2010/06/30 by Benoit Kloeckner, BENOÎT KLOECKNER
Mathematics · Physics and Astronomy · #Absolute continuity #Action (physics) #Eigenvalues and eigenvectors #Geometric Analysis and Curvature Flows #Invariant (physics) #Invariant measure #Mathematical Dynamics and Fractals #Measure (data warehouse) #Metric space #Probability measure #Quantum chaos and dynamical systems #math.DG #math.DS #math.FA
paper · pdf · doi:10.1017/s014338571100109x
published as Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2013, 33 (02), pp 529-548 · 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)
openalex publication_date 2012/02/07 · arxiv created 2015/05/21 · arxiv updated 2015/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.