 1.9

Recall a general fact by Bowen [16] that letting $QR(t):=\{x\in X|\text{ there is }\mu\in V_{f}(x)\text{ such that }h_{\mu}\leq t\}$, then $h_{top}(T,QR(t))\leq t.$ From [16] we also know that

[TABLE]

Then it is easy to check that $M\setminus\Gamma_{f}\subseteq QR(t)$ when $t=sup_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)$). Thus

Lemma 3.1**.**

For any $A\subseteq M,$ $h_{top}(A\setminus\Gamma_{f})\leq h_{top}(M\setminus\Gamma_{f})\leq sup_{\mu\in\mathcal{O}_{f}}h_{\mu}(f).$ If further $sup_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)<h_{top}(f)$ and $h_{top}(A)=h_{top}(f)$, then $h_{top}(A\cap\Gamma_{f})=h_{top}(f)$.

From [25] we know that for a $C^{0}$ generic conservative system, $f$ is topological mixing with infinite topological entropy, the periodic points are dense in the whole space, the Lebesgue measure $Leb$ is ergodic with zero metric entropy and thus $\mathcal{O}_{f}$ is composed by just the Lebesgue measure. Recently it is proved in [26] for a $C^{0}$ generic conservative system, $f$ has specification property (and also has shadowing and periodic shadowing). Note that any periodic measure is not in $\mathcal{O}_{f}$ and any periodic point is not physically-typical. Thus items (1) and (2) of Theorem A both are false.

(6) Since $sup_{\mu\in\mathcal{O}_{f}}h_{\mu}(f)=h_{Leb}(f)=0,\ h_{top}(f)=+\infty,$ by Lemma 3.1 one gets that $h_{top}(M\setminus\Gamma_{f})=0$ and $h_{top}(\Gamma_{f})=h_{top}(f)=+\infty$.

(7) Note that $Leb(G_{Leb})=1$ and $G_{Leb}\subseteq R_{f}$ so that $Leb(IR_{f})=0.$ Given a continuous function $\phi:X\rightarrow\mathbb{R}$, let

[TABLE]

Since $f$ is not uniquely ergodic, then there exists a continuous function $\phi:X\rightarrow\mathbb{R}$ such that $\inf_{\mu\in\mathcal{P}_{f}}\int\phi d\mu<\sup_{\mu\in\mathcal{P}_{f}}\int\phi d\mu.$ Then by [46] one can get that

[TABLE]

Note that $I_{\phi}\subseteq IR_{f}$ so that $h_{top}(IR_{f})=h_{top}(f).$ Then by Lemma 3.1, one has item (7).

(8) Fix small $\epsilon>0.$ By variational principle, there is an ergodic $\mu$ such that $h_{\mu}(f)>h_{top}(f)-\epsilon.$ By Birkhoff ergodic theorem $\mu(G_{\mu}\cap S_{\mu})=1$ and then by [16] $h_{top}(G_{\mu}\cap S_{\mu})\geq h_{\mu}(f)>h_{top}(f)-\epsilon.$ Note that $\mu$ is not the Lebesgue measure since $h_{\mu}(f)>0$. Then $G_{\mu}\cap S_{\mu}\subseteq SR_{f}\cap\Gamma_{f}$ and thus one has item (8).

(9) By [37] specification implies that for any invariant measure $\mu$, $h_{top}(G_{\mu})=h_{\mu}(f).$ Fix small $\epsilon>0.$ By variational principle, there is an ergodic $\mu$ such that $h_{\mu}(f)>h_{top}(f)-\epsilon.$ Since $f$ is not uniquely ergodic, one can take an invariant measure $\mu^{\prime}\neq\mu$. Choose $\theta\in(0,1)$ close 1 enough such that $h_{\nu}(f)>h_{top}(f)-\epsilon,$ where $\nu=\theta\mu+(1-\theta)\mu^{\prime}.$ Then $\nu$ is not ergodic, not the Lebesgue measure, and $h_{top}(G_{\nu})=h_{\nu}(f)>h_{top}(f)-\epsilon.$ Note that $G_{\mu}\subseteq(R_{f}\setminus SR_{f})\cap\Gamma_{f}$ and thus one has item (9). ∎

3.3. Proof of Theorem 1.8

By Theorem 1.9, item (8) implies item (3) and item (7) implies (5). For item (4), recall that from [25] for a $C^{0}$ generic conservative system, the Lebesgue measure $Leb$ is ergodic with zero metric entropy. Taking $\Lambda=G_{Leb}$, then by ergodicity of $Leb$ and Birkhoff ergodic theorem $Leb(\Lambda)=1$. And by [16] $h_{top}(\Lambda)\leq h_{Leb}(f)=0.$ ∎

Acknowlegements

X. Tian is supported by National Natural Science Foundation of China (grant no. 11671093).