整函数的最终双曲维数与多项式的Poincaré函数
Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials
摘要：
假设$ P \colon \mathbb{C} \to \mathbb{C} $是一个完整的函数。$ P $的庞加莱函数$ L \colon \mathbb{C} \to \mathbb{C} $是$ P $的排斥不动点附近的线性化坐标的整个扩展。从可测动力学的观点出发，我们提出这样的庞加莱函数作为一个丰富而自然的动力系统类，表明$ P $的可测动力学影响了$ L $的可测动力学.更准确地说，$ P $的双曲维数是$ L $的双曲维数的下界。我们的结果使我们能够描述一个大的双曲整函数的集合，具有完整的双曲维数，因此没有自然的不变测度。(The这些例子的存在只是最近才确定的，使用的是非常不同的和更不直接的直接法）。我们也给出了一个否定的答案，一个自然的问题，关于行为的最终尺寸下拟共形等价。
英文摘要：
Let $ P \colon \mathbb{C} \to \mathbb{C} $ be an entire function. A Poincaré function $ L \colon \mathbb{C} \to \mathbb{C} $ of $ P $ is the entire extension of a linearising coordinate near a repelling fixed point of $ P $. We propose such Poincaré functions as a rich and natural class of dynamical systems from the point of view of measurable dynamics, showing that the measurable dynamics of $ P $ influences that of $ L $. More precisely, the hyperbolic dimension of $ P $ is a lower bound for the hyperbolic dimension of $ L $. Our results allow us to describe a large collection of hyperbolic entire functions having full hyperbolic dimension, and hence no natural invariant measures. (The existence of such examples was only recently established, using very different and much less direct methods.) We also give a negative answer to a natural question concerning the behaviour of eventual dimensions under quasiconformal equivalence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
