Research Of Polynomial-like Mappings In Complex Dynamics

The research of complex dynamics began with the study of Newton’s method in solving complex polynomial equations. The elementary theory of complex dynamics was established by Julia and Fatou using the theory of nor-mal family in complex analysis. After that, Ahlfors and Bers introduced the quasi-conformal map and the Teichmuller space theory as the new tools in the study of complex dynamics and complex dynamics began to enter into a new stage of development. During this period of time, many important questions were resolved, one of which was accomplished by Yocooz who win the Feilds Medal in1994. At the same time, as the using of computers in complex dy-namics study become more and more comprehensive, the study of the Julia set and Mandelbrolte set had also achieved a great development. Generally speaking, the study of complex dynamics mainly focused on complex polyno-mials and rational functions, while the results about holomorphic functions and meromorphic functions are relatively much less. we mainly interested in the dynamical behaviors near the Julia set where "chaotic" dynamics occur.In1985, A.Douady and J.IIubbard first introduced the concept of polyn omial-like mapping in their work of "On the Dynamics of Polynomial-like Map-pings" and proved the so called "straightening theorem" which says that for an arbitrary polynomial-like map f there is a quasi-conformal map φ and a polynomial P of the same degree, such that f=φ-1o fo φ on a neighbor-hood of Kf and (?)φ=0on Kf. Its proof relies on the "measurable Riemann mapping Theorem". In2008,T-C.Dinh and N.Sibony[7] tried to generalize the polynomial-like mapping theory in the background of several complex vari-ables, but they didn’t get the corresponding straightening theorem. In2012, II.Inou [8]extending the local analytic conjugation to global and obtained a interesting corollary about polynomial-like map which can be viewed as a kind of generalization of the straightening theorem.This paper mainly researches the properties and applications of polynomial-like mapping in complex dynamics. Our major work are based on A.Douady and J.IIubbard’s important paper about polynomial-like mappings. This the-ory plays an important role in the research of Julia set of general functions and the renomalization theory in complex dynamics. We will first introduce the straightening theorem and then illustrate its applications in the study of Julia set of general function with several examples. Our results about the polynomial-like mapping are about two unproved conclusions:Theorem2.3.2There is at least one critical point in the immediate basin of an attractive cycle.Theorem2.3.4The set Kf is connected if and only if all the critical points of f belong to Kf. If none of the critical points belong to Kf then Kf is a Cantor set. which we give exhaustive proofs and an implicit fact that the filled Julia set isn’t empty. We also have the following theorem after summarizing the process of the proof of the straightening theorem:Theorem2.3.5Let f:U’�'U be a polynomial-like map of degree d>1. There exists a polynomial of the same degree and a quasi-conformal map φ:C�'C with φ�'(∞)=∞such that φ o f o φ-1=P in a neighborhood of the filled Julia set Kf. Moreover,(?)φ=0on Kf.In the end of this paper, we compliment some results on meromorphic solutions of linear difference equations which is a cooperative work with Cui-WeiWei. Hayman conjectured [6] that if f is a transcendental meromorphic functions and n∈N, then fn f’ takes every finite nonzero value infinitely many times. This conjecture has been solved by Hayman [7] for n>3, by Mues [8] for n=2, by Bergweiler and Eremenko [9] for n=1. As to an analog of Hayman conjecture for difference, Laine and Yang [10, Theorem2] proved:Theorem A. Let f be transcendental entire function with finite order and c be a nonzero complex constant. Then for n≥2, f(z)nf(z+c) assumes every nonzero value a∈C infinitely often.Liu Kai [11] improved Theorem A to the case of meromorphic functions, and obtain the following theorem. In the following, we assume that α(z), β(z) are small functions with respect to f.Theorem B. Let f be transcendental meromorphic function with finite order and c be a nonzero complex constant. If n≥6, then the difference polynomial f(z)nf(z+c)-α(z) has infinitely many zeros. He also obtains that:Theorem C. Let f be transcendental meromorphic function with finite order and c be a nonzero complex constant. If n≥>7, then the difference polynomial f(z)n[f(z+c)-f(z)]-a(z) has infinitely many zeros.We extend Theorem C to a general situation and obtain the following theorem:Theorem3.1.1. Let f be transcendental meromorphic function with finite order and c be a nonzero complex constant. If n≥m+7, then the difference polynomial f(z)n[f(z+c)-f(z)]m-α(z) has infinitely many zeros.And we also consider the zeros of other difference polynomials:Theorem3.1.2. Letf be transcendental meromorphic function with finite order and c be a nonzero complex constant. If n>11. then the differ-ence polynomial f(z)n[f(z+c)-f(z)]f(z+c)-α(z) has infinitely many zeros.Using similar method of proof of Theorem4.1.1, we can easily obtain The-orem4.1.3:Theorem3.1.3. Let f be transcendental meromorphic function with finite order and c be a nonzero complex constant. If n+m≥3, then the difference polynomial f(z)n[f(z)-1]mf(z+c)-a(z) has infinitely many zeros.Recently, many authors have considered the uniqueness problems of d-ifference polynomials of meromorphic, such as Liu [11] and Qi [12]. They considered the uniqueness problem when two difference polynomials share one value and obtain the following results:Theorem D ([12]). Let f and g be transcendental meromorphic functions with finite order and c be a nonzero complex constant. If n≥6, f(z)nf(z+c) and has g(z)ng(z+c) share z CM, then f=t1g for a constant t1that satisfies t1n+1=1.Theorem E ([11]). Let f and g be transcendental entire functions with finite order and c be a nonzero complex constant and∈N. If n≥14, f(z)nf(z+c) and has g(z)ng(z+c) share1CM, then f=tg or fg=t where tn+1=1. The proofs of other difference polynomials are much more difficult. But under some other conditions, we can give the same results. We obtain the following theorems:Theorem3.1.4. Let f(z) and g(z) be transcendental entire functions with finite order and c be a nonzero complex constant and n∈N. If n> m+6, f(z)n[f(z)m-a]f(z+c) and has g(z)n[g(z)m-a]g(z+c) share1CM, then f=lg where Lm=1.Theorem3.1.5. Let J’(z) and g(z) be transcendental meromorphic func-tions with finite order≥1and max{λ(f), λ(g)}<min{σ(f),σ((g)}, c be a nonzero complex constant and n j N. If n≥17, f(z)n[f(z)-1]f(z+c) and has g(z)n[g(z)-1]g(z+c) share1CM, then f≡g.