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Asymptotic entropy, periodic orbits, and pseudo-Anosov maps

Published online by Cambridge University Press:  01 April 1998

JAROSLAW KWAPISZ  and
RICHARD SWANSON
JAROSLAW KWAPISZ
Affiliation:
Mathematics Department, SUNY at Stony Brook, NY 11794-3651, USA Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
RICHARD SWANSON
Affiliation:
Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717-0240, USA
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Abstract

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [KS] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Hölder $C^1$ diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a ($C^0$) homeomorphism with nonzero asymptotic entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 2 , April 1998 , pp. 425 - 439
Copyright
1998 Cambridge University Press

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