Structure of attractors for boundary maps associated to Fuchsian groups

2016/10/01 by Svetlana Katok, Ilie Ugarcovici, Katok, Svetlana +1
Mathematics · #37D40 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37D40

paper · pdf · doi:10.48550/arxiv.1610.00167

27 pages, 12 figures; revised version accepted for publication in Geometriae Dedicata

arxiv created 2017/05/03 · arxiv updated 2017/05/05

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 Poincaré 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.

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