Iterations, Denjoy-Wolff points and ergodic properties of point-mass singular inner functions (Q2367771)

The authors study in detail the iteration behavior of the simplest singular inner functions on the disk \(\mathbb{D}:=\{z\in\mathbb{C}: | z|<1\}\). These have the form \(M(z):=\exp(-\alpha(\zeta+z)(\zeta- z)^{-1})\), \(\alpha>0\), \(\zeta\in\partial\mathbb{D}\). It is a classical fact (A. Denjoy and J. Wolff) that the full sequence of iterates \((M^{[n]})\) converges locally uniformly in \(\mathbb{D}\) to a constant \(p\) which either lies on \(\partial\mathbb{D}\) or is the unique fixed point of \(M\) in \(\mathbb{D}\). It is also known that each iterate \(M^{[n]}\) is itself a singular inner function, and hence has the form \(e^{i\gamma_ n}\exp\bigl(- \int(\zeta+z)(\zeta-z)^{-1}d\nu_ n(\zeta)\bigr)\) for some positive measure \(\nu_ n\) on \(\partial\mathbb{D}\) which is singular with respect to Lebesgue measure. The authors explicitly determine \(\gamma_ n\) and \(\nu_ n\). Writing \(\zeta=e^{i\phi}\) with \(|\phi|\leq \pi\), they determine in terms of \(\alpha\) and \(\phi\) when \(p\in\mathbb{D}\) and when \(p\in\partial\mathbb{D}\). Namely, \(p\in\mathbb{D}\) if \(\alpha>2\) or if \(0<\alpha\leq 2\) and \(|\phi|< f(\alpha):=\sqrt{\alpha(2-\alpha)}+2 \sin^{- 1}\sqrt{\alpha/2}\); otherwise \(p\in\partial\mathbb{D}\). They fix \(\alpha\) and describe rather explicitly the path \(p\) traces out as \(\zeta\) runs over \(\partial\mathbb{D}\). Finally, they show that \(M\) is ergodic on \(\partial \mathbb{D}\) if \(p\in \mathbb{D}\) and is not ergodic if \(0<\alpha\leq 2\) and \(f(\alpha)<|\phi|\leq \pi\). The case where \(0<\alpha\leq 2\) and \(|\phi|\) has the critical value \(f(\alpha)\) remains undecided. However, the necessary and sufficient condition that \(M\) have no non- trivial, absolutely continuous invariant measure on \(\partial\mathbb{D}\) is that \(p\in\partial\mathbb{D}\).