A Non-Autonomous Version Of The Denjoy-Wolff Theorem

TL;DR

This paper extends the Denjoy-Wolff Theorem to generalized Loewner chains, revealing new dynamical behaviors such as $oldsymbol{ ext{omega}}$-limits on arcs, contrasting with classical convergence results.

Contribution

It introduces a non-autonomous version of the Denjoy-Wolff Theorem applicable to generalized Loewner chains, uncovering novel asymptotic phenomena.

Findings

Identification of new $oldsymbol{ ext{omega}}$-limit behaviors

Demonstration of multiple asymptotic possibilities in generalized chains

Contrast with classical convergence results

Abstract

The aim of this work is to establish the celebrated Denjoy-Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, $\omega$-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.