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Optimal transport and dynamics of expanding circle maps acting on measures

Published online by Cambridge University Press:  07 February 2012

BENOÎT KLOECKNER
BENOÎT KLOECKNER*
Affiliation:
UJF-Grenoble 1, CNRS UMR 5582 Institut Fourier, Grenoble, F-38401, France
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Abstract

In this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 529 - 548
Copyright
©2012 Cambridge University Press

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