Hausdorff dimension for ergodic measures of interval exchange transformations

TL;DR

This paper demonstrates that for certain minimal interval exchange transformations, the Hausdorff dimension of ergodic measures can be made arbitrarily small, including zero, and provides bounds for specific cases.

Contribution

It establishes the existence of minimal interval exchange transformations with ergodic measures of arbitrarily small Hausdorff dimension and derives bounds for particular scenarios.

Findings

Existence of transformations with Hausdorff dimension arbitrarily close to zero.

Bounds on Hausdorff dimension between 1/(2r+4) and 1/r for r > 1.

Demonstrates variability of measure complexity in interval exchange transformations.

Abstract

I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\frac 1 {2r+4}$ and $\frac 1 r$ for any r greater than 1.