The Invariant Measures of some Infinite Interval Exchange Maps

TL;DR

This paper classifies invariant measures for certain infinite interval exchange transformations and related straight-line flows on translation surfaces, revealing new measure characterizations and ergodic properties in specific geometric contexts.

Contribution

It provides a classification of ergodic invariant measures for infinite IETs and straight-line flows on translation surfaces, especially those with nilpotent group actions.

Findings

Invariant measures characterized for a broad class of infinite IETs.

All ergodic measures are Maharam measures when surfaces admit a nilpotent group action.

Unique ergodicity established for finite-area translation surfaces in studied directions.

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 for 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 finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.