On measures and semiconjugacies for affine interval exchange transformations

TL;DR

This paper investigates the Hausdorff dimensions of invariant and conformal measures for affine interval exchange transformations semi-conjugated to hyperbolic periodic IETs, providing explicit formulas and regularity results.

Contribution

It offers a precise formula for Hausdorff dimension in specific AIET cases and analyzes the regularity of semi-conjugacies, advancing understanding of their measure-theoretic properties.

Findings

Hausdorff dimension formula for AIET invariant measures

Explicit H"older exponent formulas for semi-conjugacies

Dimension is strictly between 0 and 1

Abstract

In this article, we study affine interval exchange transformations (AIETs) which are semi-conjugated to interval exchange transformations (IETs) of hyperbolic periodic type. More precisely, we study the Hausdorff dimension of their invariant measures, as well as the Hausdorff dimension of conformal measures of self-similar interval exchange transformations, and implicit relations between them. Among the highlights of this paper, we provide a precise formula for the Hausdorff dimension when the vector of the logarithm of slopes is of central-stable type with respect to the renormalization matrix. This dimension turns out to be strictly between $0$ and $1$. Moreover, we study the regularity of the semi-conjugacy between an AIET and an IET in the periodic case, deriving explicit formulas for their supremal H\"older exponents.