On The Accumulation Sets Of Exponential Escaping Rays And The Julia Sets Of The Singularly Perturbed Rational Maps

This doctoral thesis mainly includes the following two parts:The first part is the study on the accumulation sets of exponential escaping rays. As a canonical object of the study of transcendental entire dynamical systems, the dynamics of exponential maps has been attracted much attention. One of the important topics is the study on the non-landing escaping rays. It was known that the types of non-landing escaping rays constructed before all have unbounded accumulation sets. More accurately, their accumulation sets are all indecomposable continua which are unbounded in the complex plane, and must contain some escaping rays.In this thesis, the author constructs the exponential escaping rays whose accumu-lation sets are compact subsets in complex plane. Based on this, the author constructs three new types of escaping rays by introducing the folding model. For each concerned escaping ray, the author defined the running-back sequence for it. Then according to the combinatorial characteristics of the running-back sequence, the author classifies their accumulation sets into three cases:(1) an indecomposable continuum containing part of the ray; (2) an indecomposable continuum disjoint from the ray; (3) a Jordan arc.The second part is the dynamics of a family of singularly perturbed rational maps. As a singularly perturbed rational maps of Pn(z)=zn, the Julia sets of the maps constructed in this thesis are Cantor sets of circles. However, the dynamics of the maps on their Julia sets are not topologically conjugate to the dynamics of the traditional McMullen maps on their corresponding Cantor sets of circles.On the one hand, the author studies the case that one of the free critical points escapes to the superattracting basin of 0 or ∞ (hyperbolic case), and give a classifica-tion of their Julia sets according to the escaping time of the free critical points that will be iterated to the superattracting basin of 0 or oo. The Julia set obtained by this fam-ily can be a quasicircle, a Cantor set of circles, a Sierpiriski carpet and a degenerated Sierpiriski carpet. It can be seen that this family produces rich dynamical behaviors. Moreover, the author also gives specific parameters to guarantee that the corresponding cases happen indeed. In particular, the author gives a description on the regularity of the boundary of the immediate attracting basin of infinity. The author proves that the boundary of the immediate attracting basin of ∞ must be a quasicircle. For the case that the Julia set is a quasicircle, the author gives the exact range of the real parameter. For the Cantor set of circles case, the author gives a necessary and sufficient condition about the degree of the map for the existence of Cantor set of circles.On the other hand, the author also studies the connectivity of the Julia sets of the maps in all cases. Specifically, the author gives a necessary and sufficient conditions for the disconnected Julia sets by discussing whether the free critical orbits are escaping to the superattracting basin of 0 or ∞:the Julia set is disconnected if and only if it is a Cantor set of circles. Equivalently, this family has a critical value in the superattracting basin of 0 or oo, but the corresponding critical points does not contained in this basin. This result can be regarded as an analogy of the connectivity of the Julia set of the classical quadratic polynomials.