Geometriae Dedicata

, Volume 191, Issue 1, pp 171–198

# Structure of attractors for boundary maps associated to Fuchsian groups

• Svetlana Katok
• Ilie Ugarcovici
Original Paper

## 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 Zagier.

## Keywords

Fuchsian groups Reduction theory Boundary maps Attractor

37D40

## Notes

### Acknowledgements

We thank the anonymous referee for several useful comments and suggestions.

The second author was partially supported by the Simons Foundation (Grant No. 281407).

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