Strict inequalities for the entropy of transitive piecewise monotone maps

Michał Misiurewicz; Peter Raith

doi:10.3934/dcds.2005.13.451

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2005, Volume 13, Issue 2: 451-468. Doi: 10.3934/dcds.2005.13.451

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Strict inequalities for the entropy of transitive piecewise monotone maps

Michał Misiurewicz^1, and
Peter Raith^2,

1.
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216
2.
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A 1090 Wien

Received: June 30, 2004

Revised: January 31, 2005

Published: January 2005

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Abstract

Let $T:[0,1]\to [0,1]$ be a piecewise differentiable piecewise monotone map, and let $r>1$. It is well known that if $|T'|\le r$ (respectively $|T'|\ge r$) then $h_{t o p}(T)\le$ log $r$ (respectively $h_{t o p}(T)\ge$ log $r$). We show that if additionally $|T'| < r $ (respectively $ |T'| > r $) on some subinterval and $T$ is topologically transitive then the inequalities for the entropy are strict. We also give examples that the assumption of piecewise monotonicity is essential, even if $T$ is continuous. In one class of examples the dynamical dimension of the whole interval can be made arbitrarily small.

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Mathematics Subject Classification: Primary: 37E05, 37B40.

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