On Non-uniform Hyperbolicity Assumptions And Dimensions Of Julia Sets In Complex Dynamics

The thesis consists of two topics of non-uniform hyperbolicity assumptions and dimensions of Julia sets in complex dynamics. In the first part, we give essentially equivalent formulations of the backward contracting property defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values. In the second part, we study the relations of fractal dimensions, by investigating the properties of conformal measures on the Julia set.The thesis is organized as follows:In chapter 1, we briefly recall the origin, developments and main objective in complex dynamics, and then we introduce the backgrounds and main results of the thesis.In chapter 2, we review some preliminary notions and results in complex analysis and complex dynamics, which will be used in our thesis.In chapter 3, we first describe a Yoccoz puzzle partition, by making use of equipo-tential curves for the Green function and all bounded periodic Fatou components, ex-ternal rays and internal rays, and then we give the precise form of complex bounds theorem.In chapter 4, we give essentially equivalent formulation of the backward contract-ing property defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least 2 which are at most finitely renormalizable and have only hyperbolic periodic points.In chapter 5, we prove that a rational map f of degree at least 2 which has no parabolic points and J(f)≠(?), satisfies the summability condition with exponentβ∈(0,1] if and only if there existδ0> 0 and a function r:(0,δ0)→(1,+∞) such that f is backward contracting with growth function r(δ) andIn chapter 6, we show that the upper box dimension of the Julia set J(f) is equal to the hyperbolic dimension for a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points and satisfies backward contraction property with an arbitrarily large constant.In chapter 7, we consider a polynomial f of degree at least 2 which is at most finitely renormalizable and has only hyperbolic periodic points. Assuming that all crit-ical points of f in J(f) are reluctantly recurrent, we show that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension. Moreover, if there is only one critical point c in the Julia set J(f) and J(f) is not equal to the closure of the forward orbit of the critical point, then the box dimension of the Julia set J(f) is equal to the hyperbolic dimension.