An analytic endomorphism of the unit disk is called an inner function if its 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 (preimage multiplicities vs singularity set);
• transfer operators (spectral gaps and perturbations);
• a conditional central limit theorem for their stochastic processes.

If there's still time at the end I'll mention some questions concerning the multidimensional situation. Based on joint work with Mahendra Nadkarni: arXiv:2305.15278.

The talk is joint with the mini-conference Ergodic group actions and unitary representations at IMPAN-BC.
Meeting ID: 852 4277 3200 Passcode: 103121