Chris Judge Indiana University Bloomington

Shrinking targets for affine automorphisms of Riemann surfaces

Abstract:

For a given ergodic dynamical system (T,X,μ) and a given measurable set A, the set of points x whose iterates intersect A infinitely has full measure. We think of A as a 'target'. To make the carnival game more difficult, we replace A with a sequence of sets A_n whose measure tends to zero. One then asks how fast the A_n can shrink and we can still hit them— Tⁿ x in A_n—with x in a full measure set. We will consider the shrinking target problem for the action of the group of self-diffeomorphsims of a Riemann surface that are affine with respect to a holomorphic 1-form on X. We use a mean ergodic theorem for SL₂(R) and work of Avila-Gouëzel to obtain estimates on the size of the targets. This is joint work with Josh Southerland.