Examples in Discrete Iteration of Arbitrary Intervals of Slopes

TL;DR

This paper constructs explicit parabolic self-maps of the upper half-plane with prescribed slope intervals, analyzing their orbit behavior and measure properties to understand their boundary regularity.

Contribution

It provides a novel explicit construction of self-maps with arbitrary slope intervals and analyzes their dynamical and measure-theoretic properties.

Findings

Constructed explicit self-maps with any given slope interval within [0,π]

Analyzed orbit wandering towards the Denjoy-Wolff point

Studied the regularity of the associated Herglotz measure

Abstract

Given a compact interval $[a,b] \subset [0,\pi]$, we construct a parabolic self-map of the upper half-plane whose set of slopes is $[a,b]$. The nature of this construction is completely discrete and explicit: we explicitly construct a self-map and we explicitly show in which way its orbits wander towards the Denjoy-Wolff point. We also analyze some properties of the Herglotz measure corresponding to such example, which yield the regularity of such self-map in its Denjoy-Wolff point.