Monotonicity properties of hyperbolic projections in holomorphic iteration

TL;DR

This paper investigates the monotonicity properties of hyperbolic projections of holomorphic orbits in the unit disc, providing a geometric framework for analyzing such projections under specific assumptions.

Contribution

It introduces a purely hyperbolic-geometric approach to study projections of holomorphic orbits, extending understanding of their monotonicity properties.

Findings

Projections exhibit monotonicity under certain conditions

Framework applicable to arbitrary sequences and curves

Provides geometric tools for future analysis

Abstract

We consider hyperbolic projections of orbits of holomorphic self-maps of the unit disc, onto curves landing on the unit circle with a given angle. We show that under certain, necessary, assumptions, the projections exhibit monotonicity properties akin to those present in continuous dynamics. Our techniques are purely hyperbolic-geometric in nature and provide the general framework for analysing projections of arbitrary sequences onto curves.