Flexibility of Lyapunov exponents for expanding circle maps

Alena Erchenko

doi:10.3934/dcds.2019098

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2019, Volume 39, Issue 5: 2325-2342. Doi: 10.3934/dcds.2019098

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Flexibility of Lyapunov exponents for expanding circle maps

Alena Erchenko

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: May 2017

Revised: July 2018

Published: January 2019

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Abstract

Let $ g $ be a smooth expanding map of degree $ D $ which maps a circle to itself, where $ D $ is a natural number greater than $ 1 $. It is known that the Lyapunov exponent of $ g $ with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to $ \log D $ which, in addition, is less than or equal to the Lyapunov exponent of $ g $ with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree $ D $ on a circle.

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Mathematics Subject Classification: Primary: 37E10; Secondary: 37A05.

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Figure 1. A representative of the SUSD-circle maps of degree $ 2 $

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References

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