# Structure of attractors for boundary maps associated to Fuchsian groups

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## 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## Mathematics Subject Classification (2010)

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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