L-infinity-ESTIMATION OF GENERALIZED THUE-MORSE TRIGONOMETRIC POLYNOMIALS AND ERGODIC MAXIMIZATION

Aihua Fan (1) , Joerg Schmeling , Weixiao Shen

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

Given an integer q >= 2 and a real number c is an element of [0, 1), consider the generalized Thue-Morse sequence (t(n)((q;c)))(n >= 0) defined by t(n)((q;c)) = e(2 pi icsq(n)),( )where s(q)(n) is the sum of digits of the q-expansion of n. We prove that the L-infinity-norm of the trigonometric polynomials sigma((q;c))(N) (x) := Sigma(N-1 )(n=0)t(n)((q;c)) e(2 pi inx), behaves like N-gamma((q;c)), where gamma(q; c) is equal to the dynamical maximal value of log(q) vertical bar sin q pi(x + c)/sin pi(x + c)vertical bar relative to the dynamics x bar right arrow qx mod 1 and that the maximum value is attained by a q-Sturmian measure. Numerical values of gamma(q; c) can be computed.

Domaines: Mathématiques [math]

Aihua Fan, Joerg Schmeling, Weixiao Shen. L-infinity-ESTIMATION OF GENERALIZED THUE-MORSE TRIGONOMETRIC POLYNOMIALS AND ERGODIC MAXIMIZATION. Discrete and Continuous Dynamical Systems - Series A, 2021, 41 (1), pp.297-327. ⟨10.3934/dcds.2020363⟩. ⟨hal-03621994⟩