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

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 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 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 quasi-decay 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 quasi-decaying. In particular, we prove the above-mentioned result about Patterson--Sullivan 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 quasi-decaying. In subsequent parts, we will continue to exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying".