文摘
We investigate under which conditions a given invariant measure \(\mu \) for the dynamical system defined by the Gauss map \(x \mapsto 1/x \,\,{\mathrm {mod}}\,1\) is a Rajchman measure with polynomially decaying Fourier transform $$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$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. Mathematics Subject Classification 42A38 11K50 37C30 60F10 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (38) References1.Alkauskas, G.: The Minkowski \(?(x)\) function and Salem’s problem. C. R. Math. Acad. Sci. 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Phys. 258(2), 455–474 (2005)MathSciNetCrossRefMATH About this Article Title Fourier transforms of Gibbs measures for the Gauss map Journal Mathematische Annalen Volume 364, Issue 3-4 , pp 983-1023 Cover Date2016-04 DOI 10.1007/s00208-015-1241-9 Print ISSN 0025-5831 Online ISSN 1432-1807 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords 42A38 11K50 37C30 60F10 Industry Sectors Finance, Business & Banking Authors Thomas Jordan (1) Tuomas Sahlsten (2) Author Affiliations 1. School of Mathematics, University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, England, UK 2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel Continue reading... To view the rest of this content please follow the download PDF link above.