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February  2005, 13(2): 451-468. doi: 10.3934/dcds.2005.13.451

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, Austria

Received  July 2004 Revised  February 2005 Published  April 2005

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.
Keywords: transitivity., piecewise monotone maps, Interval maps, topological entropy.
Mathematics Subject Classification: Primary: 37E05, 37B4.
Citation: Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
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