Siegel Discs in Complex Dynamics

Tarakanta Nayak, Research Scholar

Department of Mathematics, IIT Guwahati

1 Introduction and Definitions

A dynamical system is a physical setting together with rules for how the setting changes or

evolves from one moment of time to the next or from one stage to the next. A basic goal of

the mathematical theory of dynamical systems is to determine or characterize the long term

behaviour of the system. The simplest model of a dynamical process supposes that (n+1)-

th state, zn+1 can be determined solely from a knowledge of the previous state zn, that is

zn+1 = f(zn) where f is a function. These systems are often called Discrete Dynamical

systems. We shall deal with one such kind of systems namely, Complex Dynamical Systems

In the study of Complex Dynamical Systems, the evolution of the system is realized by

iteration of entire or meromorphic complex functions f : C → Cˆ. For an initially chosen

point z0 ∈ C, the long term behavior of iterates {fn(z0)} is of primary importance. For

being more precise the following definition is required.

Definition 1.1. The family T of functions defined on the plane is said to be normal at

z ∈ C if every sequence extracted from T has a subsequence which converges uniformly

either to a bounded function or to ∞ on each compact subset of some neighbourhood of z.

In the present context, the family T is the sequence of iterates {fn}n>0.

A function f : C → Cˆ is said to be rational if it is of the form p(z)

q(z) where both p(z)

and q(z) are complex polynomials not having any common factor. The degree of f(z) is

defined by max{degree p(z), degree q(z)}). Any other function on C which is not rational

is called transcendental.

By a function, it shall be meant to be a rational function of degree more than one or a

transcendental function through out the article.

The set where {fn} is normal, is widely known as Fatou set(or stable set) of f, denoted

by F(f). The complement of F(f) in the extended complex plane is known as Julia set. A

detailed description of Fatou set follows.

Fatou Components:

The Fatou set of a function is open by definition. A Fatou component is a maximal

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connected open subset of F(f). A component U of F(f) is n-periodic if n is the smallest

natural number to satisfy fn(U) ⊆ U. Then {U, f(U), f 2(U)...fn−1(U)} is called an nperiodic cycle. If n = 1, U is called an invariant component. A Fatou component U is said

to be completely invariant if it is invariant and satisfies f −1(U) ⊆ U. A component U of

F(f) is said to be pre-periodic if there exists a natural number k > 1 such that f k(U) is

periodic.