On measures and semiconjugacies for affine interval exchange transformations
P. Berk, K. Frączek, Ł. Kotlewski, F. Trujillo
TL;DR
This work develops a comprehensive dimension and regularity theory for AIETs semi-conjugate to self-similar hyperbolic IETs, revealing a sharp dichotomy based on the log-slope vector’s Oseledets-type placement. It provides explicit formulas for the Hausdorff dimension of invariant measures in the central-stable regime, and for conformal measures, expressed via Perron-Frobenius eigenvalues of associated self-similarity matrices; stability yields dim_H=1, instability yields dim_H=0, and central-stable yields 0<dim_H<1 with dim_H(μ) = ρ_T / G(T, ω). The analysis hinges on a robust renormalization framework built from Rauzy-Veech induction, Cantor/continuous models, and suspended dynamics, enabling effective computation of Hölder exponents for the semi-conjugacy and its inverse. The results connect to Ledrappier-Young-type formulas in this interval-map setting and provide a flexible toolkit to study the dimensional and regularity properties of GIETs beyond circle-map cases, with implications for wandering intervals and rigidity phenomena. Overall, the paper advances a unified approach to quantify information content, dimension, and regularity in AIETs of hyperbolic periodic type via a synergy of renormalization, conformal measures, and graph-based optimization.
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 Hölder exponents.