AI 中文总结
本文对单位圆盘内带不动点的有限Blaschke乘积及其自由半群作用的混合速率完成分类，揭示混合速率的非刚性及刚性条件，明确双指数混合的尖锐指数值。
AI 中文摘要
我们研究解析可观测量上的双指数速率混合，并探究映射在多大程度上由其混合速率所决定。对于单位圆盘内带一个不动点的有限Blaschke乘积，以及它们生成的自由半群作用，我们给出了完整分类：混合速率（无混合、指数混合或双指数混合）由生成元在该不动点处的乘子决定，不变测度是带有该极点的调和测度。在双指数区域，指数等于log p，其中p是生成元在该不动点处的最小局部次数，我们证明该值是尖锐的。因此，该速率不具有刚性：它在C¹-扰动下不稳定，且不蕴含与仿射模型的C¹-共轭。当指数达到次数的最大值时，刚性成立：次数为q且指数达到log q的映射与仿射模型是Möbius共轭的。
英文摘要
We study mixing at a double exponential rate on analytic observables and ask how much of a map is remembered by its rate of mixing. For finite Blaschke products of the circle with a fixed point in the unit disk, and for the free semigroup actions they generate, we give a complete classification: the rate of mixing (no mixing, exponential, or double exponential) is determined by the multiplier of the generators at that fixed point, the invariant measure being the harmonic measure with a pole there. In the double exponential regime, the exponent equals $\log p$, where $p$ is the minimal local degree of the generators at the fixed point, and we show that this value is sharp. Consequently, the rate is not rigid: it is not stable under $C^1$-perturbations and does not imply $C^1$-conjugacy to affine models. Rigidity holds when the exponent is maximal for the degree: a map of degree $q$ whose exponent attains $\log q$ is Möbius conjugate to an affine model.
Comments35 pages
