PlanetMath: topological entropy

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topological entropy

(Definition)

Let be a compact metric space and $f\colon X\to X$ a continuous map. For each $n\geq 0$ , we define a new metric by

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

Two points are $\epsilon$ -close with respect to this metric if their first

iterates are $\epsilon$ -close. For $\epsilon>0$ and $n\geq 0$ we say that $F\subset X$ is an $(n,\epsilon)$ -separated set if for each pair

of points of

we have $d_n(x,y)>\epsilon$ . Denote by $N(n,\epsilon)$ the maximum cardinality of an $(n,\epsilon)$ -separated set (which is finite, because

is compact). Roughly, $N(n,\epsilon)$ represents the number of “distinguishable” orbit segments of length

, assuming we cannot distinguish points that are less than $\epsilon$ apart. The topological entropy of

is defined by

$\displaystyle h_{top}(f)=\lim_{\epsilon\to 0} \left(\limsup_{n\to \infty} \frac{1}{n}\log N(n,\epsilon)\right).$

It is easy to see that this limit always exists, but it could be infinite. A rough interpretation of this number is that it measures the average 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 equivalent) definition to the one presented here. The definition we give here is due to Bowen and Dinaburg.

"topological entropy" is owned by Koro.

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Other names:

entropy

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Cross-references: equivalent, exponential growth, average, measures, interpretation, infinite, limit, easy to see, length, segments, orbit, number, represents, finite, cardinality, iterates, points, metric, continuous map, metric space, compact
There are 2 references to this entry.

This is version 4 of topological entropy, born on 2004-08-03, modified 2006-06-11.
Object id is 6068, canonical name is TopologicalEntropy.
Accessed 4133 times total.

Classification:

AMS MSC:

37B40 (Dynamical systems and ergodic theory :: Topological dynamics :: Topological entropy)

Pending Errata and Addenda

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