Fourier decay and spectral gaps for the stationary measures on the circle
Student No.:40
Time:Fri 09:50-11:00, Aug.10
Instructor:李嘉伦 Li Jialun  
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-8-10
Ending Date:2018-8-10

Let μ be a probability measure on SL2(R) such that the support of μ is not concentrated on a solvable subgroup of SL2(R). By a theorem of Furstenberg, there exists a unique stationary measure on the circle. We consider the Fourier coefficients of the stationary measure and we will prove that the Fourier coefficients tend to 0 at infinite with a polynomial speed. The main ingredients of the proof are the Large Deviation Principle and the discretized sum-product estimate of Bourgain. We will also talk about its application to spectral gaps and higher dimensional generalizations.