Singular Vectors on Fractals and Homogeneous Flows

Abstract: The theory of Diophantine approximation is underpinned by Dirichlet’s fundamental theorem. Broadly speaking, the main questions in the theory concern quantifying the prevalence of points with exceptional behavior with respect to Dirichlet’s result. The work of Dani and Kleinbock-Margulis connects these questions to the recurrence behavior of certain flows on homogeneous spaces. For example, divergent orbits of such flows correspond to so-called singular vectors. After a brief overview of the subject and the motivating questions, I will discuss new results giving a sharp upper bound on the Hausdorff dimension of divergent orbits of certain diagonal flows emanating from fractals on the space of unimodular lattices. Time permitting, connections to the theory of projections of self-similar measures will be presented.

dynamical systemsprobability

Audience: researchers in the topic

( slides )

Horowitz seminar on probability, ergodic theory and dynamical systems