Dynamical Resistance Day

Title: Is your favorite dynamically defined measure strongly extremal? (video)

(Strongly) extremal measures, in the sense of metric Diophantine approximation, are locally finite Borel measures in a finite-dimensional Euclidean space for which most points are not very well (multiplicatively) approximable by ones with rational coordinates. This notion was defined by Sprindžuk, whose conjecture that the Lebesgue measure of any nondegenerate manifold is extremal was proven by Kleinbock-Margulis (Annals, 1998), and later strengthened by Kleinbock—Lindenstrauss-Weiss (Selecta, 2004). We present a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which isolates and studies a wide class of strongly extremal measures that we call weakly quasi-decaying. Applications include improving the results of Kleinbock-Margulis and Kleinbock-Lindenstrauss-Weiss; as well as exhibiting several new examples of quasi-decaying measures, many coming from conformal dynamical systems, in support of the thesis that “almost any measure from dynamics and/or fractal geometry is quasi-decaying”. We also discuss examples of non-extremal dynamically-defined measures illustrating where the theory must halt. The talk will be accessible to students and faculty interested in some convex combination of dynamics, Diophantine approximation and fractal geometry; with the hope of presenting open questions and directions that have yet to be explored.

References:

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. (Selecta, 2018) https://doi.org/10.1007/s00029-017-0324-8 and https://arxiv.org/abs/1504.04778

Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems. (ETDS, 2020)
https://doi.org/10.1017/etds.2020.46 and https://arxiv.org/abs/1508.05592