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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the no-wandering-domain theorem is a result on
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, proven by
Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University ...
in 1985. The theorem states that a
rational map In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal ...
''f'' : Ĉ → Ĉ with deg(''f'') ≥ 2 does not have a wandering domain, where Ĉ denotes the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
. More precisely, for every
component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...
''U'' in the
Fatou set In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values ...
of ''f'', the sequence :U,f(U),f(f(U)),\dots,f^n(U), \dots will eventually become periodic. Here, ''f'' ''n'' denotes the ''n''-fold iteration of ''f'', that is, :f^n = \underbrace_n . The theorem does not hold for arbitrary maps; for example, the transcendental map f(z)=z+2\pi\sin(z) has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.


References

*
Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 200 ...
and Theodore W. Gamelin, ''Complex Dynamics'', Universitext: Tracts in Mathematics,
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, New York, 1993, * Dennis Sullivan, ''Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains'',
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
122 (1985), no. 3, 401–18. * S. Zakeri,
Sullivan's proof of Fatou's no wandering domain conjecture
' Ergodic theory Limit sets Theorems in dynamical systems Complex dynamics {{chaos-stub