Topological Entropy and Chaos

枫叶之影發表於2024-07-13

Topological Entropy and Li-Yorke Chaos

"Topological entropy of maps on the real line"

Let \(X\) be a Hausdorff topological space and let \(f: X\to X\) be a continuous self-map
on \(X\). The pair \((X,f)\) is called a dynamical system.
A subset \(K\subset X\) is said to be invariant by \(f\) if \(f(K)\subset K\) and it is strictly invariant by \(f\) if \(f(K)=K\).

We summarize some properties of the topological entropy below.

Theorem 1. Let \(X\) and \(Y\) be two (metric) compact topological sets and let \(f: X \to X\) and \(g: Y \to Y\) be two continuous maps. Then the following properties are held:

(a) Let \(\phi:X\to Y\) be continuous such that \(g\circ \phi=\phi\circ f\). Then:
(a1) If the map \(\phi\) is injective, then \(h(f)\le h(g)\).
(a2) If the map \(\phi\) is surjective, then \(h(f)\ge h(g)\).
(a3) If the map \(\phi\) is bijective, then \(h(f)=h(g)\).

(b) Suppose that \(X=\cup_{i=1}^nX_i\), where \(X_i\) are compact and invariants by \(f\). Then \(h(f)=\max\{h(f|_{x_i})\}\)

(c) For any integer \(n \geqslant 0\) it is hold \(h\left(f^n\right)=n h(f)\).

(d) Let \(f \times g: X \times Y \rightarrow X \times Y\) be defined by \((f \times g)(x, y)=(f(x), g(y))\) for all \((x, y) \in\) \(X \times Y\). Then \(h(f \times g)=h(f)+h(g)\).

(e) If \(f\) is a homeomorphism, then \(h(f)=h\left(f^{-1}\right)\).

(f) Let \(\varphi: X \rightarrow Y\) be a continuous surjective map such that \(\varphi \circ f=g \circ \varphi\). Then \(\max \left\{h(g), \sup \left\{h\left(f, \varphi^{-1}(y)\right): y \in Y\right\}\right\} \leqslant h(f) \leqslant h(g)+\sup \left\{h\left(f, \varphi^{-1}(y)\right): y \in Y\right\}\).

(g) If \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) are continuous, then \(h(f \circ g)=h(g \circ f)\).

(h) Let \(f: X \rightarrow Y, g: Y \rightarrow X\) be continuous and let \(F: X \times Y \rightarrow X \times Y\) be defined by \(F(x, y)=(g(y), f(x))\) for all \((x, y) \in X \times Y\). Then

\[\begin{equation} h(F)=h(f \circ g)=h(g \circ f). \end{equation} \]

(i) If \(X_\infty=\cap_{n\ge 0}f^n(X)\) then \(h(f)=h(f|_{X_\infty})\).

(j) \(h(f)=h(f_{\Omega(f)})\) where \(x\in \Omega(f)\) if for all neighborhood \(U\) of \(x\) there is \(n\ge 0\) such that \(f^n(U)\cap U\ne \emptyset\)(\(\Omega(f)\) is called non-wandering set of \(f\)).

A dynamical system (X,f) is called minimal if X does not contain any non-empty, proper,  
closed sf-invariant subset. 

In such a case we also say that the map f itself is minimal.


The following conditions are equivalent: 
-$(X,f)$ is minimal,
-every orbit is dense in $X$, 
-$\omega_f(x)=X$ for every $x\in X$.

Definition of topological entropy on matric space

For continuous maps on a metric space \((X,f)\) the topological entropy of \(f\) is defined by

\[\begin{equation} {\rm ent}(f):=\sup\{h(f|_K): K\subset X, \text{compact and invariant by}\, f\}. \end{equation} \]

By Theorem 1(i) we have

\[\begin{equation} {\rm ent}(f)=\sup\{h(f|_K):K\in \mathcal{K}(X,f)\} \end{equation} \]

where \(\mathcal{K}(X,f)\) is the family of all the compact subsets of \(X\) which are strictly invariant by \(f\). Notice that this definition makes sense when \(X\) is matric or simply a topological space.

Explanation/interpretation (3):

Any compact \(f\)-invariant set \(K\) determines uniquely a strictly \(f\)-invariant closed set \(K_\infty=\cap_{n\ge 0}f^n(K)\in \mathcal{K}(X,f)\) such that \(h(f|_K)=h(f|_{K_\infty})\), so

\[\begin{equation}\begin{aligned} &\{K_\infty=\cap_{n\ge 0}f^n(K):K \text{ compact and}\, f-\text{invariant}\}\\ & \subset \{K: K \text{ compact and}\, f-\text{invariant}\}. \end{aligned} \end{equation} \]

Therefore,

\[\begin{equation} \begin{aligned} {\rm ent}(f)&=\sup \{h(f|_K):K\subset X, K \text{compact and} \, f-\text{invariant}\}\\ &=\sup \{h(f|_{K_\infty}):K\subset X, K\text{compact and} \, f-\text{invariant}\}\\ &\le \sup \{h(f|_{K_\infty}):K_\infty\in \mathcal{K}(X,f)\}\\ &\le {\rm ent}(f), \end{aligned} \end{equation} \]

i.e., (3) holds.

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 (https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7737952BD34F742FC1118C8353DB3CE0/S0143385700006817a.pdf/d-function-of-a-minimal-set-and-an-extension-of-sharkovskiis-theorem-to-minimal-sets.pdf)


 - http://www.scholarpedia.org/article/Minimal_dynamical_systems#Minimality_of_a_map_and_its_iterates (minimal system)

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