Decrease of Fourier coefficients of stationary measures

Abstract

Let $\mu$ be a Borel probability measure on ${\mathrm{SL}}_2(\mathbb{R})$ with a finite exponential moment, and assume that the subgroup ${\Gamma }_{\mu }$ generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique $\mu -$stationary measure on ${\mathbb{P}}_{\mathbb{R}}^1$. We prove that the Fourier coefficients $\hat{\nu }(k)$ of $\nu$ converge to $0$ as $|k|$ tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.

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