Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Decrease of Fourier coefficients of Furstenberg measures and renewal theory

Dr. Jialun Li talk

**Date:** 13.05.19 **Time:** 14.00 - 15.00 **Room:** ETH HG G 43

Let μ be a Borel probability measure on SL2(R) with a finite exponential moment, such that the support of μ generates a Zariski dense subgroup in SL2(R). We can define a unique probability measure on the circle, which is called the μ stationary measure or Furstenberg measure. We will prove, using Bourgain's discretized sum-product estimate, that the Fourier coefficients of this measure go to zero with a polynomial speed. Starting from this result, we can obtain a spectral gap of the transfer operator, whose properties enable us to get an exponential error term in the renewal theorem in the context of random products of matrices.