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
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
Heaviness in symbolic dynamics: Substitution and Sturmian systems
David Ralston
Discrete & Continuous Dynamical Systems - S 2009, 2(2): 287-300 doi: 10.3934/dcdss.2009.2.287
Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in [11]. After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial rational eigenvalues. To contrast, Sturmian sequences are examined, including a new condition for a sequence to be Sturmian.
keywords: Heaviness Morse Sequence. Sturmian Sequences

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