# Decrease of Fourier coefficients of stationary measures

## TL;DR

This paper proves that the Fourier coefficients of the unique stationary measure on the real projective line, associated with a Zariski dense subgroup of SL(2,R), tend to zero as the frequency increases, using a renewal theorem approach.

## Contribution

It establishes the decay of Fourier coefficients for stationary measures on projective space under Zariski dense subgroups, extending understanding of harmonic analysis on these measures.

## Key findings

- Fourier coefficients of the stationary measure tend to zero at infinity.
- The proof uses a generalized renewal theorem for the Cartan projection.
- Results apply to measures with finite exponential moments and Zariski dense support.

## 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 $\mathbb P^1_{\mathbb R}$. We prove that the Fourier coefficients $\widehat{\nu}(k)$ of $\nu$ converge to $0$ as $|k|$ tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.