Abstract: An analytic endomorphism of the unit disk is called an inner
function if it’s boundary limit defines a transformation of the circle
- which is necessarily Lebesgue nonsingular. I’ll review the ergodic
theory of inner functions & then present results on:
– their structure;
– spectral gaps for their transfer operators; and
– a conditional central limit theorem;
 recently obtained with Mahendra Nadkarni see  arXiv:2305.15278