Fourier decay of equilibrium states for bunched attractors

Abstract

Let $M$ be a closed manifold, and let $f:M\to M$ be a $C^{2+\alpha }$ Axiom A diffeomorphism. Suppose that $f$ has an attractor $\Omega$ with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on $\Omega$. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.

arXiv commentThis is a first work on trying to understand the Fourier transform of equilibrium states for Axiom A diffeomorphisms using Markov partitions. This is part of the author's PhD and might not be published soon, as we are working on another approach that might gives better results