Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum
Julien Grivaux Pascal Hubert
Journal of Modern Dynamics 2014, 8(1): 61-73 doi: 10.3934/jmd.2014.8.61
We construct explicit closed $\mathrm{GL}(2; \mathbb{R})$-invariant loci in strata of meromorphic quadratic differentials of arbitrarily large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.
keywords: Meromorphic quadratic differentials degenerate Lyapunov spectrum.
Infinite translation surfaces with infinitely generated Veech groups
Pascal Hubert Gabriela Schmithüsen
Journal of Modern Dynamics 2010, 4(4): 715-732 doi: 10.3934/jmd.2010.4.715
We study infinite translation surfaces which are $\ZZ$-covers of finite square-tiled surfaces obtained by a certain two-slit cut and paste construction. We show that if the finite translation surface has a one-cylinder decomposition in some direction, then the Veech group of the infinite translation surface is either a lattice or an infinitely generated group of the first kind. The square-tiled surfaces of genus two with one zero provide examples for finite translation surfaces that fulfill the prerequisites of the theorem.
keywords: Veech groups infinite translation surfaces holomorphic differentials.
Dynamics on the infinite staircase
W. Patrick Hooper Pascal Hubert Barak Weiss
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4341-4347 doi: 10.3934/dcds.2013.33.4341
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.
keywords: ergodicity Dynamics Maharam measure infinite staircase infinite lattice surface.

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