The linear request problem

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.