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Dolgopyat's method
and the fractal uncertainty principle
Semyon Dyatlov and Long Jin |
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Vol. 11 (2018), No. 6, 1457–1485
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Abstract |
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We show a fractal uncertainty principle with exponent , , for Ahlfors–David regular subsets of of dimension . This is an improvement over the volume bound , and is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex cocompact hyperbolic surfaces and obtain a new spectral gap for open quantum baker maps. |
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Keywords
resonances, fractal uncertainty principle
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Mathematical Subject Classification 2010
Primary: 28A80, 35B34, 81Q50
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Milestones
Received: 23 February 2017
Revised: 26 October 2017
Accepted: 12 January 2018
Published: 3 May 2018
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Authors |
| Semyon Dyatlov | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
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| Long Jin | |
| Department of Mathematics Purdue University West Lafayette, IN United States |
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