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The invariant measures of some infinite interval exchange maps

W Patrick Hooper

Geometry & Topology 19 (2015) 1895–2038

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions to which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1 2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are of finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multitwists in a pair of cylinder decompositions in nonparallel directions.

interval exchange, IET, ergodic, measure classification, Veech group, translation surface, skew product, Maharam measure, infinite ergodic theory, Wind-tree, renormalization
Mathematical Subject Classification 2010
Primary: 37E05
Secondary: 37E20, 37A40
Received: 25 February 2013
Revised: 5 July 2014
Accepted: 5 January 2015
Published: 29 July 2015
Proposed: Leonid Polterovich
Seconded: David Gabai, Walter Neumann
W Patrick Hooper
Department of Mathematics
City College of New York
160 Convent Ave
New York, NY 10031