Ergodic Optimization Of Generalized Thue-Morse Sequences

Let q? 2 be an integer and c ?[0,1)are real number.The generalized Thue-Morse sequence(tn(q;c))n?0 is defined by tn(q;c)=e2?icSq(n),where Sq(n)is the sum of digits in the q-expansion of n.We are interested in the Gelfond exponent ?(q;e)which is the best ?(q;c)such that(?)tn(q;c)ee2?sin??=O(N?).It was proved[4]that ?(q;c)=?(c)/logq,where ?(c)is the ergodic maximal value of log |sinq?(x+c)/sin?(x+c|relative to the dynamic x(?)qx mod 1.This thesis will present theoretic result and numerical computations of the Gel-fond exponents.