Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Dynamics on the infinite staircase

Pages: 4341 - 4347, Volume 33, Issue 9, September 2013      doi:10.3934/dcds.2013.33.4341

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W. Patrick Hooper - Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY, USA 10031, United States (email)
Pascal Hubert - LATP, case cour A, Faculté des sciences Saint Jérôme, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20, France (email)
Barak Weiss - Ben Gurion University, Be'er Sheva, Israel 84105, Israel (email)

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.

Keywords:  Dynamics, infinite staircase, ergodicity, Maharam measure, infinite lattice surface.
Mathematics Subject Classification:  Primary: 37D50; Secondary: 14H30, 30F30, 30F35, 37F30.

Received: July 2010;      Revised: February 2011;      Available Online: March 2013.