# American Institute of Mathematical Sciences

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

## Strict inequalities for the entropy of transitive piecewise monotone maps

 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.
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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