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On topological entropy of maps

Published online by Cambridge University Press:  19 September 2008

Mike Hurley
Mike Hurley
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA (email: [email protected])
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Abstract

We introduce an ‘entropy’ hm(f) for a continuous mapping of a compact metric space to itself which is denned in terms of (n, ∈)-separated subsets of inverse images of individual points. This invariant is compared with the inverse-image entropy h_(f) introduced recently by Langevin and Walczak. The two main results are: (1) the inequality hm(f) ≤ h(f) ≤ hm(f) + h_(f) relating hm, h_ and the topological entropy h(f); (2) if pseudo-orbits are used in place of orbits in the definition of hm then the quantity that results is equal to the topological entropy. We actually establish an inequality that at least formally is slightly stronger than (1) by defining a variant of h_ which we call hi; it is trivial to show that hi ≤ h_, and we show that h ≤ hi + hm, from which (1) follows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 557 - 568
Copyright
Copyright © Cambridge University Press 1995

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References

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