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Entropy of snakes and the restricted variational principle

Published online by Cambridge University Press:  19 September 2008

M. Misiurewicz  and
J. Tolosa
M. Misiurewicz
Affiliation:
Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland and Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
J. Tolosa
Affiliation:
NAMS, Stockton State College, Pomona, NJ 08240, USA
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Abstract

For interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 791 - 802
Copyright
Copyright © Cambridge University Press 1992

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