Orbits of non-elliptic disc automorphisms

TL;DR

This paper characterizes the limit behavior of orbits of thin Blaschke products under non-elliptic automorphism-induced composition operators, revealing connections to model spaces and describing eigenfunctions in various Hardy spaces.

Contribution

It provides an explicit description of the closed subspace generated by orbit limit points and characterizes eigenfunctions for these composition operators across Hardy spaces.

Findings

Explicit description of the orbit-generated subspace in $H^2$

Connection between orbit limits and model spaces

Constructive characterization of $C_$-eigenfunctions in $H^p$

Abstract

Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace $H^2$ generated by the limit points in the $H^2$ norm of the orbit of a thin Blaschke product $B$ under composition operators $C_\phi$ induced by non-elliptic automorphisms. This description exhibits a surprising connection to model spaces. Finally, we give a constructive characterization of the $C_\phi$-eigenfunctions in $H^p$ for $1\le p\le \infty$.