Fourier transforms of Gibbs measures for the Gauss map

By T. Jordan, Tuomas Sahlsten. 2013

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We investigate under which conditions a given invariant measure $$\mu $$μ for the dynamical system defined by the Gauss map $$x \mapsto 1/x \,\,{\mathrm {mod}}\,1$$x↦1/xmod1 is a Rajchman measure with polynomially decaying Fourier transform $$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$|μ^(ξ)|=O(|ξ|-η),as|ξ|→∞.We show that this property holds for any Gibbs measure $$\mu $$μ of Hausdorff dimension greater than 1 / 2 with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than 1 / 2 on badly approximable numbers, which extends the constructions of Kaufman and Queffélec–Ramaré. Our main result implies that the Fourier–Stieltjes coefficients of the Minkowski’s question mark function decay to 0 polynomially answering a question of Salem from 1943. As an application of the Davenport–Erdős–LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman–Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00b361dbe54c40c845e5d36da9ade5191a02523b