A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing

TL;DR

This paper investigates the regularity of conjugacies between generalized interval exchange transformations (GIETs) and their inverses, revealing unexpected asymmetric behaviors in their smoothness properties under specific hyperbolic conditions.

Contribution

It demonstrates that, contrary to expectations, the conjugacy can become arbitrarily irregular while its inverse remains uniformly H"older, using thermodynamic formalism and zero-temperature limits.

Findings

Conjugacy can be arbitrarily irregular in certain GIETs.

Inverse conjugacy remains uniformly H"older despite irregularity.

Sharp asymptotics for Hausdorff dimensions and H"older exponents are obtained.

Abstract

Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is $\infty$-complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their H\"older regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly H\"older. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal H\"older exponents of the conjugacy and its inverse.