论文标题
多项式的整个功能和庞加莱功能的最终双曲线维度
Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials
论文作者
论文摘要
令$ p \ colon \ mathbb {c} \ to \ mathbb {c} $为整个功能。 poincaré函数$ l \ colon \ mathbb {c} \ to \ mathbb {c} $ $ p $是$ p $的固定点附近的线性化坐标的整个扩展。我们从可衡量的动力学的角度提出了这种庞加莱作为丰富而自然的动力系统类别的功能，表明$ p $的可测量动力会影响$ l $的动力。更确切地说，$ P $的双曲线尺寸是$ L $的双曲线尺寸的下限。 我们的结果使我们能够描述大量具有完整双曲线维度的双曲线整个功能，因此没有自然的不变措施。 （此类示例的存在直到最近才建立，使用非常不同的直接方法。）我们还为关于准文献等效性下最终维度的行为的自然问题给出了负面答案。
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.
