Speaker: Prof.Aihua Fan (Central China Normal University&University of Amiens)

Title:Growth of $L^\infty$-norm of Thue-Morse polynomials and Ergodic optimization

Time: Fri, June.15, 2018 , PM:3:00-4:00.

Location: Room 4318, Building No.4, Wushan Campus

Abstract:

  Thue-Morse sequence and its generalizations, which are all $2$-multiplicative, define what we call Thue-Morse (trigonometric) polynomials. Such $2$-multiplicativity (and more general $q$-multiplicativity) was introduced by A.O. Gelfond who were interested in number theory. The estimates of $L^p$-norms are problems to be solved and few results exist.We study the $L^\infty$-norm from the point of view of dynamical systems.Here the angle-doubling system is involved. We prove that the $L^\infty$-norm grows polynomially like $O(N^\gamma)$ and the best exponent $\gamma$ is simply related to the maximum value of a dynamical maximization, which is attained by a Sturmian measure. This is a part of joint work with Joerg Schmeling and Wexiao Shen. Furthermore, it can be proved that Thue-Morse sequence and its generalizations are all Gowers uniform of all orders. This is a joint work with Jakub Konieczny.