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

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.