Chris Judge

Shrinking targets for affine automorphisms of Riemann surfaces

For a given ergodic dynamical system (T,X,\mu) and a given measurable set A, the set of points x whose iterates intersect A infinitely often has full measure. Here A is a 'target'. To make this carnival game more difficult, we replace A with a sequence of sets A_n whose measure tends to zero. One asks whether the set of points x so that for infinitely many n we have T^n x in A_n has full measure. We will consider the shrinking target problem for the action 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 SL_2(R) and work of Avila-Gouezel to obtain estimates on the asymptotics of the targets. This is joint work with Josh Southerland.