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{{About|entropy in geometry and topology|other uses|Entropy (disambiguation)}}

In [[mathematics]], the '''topological entropy''' of a topological [[dynamical system]] is a nonnegative real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by [[Roy Adler|Adler]], Konheim and McAndrew. Their definition was modelled after the definition of the [[Kolmogorov–Sinai entropy|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 [[periodic orbit|orbits]] of the iterates. An important '''variational principle''' relates the notions of topological and measure-theoretic entropy.

== Definition ==
A [[Topological dynamics|topological dynamical system]] consists of a [[Hausdorff topological space]] ''X'' (usually assumed to be [[compact space|compact]]) and a [[continuous function (topology)|continuous]] self-map ''f''. Its '''topological entropy''' is a nonnegative real number that can be defined in various ways, which are known to be equivalent.

=== Definition of Adler, Konheim, and McAndrew ===
Let ''X'' be a compact Hausdorff topological space. For any finite open [[cover (topology)|cover]] ''C'' of ''X'', let ''H''(''C'') be the [[logarithm]] (usually to base 2) of the smallest number of elements of ''C'' that cover ''X''.<ref>Since ''X'' is compact, ''H''(''C'') is always finite, even for an infinite cover ''C''. The use of arbitrary covers yields the same value of entropy.</ref> For two covers ''C'' and ''D'', let

: <math> C \vee D </math>

be their (minimal) common refinement, which consists of all the non-empty intersections of a set from ''C'' with a set from ''D'', and similarly for multiple covers. For any [[continuous map]] ''f'': ''X'' → ''X'', the following limit exists:

: <math> H(C,f) = \lim_{n\to\infty}
\frac{1}{n} H(C\vee f^{-1}C\vee \ldots\vee f^{-n+1}C). </math>

Then the '''topological entropy''' of ''f'', denoted ''h''(''f''), is defined to be the [[supremum]] of ''H''(''C'', ''f'') over all possible finite covers ''C'' of ''X''.

==== Interpretation ====
The parts of ''C'' may be viewed as symbols that (partially) describe the position of a point ''x'' in ''X'': all points ''x'' ∈ ''C''''i'' are assigned the symbol ''C''''i'' . Imagine that the position of ''x'' is (imperfectly) measured by a certain device and that each part of ''C'' corresponds to one possible outcome of the measurement. The integer <math>H(C\vee f^{-1}C\vee \ldots\vee f^{-n+1}C)</math> then represents the minimal number of "words" of length ''n'' needed to encode the points of ''X'' according to the behavior of their first ''n'' − 1 iterates under ''f'', or, put differently, the total number of "scenarios" of the behavior of these iterates, as "seen" by the partition ''C''. Thus the topological entropy is the average (per iteration) amount of [[information]] needed to describe long iterations of the map ''f''.

=== Definition of Bowen and Dinaburg ===
This definition uses a [[Metric (mathematics)|metric]] on ''X'' (actually, [[uniform structure]] would suffice). This is a weaker definition than that of Adler, Konheim, and McAndrew, as it requires additional, unnecessary structure on the topological space. However, in practice, the Bowen-Dinaburg topological entropy is usually much easier to calculate.

Let (''X'', ''d'') be a [[compact space|compact]] [[metric space]] and ''f'': ''X'' → ''X'' be a [[continuous map]]. For each [[natural number]] ''n'', a new metric ''d''''n'' is defined on ''X'' by the formula

:<math>d_n(x,y)=\max\{d(f^i(x),f^i(y)): 0\leq i<n\}.</math>

Given any ''ε'' > 0 and ''n'' ≥ 1, two points of ''X'' are ''ε''-close with respect to this metric if their first ''n'' iterates are ''ε''-close. This metric allows one to distinguish in a neighborhood of an orbit the points that move away from each other during the iteration from the points that travel together. A subset ''E'' of ''X'' is said to be '''(''n'', ''ε'')-separated''' if each pair of distinct points of ''E'' is at least ''ε'' apart in the metric ''d''''n''.
Denote by ''N''(''n'', ''ε'') the maximum [[cardinality]] of an (''n'', ''ε'')-separated set. The '''topological entropy''' of the map ''f'' is defined by

:<math>h(f)=\lim_{\epsilon\to 0} \left(\limsup_{n\to \infty} \frac{1}{n}\log N(n,\epsilon)\right).</math>

==== Interpretation ====
Since ''X'' is compact, ''N''(''n'', ''ε'') is finite and represents the number of distinguishable orbit segments of length ''n'', assuming that we cannot distinguish points within ''ε'' of one another. A straightforward argument shows that the limit defining ''h''(''f'') always exists in the [[extended real line]] (but could be infinite). This limit may be interpreted as the measure of the average exponential growth of the number of distinguishable orbit segments. In this sense, it measures complexity of the topological dynamical system (''X'', ''f''). Rufus Bowen extended this definition of topological entropy in a way which permits ''X'' to be noncompact.

== Notes ==
<references/>

== See also ==
* [[Milnor–Thurston kneading theory]]
* For the measure of correlations in systems with [[topological order]] see [[Topological entanglement entropy]]

== References ==
* {{cite journal | last1=Adler | first1=R.L. | last2=Konheim | first2=Allan G. | last3=McAndrew | first3=M.H. | title=Topological entropy | year=1965 | url=http://links.jstor.org/sici?sici=0002-9947(196502)114%3A2%3C309%3ATE%3E2.0.CO%3B2-N | journal=[[Transactions of the American Mathematical Society]] | volume=114 | number=2 | pages=309–319 | zbl=0127.13102 }}
*{{SpringerEOM|author=[[Dmitri Anosov]]|id=T/t093040}}
* Roy Adler, Tomasz Downarowicz, Michał Misiurewicz, [http://www.scholarpedia.org/article/Topological_entropy Topological entropy] at [[Scholarpedia]]
* {{cite book | last=Walters | first=Peter | title=An introduction to ergodic theory | series=[[Graduate Texts in Mathematics]] | volume=79 | publisher=[[Springer-Verlag]] | year=1982 | isbn=0-387-95152-0 | zbl=0475.28009 }}

{{PlanetMath attribution|id=6068|title=Topological Entropy}}

[[Category:Entropy and information]]
[[Category:Ergodic theory]]
[[Category:Topological dynamics]]

Geometric function theory - Revision history

en>Mogism: /* Univalent function */Cleanup/Typo fixing, typo(s) fixed: i.e → i.e. , german → German using AWB