Fourier transforms of Gibbs measures for the Gauss map
Abstract
We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.3619
- arXiv:
- arXiv:1312.3619
- Bibcode:
- 2013arXiv1312.3619J
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Number Theory;
- 42A38 (Primary);
- 11K50;
- 37C30;
- 60F10 (Secondary)
- E-Print:
- v3: 29 pages