Minimizing entropy for translation surfaces

Colognese, Paul; Pollicott, Mark

doi:10.1090/ecgd/374

Minimizing entropy for translation surfaces
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by Paul Colognese and Mark Pollicott

Conform. Geom. Dyn. 26 (2022), 97-110

DOI: https://doi.org/10.1090/ecgd/374

Published electronically: August 17, 2022

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Abstract:

In this note we consider the entropy by Dankwart [On the large-scale geometry of flat surfaces, 2014, PhD thesis. https://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf] of unit area translation surfaces in the $SL(2, \mathbb R)$ orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices. We also provide a method for approximating the entropy of surfaces in the orbits.

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