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Piecewise monotone maps without periodic points: rigidity, measures and complexity

Published online by Cambridge University Press:  09 March 2004

JÉRÔME BUZZI  and
PASCAL HUBERT
JÉRÔME BUZZI
Affiliation:
Centre de Mathématiques de l'Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., 91128 Palaiseau cedex, France (e-mail: [email protected])
PASCAL HUBERT
Affiliation:
Institut de Mathématiques de Luminy, U.P.R. 9016 du C.N.R.S., 163 av. de Luminy, 13288 Marseille cedex 20, France (e-mail: [email protected])
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Abstract

We consider piecewise monotone maps of the interval with zero entropy or no periodic points. First, we give a rigid model for these maps: the interval translations mappings, possibly with flips. It follows, for example, that the complexity of a piecewise monotone map of the interval is at most polynomial if and only if this map has a finite number of periodic points up to monotone equivalence. Second, we study the invariant and ergodic measures of a piecewise monotone map with zero entropy and prove that their number is bounded by twice the number of monotony intervals; for a piecewise increasing map their number is at most the number of intervals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 2 , April 2004 , pp. 383 - 405
Copyright
2004 Cambridge University Press

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