On the Hausdorff Dimension of Measures for a Non-Uniquely Ergodic Family of Interval Exchange Transformations

TL;DR

This paper constructs a family of interval exchange transformations with the maximum number of measures and estimates their Hausdorff dimensions, advancing understanding of measure complexity in non-uniquely ergodic systems.

Contribution

It introduces a new family of transformations with maximal measures and generalizes Hausdorff dimension estimates to this broader class.

Findings

Constructed a family with loor(n/2) measures

Extended Hausdorff dimension estimates to maximal measure systems

Provided new insights into measure complexity in ergodic theory

Abstract

In this paper, based on a construction by J. Fickenscher, we construct a family of non-uniquely ergodic interval exchange transformations on $n$ intervals with the maximal possible number of measures, $\left\lfloor \frac{n}{2} \right\rfloor$. Subsequently, we generalize J. Chaika's result on estimating the Hausdorff dimension of the two measures from M. Keane's example to the family of interval exchange transformations with the maximal possible number of measures.