topological entropy

Let (X,d) be a compactPlanetmathPlanetmath metric space and f:XX a continuous mapMathworldPlanetmath. For each n0, we define a new metric dn by


Two points are ϵ-close with respect to this metric if their first n iterates are ϵ-close. For ϵ>0 and n0 we say that FX is an (n,ϵ)-separated set if for each pair x,y of points of F we have dn(x,y)>ϵ. Denote by N(n,ϵ) the maximum cardinality of an (n,ϵ)-separated set (which is finite, because X is compact). Roughly, N(n,ϵ) represents the number of “distinguishable” orbit segments of length n, assuming we cannot distinguish points that are less than ϵ apart. The topological entropy of f is defined by

htop(f)=limϵ0(lim supn1nlogN(n,ϵ)).

It is easy to see that this limit always exists, but it could be infiniteMathworldPlanetmath. A rough interpretationMathworldPlanetmathPlanetmath of this number is that it measures the averageMathworldPlanetmath exponential growth of the number of distinguishable orbit segments. Hence, roughly speaking again, we could say that the higher the topological entropy is, the more essentially different orbits we have.

Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew, with a different (but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) definition to the one presented here. The definition we give here is due to Bowen and Dinaburg.

Title topological entropy
Canonical name TopologicalEntropy
Date of creation 2013-03-22 14:31:34
Last modified on 2013-03-22 14:31:34
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Definition
Classification msc 37B40
Synonym entropy