Geometriae Dedicata

, Volume 190, Issue 1, pp 53–80 | Cite as

Cutting sequences on square-tiled surfaces

  • Charles C. Johnson
Original Paper


We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation, and we convert cutting sequences to symbolic trajectories of these interval exchanges to show that special types of combinatorial lifts of Sturmian sequences completely describe all cutting sequences on a square-tiled surface. Our results extend the list of families of surfaces where cutting sequences are understood to a dense subset of the moduli space of all translation surfaces.


Cutting sequences Dynamical systems Square-tiled surfaces Translation surfaces 

Mathematics Subject Classification (1991)

Primary 37E35 


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Wake Forest UniversityWinston-SalemUSA

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