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Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems

Part of: Diophantine approximation, transcendental number theory Complex dynamical systems Classical measure theory

Published online by Cambridge University Press:  30 June 2020

TUSHAR DAS Open the ORCID record for TUSHAR DAS [Opens in a new window] ,
LIOR FISHMAN ,
DAVID SIMMONS  and
MARIUSZ URBAŃSKI
TUSHAR DAS
Affiliation:
University of Wisconsin–La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI54601, USA (e-mail: tdas@uwlax.edu)
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX76203-5017, USA (e-mail: lior.fishman@unt.edu, urbanski@unt.edu)
DAVID SIMMONS
Affiliation:
434 Hanover Ln, Irving, TX75062, USA (e-mail: david9550@gmail.com)
MARIUSZ URBAŃSKI
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX76203-5017, USA (e-mail: lior.fishman@unt.edu, urbanski@unt.edu)
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Abstract

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

Keywords

metric Diophantine approximationgeometric measure theoryfractal geometryconformal dynamicsiterated function system (IFS)

MSC classification

Primary: 11J13: Simultaneous homogeneous approximation, linear forms 11J83: Metric theory 28A75: Length, area, volume, other geometric measure theory
Secondary: 37F35: Conformal densities and Hausdorff dimension
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2311 - 2348
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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