Renormalization for Bruin-Troubetzkoy ITMs

TL;DR

This paper introduces a new renormalization scheme for Bruin-Troubetzkoy interval translation mappings, establishing invariant measures, ergodic properties, and a connection to multidimensional continued fractions with Pisot properties.

Contribution

It develops a novel renormalization approach inspired by Rauzy induction, constructs an invariant measure, and links the dynamics to multidimensional continued fractions with Pisot properties.

Findings

Invariant measure supported on infinite type mappings

Almost everywhere unique ergodicity under the measure

Renormalization exhibits Pisot property almost always

Abstract

We study a class of interval translation mappings introduced by Bruin and Troubetzkoy, describing a new renormalization scheme, inspired by the classical Rauzy induction for this class. We construct a measure, invariant under the renormalization, supported on the parameters yielding infinite type interval translation mappings in this class. With respect to this measure, a.e. transformation is uniquely ergodic. We show that this set has Hausdorff dimension between 1.5 and 2, and that the Hausdorff dimension coincides with the affinity dimension. Finally, seeing our renormalization as a multidimensional continued fraction algorithm, we show that it has almost always the Pisot property. We discover an interesting phenomenon: the dynamics of this class of transformations is often (conjecturally: almost always) weak mixing, while the renormalizing algorithm typically has the Pisot property.