Topological entropy

A topological dynamical system consists of a Hausdorff topological space X (usually assumed to be compact) and a continuous self-map f. Its topological entropy is a nonnegative extended real number that can be defined in various ways, which are known to be equivalent.

• Topological entropy is an invariant of topological dynamical systems, meaning that it is preserved by topological conjugacy.
• Let ${\displaystyle f}$ be an expansive homeomorphism of a compact metric space ${\displaystyle X}$ and let ${\displaystyle C}$ be a topological generator. Then the topological entropy of ${\displaystyle f}$ relative to ${\displaystyle C}$ is equal to the topological entropy of ${\displaystyle f}$, i.e.

${\displaystyle h(f)=H(f,C).}$

• Let ${\displaystyle f:X\rightarrow X}$ be a continuous transformation of a compact metric space ${\displaystyle X}$, let ${\displaystyle h_{\mu }(f)}$ be the measure-theoretic entropy of ${\displaystyle f}$ with respect to ${\displaystyle \mu }$ and let ${\displaystyle M(X,f)}$ be the set of all ${\displaystyle f}$-invariant Borel probability measures on X. Then the variational principle for entropy[6] states that

${\displaystyle h(f)=\sup _{\mu \in M(X,f)}h_{\mu }(f)}$.

• In general the maximum of the quantities ${\displaystyle h_{\mu }}$ over the set ${\displaystyle M(X,f)}$ is not attained, but if additionally the entropy map ${\displaystyle \mu \mapsto h_{\mu }(f):M(X,f)\rightarrow \mathbb {R} }$ is upper semicontinuous, then a measure of maximal entropy - meaning a measure ${\displaystyle \mu }$ in ${\displaystyle M(X,f)}$ with ${\displaystyle h_{\mu }(f)=h(f)}$ - exists.

• If ${\displaystyle f}$ has a unique measure of maximal entropy ${\displaystyle \mu }$, then ${\displaystyle f}$ is ergodic with respect to ${\displaystyle \mu }$.

• Let ${\displaystyle \sigma$ :\Sigma _{k}\rightarrow \Sigma _{k}} by ${\displaystyle x_{n}\mapsto x_{n-1}}$ denote the full two-sided k-shift on symbols ${\displaystyle \{1,\dots ,k\}}$. Let ${\displaystyle C=\{[1],\dots ,[k]\}}$ denote the partition of ${\displaystyle \Sigma _{k}}$ into cylinders of length 1. Then ${\displaystyle \bigvee _{j=0}^{n}\sigma ^{-1}(C)}$ is a partition of ${\displaystyle \Sigma _{k}}$ for all ${\displaystyle n\in \mathbb {N} }$ and the number of sets is ${\displaystyle k^{n}}$ respectively. The partitions are open covers and ${\displaystyle C}$ is a topological generator. Hence

${\displaystyle h(\sigma )=H(\sigma ,C)=\lim _{n\rightarrow \infty }{\frac {1}{n}}\log k^{n}=\log k}$. The measure-theoretic entropy of the Bernoulli ${\displaystyle \left({\frac {1}{k}},\dots ,{\frac {1}{k}}\right)}$-measure is also ${\displaystyle \log k}$. Hence it is a measure of maximal entropy. Further on it can be shown that no other measures of maximal entropy exist.

• Let ${\displaystyle A}$ be an irreducible ${\displaystyle k\times k}$ matrix with entries in ${\displaystyle \{0,1\}}$ and let ${\displaystyle \sigma$ :\Sigma _{A}\rightarrow \Sigma _{A}} be the corresponding subshift of finite type. Then ${\displaystyle h(\sigma )=\log \lambda }$ where ${\displaystyle \lambda }$ is the largest positive eigenvalue of ${\displaystyle A}$.

• Milnor–Thurston kneading theory
• For the measure of correlations in systems with topological order see Topological entanglement entropy
• Mean dimension

• Adler, R.L.; Konheim, Allan G.; McAndrew, M.H. (1965). "Topological entropy". Transactions of the American Mathematical Society. 114 (2): 309–319. doi:10.2307/1994177. JSTOR 1994177. Zbl 0127.13102.
• Dmitri Anosov (2001) [1994], "Topological entropy", Encyclopedia of Mathematics, EMS Press
• Roy Adler, Tomasz Downarowicz, Michał Misiurewicz, Topological entropy at Scholarpedia
• Walters, Peter (1982). An introduction to ergodic theory. Graduate Texts in Mathematics. 79. Springer-Verlag. ISBN 0-387-95152-0. Zbl 0475.28009.

• http://www.scholarpedia.org/article/Topological_entropy

This article incorporates material from Topological Entropy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.