generalized区间交换变换与其逆变换在冻结时正则性之间的意外差异
A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing
AI总结：
研究揭示了广义区间交换变换与其逆在冻结时正则性的不对称行为，通过热力学正则化和零温极限分析，获得Hausdorff维度和Hölder指数的渐进行为。
摘要：
广义区间交换变换（GIETs）在Rauzy-Veech组合学为∞-完备时半共轭于区间交换变换（IETs）。当这种半共轭性成为同胚映射时，一个基本问题是理解共轭及其逆的正则性。与通常预期其Hölder正则性同时退化相反，我们展示了强烈不对称的行为。对于自相似IETs的双曲周期型和通过冻结（零温）极限获得的自然一参数中心族仿射IET变形，共轭变得可以任意不规则，而其逆始终保持均匀Hölder。利用重整化中的热力学正则化和零温极限，我们获得了不变和共形测度的Hausdorff维度的精确渐进行为，以及共轭及其逆的上Hölder指数的精确渐进行为。
英文摘要：
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 Hölder 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 Hölder. 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 Hölder exponents of the conjugacy and its inverse.
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