Is your favorite measure extremal in the sense of metric Diophantine approximation? We present a small stream of research, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which attempts to provide some answers to this broad question. The talk will be accessible to students and faculty interested in some convex combination of dynamics, Diophantine approximation and fractal geometry. The topic, still in its nascency, contains several open questions and directions that have yet to be fully explored.
In more detail, we obtain the extremality of several new classes of dynamically defined fractal measures, e.g., by proving Patterson–Sullivan measures of all nonplanar geometrically finite Kleinian groups, and Gibbs measures for nonplanar infinite conformal iterated function systems and non-uniformly hyperbolic rational functions are <<weakly quasi-decaying>> (a geometric sufficient condition for extremality that is significantly weaker than <<friendliness>>). Among other results, we can improve on the celebrated theorem of Kleinbock and Margulis (1998) resolving Sprindzuk’s conjecture (viz. that Lebesgue measure on any nondegenerate manifold is extremal), as well as its extension by Kleinbock, Lindenstrauss and Weiss (2004).