Elementary Results Of Dynamics In One Complex Variable

Complex dynamics has become more and more popular. It has a history dates back to the first world war, and from 80s it has become eminent in modern sciences. In this paper, we consider some elementary results of dynamics in one complex variable, concentrating mainly on Julia set, Fatou set, periodic points, properties of Julia set, classical results concerning the Fatou set, Mandelbrot set and other topics such as polynomial maps and some recent developments, etc.We give the definition of the Julia set and the Fatou set. We concern mainly analytic functions on the Riemann sphere C.Definition 1.1. Let f(z) be an analytic function on the Riemann sphere C. (?) z∈C, if{fn} is a normal family on some neighborhood of point z, then we call z is a steady point, otherwise we call z is an unsteady point. Let F(f) stands for the set of all steady points in C, we call it the Fatou set of f(z), and let J(f) stands for the set of all unsteady points in C, we call it the Julia set of f(z). Now we have C= F(f)∪J(f).Here are some basic properties of the Julia set.Proposition 1.1. F(f) is closed while J(f) is an open set.Proposition 1.2. F(f) and J(f) is fully invariant under f(z).Proposition 1.3. Let f(z) be an analytic function, and suppose degf> 1, then J(f) is nonempty.Definition 1.2. Let z0∈C be an n periodic point of f(z), and the periodic orbit is O+(z o). We callλ= (fn)'(z0) the eigenvalue of z0 and O+(z0). According A we classify O+(z0) as follows:ifλ=0, then we call O+(z0) superattracting; if0<|λ|< 1, then we call O+(z0) attracting;if|λ|= 1, then we call O+(z o) indifferent;if|λ|> 1, then we call O+(z o) repelling.Theorem 1.1. Attracting and superattracting periodic orbits are contained in the Fatou set, while repelling periodic orbits are contained in the Julia set. In fact the basin of attractionΩof the attracting and superattracting periodic orbits are contained in the Fatou set, while the boundary (?)ΩofΩis contained in the Julia set.Proof can be found in Milnor[15].There are some more properties of the Julia set.Proposition 1.4. If J(f) has interior points, then J(f)= C.Proposition 1.5. J(f) has no isolated points.Proposition 1.6. J(f) is a perfect set.Proposition 1.7. (?) z 0∈J(f), the set∪f-n(z0) is dense on J(f).There are some properties concerning the Fatou components.Theorem 1.2. The Fatou components of analytic functions are eventually periodic.Theorem 1.3. Let D be a steady components of f(z), and fn(D)= D. then there are five possibilities of D:(1) there is a superattracting periodic point of f(z) in D, every periodic orbits in D are attracted by it, and there exist at least one critical point of f(z) in∪fi(D);(2) there is an attracting periodic point of f(z) in D, every periodic orbits in D are attracted by it, and there exist at least one critical point of f(z) in∪fi(D);(3) there is a rational indifferent periodic point of f(z) in D, every periodic orbits in D are attracted by it, and there exist at least one critical point of f(z) in∪fi(D);(4) there is an irrational indifferent periodic point of f(z) in D, Dis simply connected, and f (z)|D conjugates to an irrational indifferent rotation. In this case we call D a Siegel disk;(5) D is conformal to an annuls, and f(z)|D conjugates to an rotation. In this case we call D a Herman ring. Proof can be found in D.Sullivan[18].We gave the definition of the Mandelbrot set M.Definition 1.3. M={c|J(fc) is connected}, or M={c|fc n(0)→∞}.A.Douady and J.H.Hubbard studied the Mandelbrot set in depths, In 1982 they gave the most important results of the Mandelbrot set M in [10].Theorem 1.4. The Mandelbrot set is connected.There are peculiar results in the Fatou components of the transcendental entire functions and meromorphic functions.Theorem 1.5. Let U be a Fatou components, then there exists the sixth possibilities:there exists z0∈(?)U such that if n→∞, (?)z∈U then fnp(z)→Z0,but fp(z0) is not defined. In this case we call U a Baker domain.For proof see [7] [23].