In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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. Examples include the mathematical models that describe the swinging of a ...
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 City University of New York Graduate Ce ...
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 de ...
''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 model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. More precisely, for every
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assemb ...
''U'' in the
Fatou set
In the context of complex dynamics, a branch of mathematics, 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 wit ...
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 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 2006 fo ...
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 the ...
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
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