题目：Spectral Gap for Transfer Operators of Torus Extensions over Expanding Maps

报告人：Dr.  Jianyu Chen

（University of Massachusetts, Amherst）

Abstract :  We study the spectral gap for transfer operators of the skew product $F: /TT^d/times /TT^/ell/to /TT^d/times /TT^/ell$ given by $F(x,y)=(Tx, y+/tau(x) /pmod{/ZZ^/ell})$, where $T: /TT^d/to /TT^d$ is a $C^/infty$ uniformly expanding endomorphism, and the fiber map $/tau: /TT^d/to /RR^/ell$ is a $C^/infty$ map. We construct a Hilbert space $/CW^{-s}$ for any $s<0$, which contains all the H/"older functions of H/"older exponents $|s|$ on $/TT^d/times /TT^/ell$. Applying the method of semiclassical analysis, we obtain the dichotomy: either the transfer operator has a spectral gap  on $/CW^{-s}$, or $/tau$ is an essential coboundary. In the former case, $F$ mixes exponentially fast for H/"older observables with H/"older exponents $|s|$; and in the latter case, either $F$ is not weak mixing and it is semiconjugate to a circle rotation, or $F$ is unstably mixing, i.e., it can be approximated by non-mixing skew products. This is a joint work with Huyi Hu.

时间：5月31日（周五）上午11:00-12：00

地点：565net必赢客户端本部教二楼 527 教室

联系人：王方

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