JMD
Ergodic infinite group extensions of geodesic flows on translation surfaces
David Ralston Serge Troubetzkoy
Journal of Modern Dynamics 2012, 6(4): 477-497 doi: 10.3934/jmd.2012.6.477
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases.
keywords: Translation surface essential values. ergodicity diophantine approximation
DCDS
Recurrence in generic staircases
Serge Troubetzkoy
Discrete & Continuous Dynamical Systems - A 2012, 32(3): 1047-1053 doi: 10.3934/dcds.2012.32.1047
The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.
keywords: translation surface. Recurrence
DCDS
Ergodicity of certain cocycles over certain interval exchanges
David Ralston Serge Troubetzkoy
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2523-2529 doi: 10.3934/dcds.2013.33.2523
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
keywords: Ergodicity cocycle essential value infinite ergodic theory. interval exchange

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