Optimal transport and dynamics of expanding circle maps acting on measures
Abstract
Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown that 1 is an eigenvalue of infinite multiplicity, so that the invariant measure admits many deformations into nearly invariant ones. As a consequence, we obtain counter-examples to an infinitesimal version of Furstenberg's conjecture. We also prove that this action has positive metric mean dimension with respect to the Wasserstein metric.
Keywords
Cite
@article{arxiv.1006.4491,
title = {Optimal transport and dynamics of expanding circle maps acting on measures},
author = {Benoit Kloeckner},
journal= {arXiv preprint arXiv:1006.4491},
year = {2015}
}
Comments
35 pages; v4 includes a corrigendum (Lemma 4.2 statement and proofs are corrected without influence on the main results) and an addendum (application to an infinitesimal version of Furstenberg Conjecture, Theorem 1.7 and Corollary 1.8)