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Veech surfaces with non-periodic directions in the trace field

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 Added by Thomas Schmidt
 Publication date 2009
  fields
and research's language is English




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We show that each of Veechs original examples of translation surfaces with ``optimal dynamics whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.



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88 - J. Beck , W.W.L. Chen , 2021
This paper is motivated by an interesting problem studied more than 50 years ago by Veech and which can be considered a parity, or mod 2, version of the classical equidistribution problem concerning the irrational rotation sequence. The Veech discrete 2-circle problem can also be visualized as a continuous flat dynamical system, in the form of 1-direction geodesic flow on a 2-square-b surface, a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of barriers and gates on the vertical edges, with appropriate modification of the edge identifications. A famous result of Gutkin and Veech says that 1-direction geodesic flow on any flat finite polysquare translation surface exhibits optimal behavior, in the form of an elegant uniform-periodic dichotomy. However, for irrational values of b, the 2-square-b surface is not a polysquare surface, and Veech and others have highlighted serious violations of the uniform-periodic dichotomy. Here we extend some of the results of Veech to consider cases previously not covered, and also obtain some time-quantitative description of these violations. Furthermore, we establish a far-reaching generalization of some earlier results to the class of flat finite polysquare-b-rational translation surfaces, obtained from flat finite polysquare translation surfaces in a similar way that the 2-square-b surface is constructed.
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