No wandering domain theorem
- No wandering domain theorem
In mathematics, the no wandering domain theorem is a result on dynamical systems, proved 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(2"πz") has wandering domains.
References
* Lennart Carleson and Theodore W. Gamelin, "Complex Dynamics", Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993, ISBN 0-387-97942-5 MathSciNet| id=1230383
* Dennis Sullivan, "Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains", Annals of Mathematics 122 (1985), no. 3, 401--418. MathSciNet| id=0819553
* S. Zakeri, " [http://www.math.qc.edu/~zakeri/notes/wander.pdf Sullivan's proof of Fatou's no wandering domain conjecture] "
Wikimedia Foundation.
2010.
Look at other dictionaries:
No-wandering-domain theorem — In mathematics, the no wandering domain theorem is a result on dynamical systems, 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… … Wikipedia
Wandering — can refer to: *Wandering (dementia) *Wandering, Western Australia *Shire of WanderingIt may also refer to: *Wandering Albatross *Wandering Detective *Wandering Genie *Wandering Jew *Wandering set or no wandering domain theorem *Wandering Spirit… … Wikipedia
Wandering set — In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non zero measure, then… … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… … Wikipedia
Julia set — In complex dynamics, the Julia set J(f), [Note that in other areas of mathematics the notation J(f), can also represent the Jacobian matrix of a real valued mapping f, between smooth manifolds.] of a holomorphic function f, informally consists of … Wikipedia
Dennis Sullivan — For other uses, see Dennis Sullivan (disambiguation). Dennis Sullivan Born February 12, 1941 … Wikipedia
Classification of Fatou components — In mathematics, if f = P(z) / Q(z) is a rational function defined in the extended complex plane, and if then for a periodic component U of the Fatou set, exactly one of the following holds: U contains an attracting periodic point U is parabolic U … Wikipedia
nature, philosophy of — Introduction the discipline that investigates substantive issues regarding the actual features of nature as a reality. The discussion here is divided into two parts: the philosophy of physics and the philosophy of biology. In this… … Universalium
History of Physics — History of Physics † Catholic Encyclopedia ► History of Physics The subject will be treated under the following heads: I. A Glance at Ancient Physics; II. Science and Early Christian Scholars; III. A Glance at Arabian Physics; IV.… … Catholic encyclopedia
