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Counter-examples to an infinitesimal version of the Furstenberg conjecture

Published online by Cambridge University Press:  27 November 2015

BENOÎT R. KLOECKNER Open the ORCID record for BENOÎT R. KLOECKNER [Opens in a new window]
BENOÎT R. KLOECKNER*
Affiliation:
Université Paris-Est, Laboratoire d’Analyse et de Matématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-94010, Créteil, France email [email protected]
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Abstract

In this paper we observe that one of our main results in ‘Optimal transport and dynamics of circle expanding maps acting on measures’ [Ergod. Th. & Dynam. Sys.33(2) (2013), 529–548] has an interesting consequence: an infinitesimal version of the Furstenberg conjecture is false in a very strong way. More precisely, we find deformations of the Lebesgue measure on the circle which are first-order invariant simultaneously for all integer multiplications modulo 1. We also correct an error in a lemma of the mentioned article. Both the proof and the statement must be corrected, but the main results of the article are not affected.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 564 - 571
Copyright
© Cambridge University Press, 2015 

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