Explicit exposure of Haar measure

Abstract

We solve the problem, posed some twenty years ago, of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. This is a special case of a more general result: given a nontrivial compact metrizable abelian group and a continuous endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by an explicit formula, every step is exact, and the rate of convergence is explicit. For expanding circle endomorphisms, ergodic toral endomorphisms, and the one-sided full shift, the construction can moreover be arranged so that the function is given by a Fourier series that converges uniformly in the classical frequency order, each of whose coefficients is rational and computable exactly by a finite procedure. We also show that no continuous function whose unique maximizing measure is Haar measure has absolutely summable Fourier coefficients, so that no such function lies in the Wiener algebra. For expanding maps of the circle, the Fourier coefficients of a continuous function with a unique maximizing measure of full support admit no closed form, in the sense that they satisfy no linear recurrence with polynomial coefficients; this follows from a result of independent interest, that a continuous function on the circle whose Fourier coefficients satisfy such a recurrence is Hölder continuous.

arXiv comment41 pages. Results added showing that for continuous functions exposing Haar measure, the Fourier coefficients cannot be absolutely summable, and cannot take a closed form