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Proceedings of the American Mathematical Society

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The linear request problem

Author: Benoît R. Kloeckner
Journal: Proc. Amer. Math. Soc. 146 (2018), 2953-2962
MSC (2010): Primary 37C40; Secondary 37A10, 37C15
Published electronically: March 20, 2018
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Abstract: We propose a simple approach to a problem introduced by Galatolo and Pollicott, which can be called a linear request problem; in its general formulation, it consists of finding a first-order perturbation of a dynamical system such that its physical measure changes in a prescribed direction. Our method needs the physical measure to be absolutely continuous with smooth positive density: instead of using transfer operators, we use the well-known fact that a change in the density of a smooth measure can be reproduced by pushing forward along a well-chosen vector field. This implies that restricting to perturbations by infinitesimal conjugacy already yields a solution to the linear request problem, allowing us to work in any dimension and to dispense from additional dynamical hypotheses. In particular, we don't need to assume hyperbolicity to obtain a solution, but if the map is Anosov, we obtain the existence of an infinite-dimensional space of solutions.

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Additional Information

Benoît R. Kloeckner
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Matématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-94010, Créteil, France

Received by editor(s): June 23, 2017
Received by editor(s) in revised form: August 29, 2017, and September 15, 2017
Published electronically: March 20, 2018
Additional Notes: The author was supported by the Agence Nationale de la Recherche, grant ANR-11-JS01-0011.
Communicated by: Nimish Shah
Article copyright: © Copyright 2018 Benoît R. Kloeckner

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