In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy.

Definition

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

Properties

• 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 ${\displaystyle \mu }$.

Examples

• 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 ${\displaystyle \bigvee _{j=0}^{n-1}\sigma ^{-j}(C)}$ 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 ${\displaystyle A}$.

Notes

See also

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

References

• 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. Vol. 79. Springer-Verlag. ISBN 0-387-95152-0. Zbl 0475.28009.

External links

• 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.