Growth gap versus smoothness for diffeomorphisms of the interval

Lev Buhovski; Roman Muraviev

doi:10.3934/jmd.2008.2.629

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2008, Volume 2, Issue 4: 629-643. Doi: 10.3934/jmd.2008.2.629

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Growth gap versus smoothness for diffeomorphisms of the interval

Lev Buhovski^1, and
Roman Muraviev^1,

1.
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978

Received: January 31, 2008

Revised: June 30, 2008

Published: September 2008

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Abstract

Given a diffeomorphism of the interval, we consider the uniform norm of the derivative of its $n$-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in the geometry of the diffeomorphism group. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the modulus of continuity of its derivative. These estimates extend previous results of Polterovich--Sodin and Borichev.

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Mathematics Subject Classification: 37C05, 37E05.

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