Fourier decay of equilibrium states for bunched attractors

TL;DR

This paper proves polynomial Fourier decay in the unstable direction for invariant measures on certain attractors of Axiom A diffeomorphisms, under generic conditions, with applications to measures of maximal entropy.

Contribution

It establishes polynomial Fourier decay for invariant measures on bunched attractors with codimension one stable lamination, under generic nonlinearity and bunching conditions.

Findings

Polynomial Fourier decay in the unstable direction.

Applicable to measures of maximal entropy.

Constructs explicit solenoid satisfying assumptions.

Abstract

Let $M$ be a closed manifold, and let $f:M \rightarrow M$ 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.