No wandering domain theorem
Encyclopedia
In mathematics
, the no-wandering-domain theorem is a result on dynamical system
s, proven by Dennis Sullivan
in 1985.
The theorem states that a rational map
f : Ĉ → Ĉ with deg(f) ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere
. More precisely, for every component
U in the Fatou set of f, the sequence
will eventually become periodic. Here, f n denotes the n-fold iteration of f, that is,
The theorem does not hold for arbitrary maps; for example, the transcendental map
f(z) = z + sin(2πz) has wandering domains.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the no-wandering-domain theorem is a result on dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s, proven by Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
in 1985.
The theorem states that a rational map
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
f : Ĉ → Ĉ with deg(f) ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. More precisely, for every component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
U in the Fatou set of f, the sequence
will eventually become periodic. Here, f n denotes the n-fold iteration of f, that is,
The theorem does not hold for arbitrary maps; for example, the transcendental map
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
f(z) = z + sin(2πz) has wandering domains.