超越函数的无限熵等分布测度

中文摘要 AI 辅助

在20世纪80年代，Lyubich和Freire-Lopes-Mañé证明了对于任意次数d≥2的有理函数，其原像和周期点都等分布到唯一的极大熵log(d)测度。他们的结果提供了对迭代有理函数动力学的基本理解，此后已被推广到许多不同背景，包括高维多项式映射和有理映射的类别。在本文中，我们脱离代数范畴，旨在证明复平面中具有无限拓扑熵的超越函数的类似命题。我们引入了两种在超越背景下构造不变测度的方法，即通过嵌入符号动力系统和通过与适当选择的权重相关的传递算子。在后一种方法中，我们分离出权重的三个性质——正规性、紧致性和不可约性——它们共同蕴含收敛到不变测度。我们为每种方法提供了例子，由三类超越整函数给出：不相交型映射、强多项式类映射以及一类受Baker的多连通游荡域构造和Bishop的Hausdorff维数为1的Julia集构造启发的映射，我们称之为Baker-Bishop映射。对于这些类别中的每一个，我们证明相对于仔细选择的权重，原像等分布到一个无限熵的不变混合概率测度。对于Baker-Bishop映射和不相交型映射，我们还证明了周期点的等分布。与有理数背景相反，我们构造的测度不是唯一的：通过改变权重，可以获得无穷多个不同的测度。

英文摘要

In the 1980s Lyubich and Freire-Lopes-Mañé proved that for any rational function of degree d \geq 2, both preimages and periodic points equidistribute to the unique measure of maximal entropy log(d). Their results provide a fundamental understanding of the dynamics of iterated rational functions, and have since been generalized to many different contexts, including classes of higher-dimensional polynomial and rational maps. In the current paper we depart from the algebraic category and aim to prove analogous statements for transcendental functions in the complex plane, which have infinite topological entropy. We introduce two different methods for constructing invariant measures in the transcendental setting, namely via embedded symbolic dynamical systems and via transfer operators associated to suitably chosen weights. In the latter case we isolate three properties of the weights -normality, tightness, and irreducibility- which together imply convergence to an invariant measure. We provide examples for each method, given by three classes of transcendental entire functions: disjoint-type maps, strongly polynomial-like maps, and a class of maps inspired by Baker's construction of multiply connected wandering domains and by Bishop's construction of Julia sets of Hausdorff dimension 1, which we call Baker-Bishop maps. For each of these classes we prove that with respect to carefully chosen weights, preimages equidistribute to an invariant mixing probability measure of infinite entropy. For Baker-Bishop maps and disjoint-type maps we also prove equidistribution of periodic points. In contrast to the rational setting, the measures we construct are not unique: by varying the weights one obtains infinitely many distinct measures.

发表机构

• University of Rome Tor Vergata（罗马托尔维加塔大学）

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