Geometry & Topology

The invariant measures of some infinite interval exchange maps

W Patrick Hooper

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Abstract

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.

Article information

Source
Geom. Topol., Volume 19, Number 4 (2015), 1895-2038.

Dates
Received: 25 February 2013
Revised: 5 July 2014
Accepted: 5 January 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858798

Digital Object Identifier
doi:10.2140/gt.2015.19.1895

Mathematical Reviews number (MathSciNet)
MR3375521

Zentralblatt MATH identifier
1371.37076

Subjects
Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37E20: Universality, renormalization [See also 37F25] 37A40: Nonsingular (and infinite-measure preserving) transformations

Keywords
interval exchange IET ergodic measure classification Veech group translation surface skew product Maharam measure infinite ergodic theory Wind-tree renormalization

Citation

Hooper, W Patrick. The invariant measures of some infinite interval exchange maps. Geom. Topol. 19 (2015), no. 4, 1895--2038. doi:10.2140/gt.2015.19.1895. https://projecteuclid.org/euclid.gt/1510858798


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