Projects per year
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 ('98) resolving Sprind\v{z}uk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), 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 PattersonSullivan 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 Part I, 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 quasidecay implies strong extremality in the matrix approximation framework, as well as proving the "inherited exponent of irrationality" version of this theorem. In Part II, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasidecaying. In particular, we prove the abovementioned result about PattersonSullivan measures, and we show that Gibbs measures (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasidecaying. In subsequent parts, we will continue to exhibit numerous examples of quasidecaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasidecaying".
Original language  English 

Number of pages  28 
Publication status  Submitted  23 Aug 2015 
Bibliographical note
Link to Part I: arXiv:1504.04778Keywords
 math.DS
 math.NT
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research