Structure of attractors for boundary maps associated to Fuchsian groups

TL;DR

This paper investigates the dynamical behavior of boundary maps linked to Fuchsian groups, revealing cycle properties and finite attractors, which enhance understanding of their complex dynamics and reduction algorithms.

Contribution

It demonstrates the cycle property for certain boundary maps and proves the existence of finite rectangular attractors for their natural extensions.

Findings

Cycle property holds for a family of boundary maps.

Finite rectangular attractors exist for natural extension maps.

Results relate to 'good' reduction algorithms as suggested by Zagier.

Abstract

We study dynamical properties of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups. These maps are defined on the unit circle (the boundary of the Poincar\'e disk) by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure. These two properties belong to the list of "good" reduction algorithms, equivalence or implications between which were suggested by Don Zagier.