Geometry & Topology

The invariant measures of some infinite interval exchange maps

W Patrick Hooper

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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