Fourier transforms of Gibbs measures for the Gauss map

Thomas M Jordan, Tuomas V A Sahlsten

Research output: Contribution to journalArticle (Academic Journal)peer-review

14 Citations (Scopus)
297 Downloads (Pure)

Abstract

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map x↦1/xmod1 is a Rajchman measure with polynomially decaying Fourier transform

|μˆ(ξ)|=O(|ξ|−η),as|ξ|→∞.

We show that this property holds for any Gibbs measure 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.
Original languageEnglish
Pages (from-to)983-1023
Number of pages41
JournalMathematische Annalen
Volume364
Issue number3
Early online date14 Jun 2015
DOIs
Publication statusPublished - Apr 2016

Bibliographical note

v3: 29 pages; peer-reviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. Ann

Keywords

  • 42A38
  • 11K50
  • 37C30
  • 60F10

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