The Study Of Some Questions Concerning The Topological And Geometric Properties Of Julia Sets Of Singularly Perturbed Rational Maps

This doctoral thesis consists of three parts:The first part is the study concerning the buried components of Julia sets of singularly perturbed rational functions.As an important object of research in complex analytic dynamical systems,the dynamics of rational functions have always been of great interest.One of the typical research topics is the geometric and topological properties(such as the connectivity and local connectivity of Julia sets and the area of Julia sets)and its Hausdorff dimension,etc.Before this study,there are very beautiful and important results on Julia sets of rational functions.A t the same time they also argued that there are buried points or buried components of Julia sets of some specially rational functions and gave the conditions for existence of their Julia sets with this property.Then the concrete expressions of those rational functions are given.Furthermore,those buried components on its Julia set are not quasiconformally conjugate to the connected Julia set of another function on the Julia set.In this thesis,the author proves the existence of such a rational function such that its connected Julia set can be embedded in the Julia set of another rational functions with higher degree in the buried manner.To be more precise,the author obtains a semi-buried component by means of the quasiconformal surgery and the first perturbation,and then obtains a buried component by means of the quasiconformal surgery and perturbation again.The degree of this rational function with higher degree is estimated.The second part is the study of a family of singularly perturbed rational maps with Julia Cantor set of circles.As the maps constructed is a singularly perturbed rational maps of P-n(z)=z-n,the Julia sets of those maps in this thesis are Cantor sets of circles.But the dynamics of the maps constructed on their Julia sets which are Cantor sets of circles are not topologically conjugate to the dynamics of any known maps(including McMullen maps)on their corresponding Julia sets which are Cantor sets of circles.Firstly,the author studies the case that one of the free critical points is attracted by the super-attracting orbit 0(?)?(hyperbolic case),and classify their Julia sets according to the iterating time of the free critical points which will be escape to the super-attracting basin of 0 or ?.The Julia set given by this family can be one of the following four types:a quasi-circle,a Cantor set of circles,a Sierpinski carpet and a degenerated Sierpinski carpet.It can be concluded that this family possess rich dvnamical behaviors.Secondly,the author gives a description on the regularity of the boundary of its Fatou components in the hyperbolic case.The author proves that the boundary of each Fatou component must be a quasicircle and estimates the Hausdorff dimension of its Julia sets in the case.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 and estimates the Hausdorff dimension of its Julia sets again.Finally,the author also studies the connectivity of the Julia sets of the maps in the case that one of the free critical points is not attracted by the superattracting orbit 0(?)?c.For the case that one of the free critical points of the rational maps constructed is not attracted by the superattracting orbit 0(?)?,the author prove that its Julia sets are connected.Combining the hyperbolic case,it is easy to see that the Julia set is disconnected if and only if it is a C antor set of circles.The third part is the study considering the sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for anti-ferromagnetic coupling,it is shown that these sets are the Julia sets of a family of rational mappings.In this part,we prove that they can be buried points of J(T?(z))for some ??R.Further,the asymptotic formula of the Hausdorff dimension of the Julia set is given as ??? which gives a lower bound of the Hausdorff dimension of the Julia set J(T?(z)).Finally,other topological structures of Julia set are discussed completely.