Growth of $L^infty$-norm of Thue-Morse polynomials and Dynamical maximization
主 题: Growth of $L^infty$-norm of Thue-Morse polynomials and Dynamical maximization
报告人: Professor Aihua Fan (University of Picardie, France)
时 间: 2018-06-06 15:00-16:00
地 点: Room 1479, Sciences Building No. 1
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 theoretic questions. 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.