Abstract: Let F be an inner function on the unit disk with an attracting fixed point at the origin. We view F as a measure-theoretic dynamical system on the unit circle. It is a classical fact that the Lebesgue measure is invariant under F. Elementary arguments involving harmonic functions by Shub-Sullivan (1985) and Pommerenke (1981) show that F is ergodic and mixing with respect to Lebesgue measure.
In the first part of the talk, we will discuss how one can use thermodynamic formalism to obtain more refined statistical properties of F such as the central limit theorem. Typically, to obtain a spectral gap, one uses the fact that F is expanding. This is true for finite Blaschke products, but not true for general inner functions since they can be very wild on the unit circle. We instead derive a spectral gap from some well known results on composition operators on spaces of holomorphic functions.
In the second part of the talk, we discuss orbit counting for inner functions of finite entropy, that is, with derivative in the Nevanlinna class. To do this, we follow the approach of McMullen (2008), who defined geodesic flows on Riemann surface laminations associated to finite Blaschke products and showed that they were ergodic. (According to Sullivan’s dictionary, these are analogous to unit tangent bundles of Riemann surfaces.) Our key insight is that backward iteration with respect to an inner function is essentially linear along almost every inverse orbit. This allows us to construct Riemann surface laminations for inner functions. One can then show ergodicity using Hopf’s method.
(This is joint work with Mariusz Urbański.)