Dimensions of the Julia sets of rational maps with the backward contraction property

Abstract

Abstract. Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f ) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.

A brief introduction to one-dimensional complex dynamics

“…Hausdorff 维数 [711] . 之后, 共形维数和 Hausdorff 维数相等的情形在一些非双曲有理函数中也得到了 (参见文献 [29,443,580,740,800] 及其中的参考文献). 在此过程中, 拓扑压、Poincaré 级数、Lyapunov 指数、双曲维数和锥点 (conical point) 等起到重要作用 (参见综述 [741] 和专著 [586]).…”

Statistical properties of one-dimensional maps under weak hyperbolicity assumptions

“…Equalities of dimensions were shown in [LS08] for backward contracting rational maps without parabolic cycles, in [Prz98] for rational maps whose derivatives at critical values grow at least as a stretched exponential function, in [GS09, Theorem 7] for rational maps satisfying a summability condition with a small exponent and without parabolic cycles, and in [Dob06] for interval maps without recurrent critical points. These equalities were shown for a class of infinitely renormalizable quadratic polynomials in [AL08].…”

On Summability Conditions for Interval Maps

“…This condition is more convenient to use as it follows immediately that the first return maps to suitably chosen small neighbourhoods of critical points have good combinatorial and geometric properties. For instance, this notion plays an important role in [1,3].…”