Dynamics on the infinite staircase

W. Patrick Hooper; Pascal Hubert; Barak Weiss

doi:10.3934/dcds.2013.33.4341

Article Contents

2013, Volume 33, Issue 9: 4341-4347. Doi: 10.3934/dcds.2013.33.4341

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Dynamics on the infinite staircase

1.
Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY, USA 10031
2.
LATP, case cour A, Faculté des sciences Saint Jérôme, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20
3.
Ben Gurion University, Be'er Sheva, Israel 84105

Received: July 2010

Revised: February 2011

Published online: March 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D50; Secondary: 14H30, 30F30, 30F35, 37F30.

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