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 $$\varGamma _{\mu }$$ generated by the support of $$\mu $$ is Zariski dense. Let $$\nu $$ be the unique $$\mu $$ -stationary measure on $$\mathbb {P}^1$$ . 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.