Sullivan's theorem on the absence of wandering components

Sullivan's theorem on the absence of wandering components of a Fatou set is a theorem of holomorphic dynamics proved by D. Sullivan in 1985, which states that every connected component of a Fatou set is pre-periodic.

Wording

Theorem. Let be

f:{\hat {\mathbb {C} }}\to {\hat {\mathbb {C} }}

{\ displaystyle f: {\ hat {\ mathbb {C}}} \ to {\ hat {\ mathbb {C}}}}

- rational mapping of the Riemann sphere to itself

\deg f\geq 2,

{\ displaystyle \ deg f \ geq 2,}

and U is the connected component of the Fatou set

F(f)

{\ displaystyle F (f)}

. Then U is preperiodic, i.e., there are

n\geq 0,m>0

{\ displaystyle n \ geq 0, m> 0}

for which

f^{m+n}(U)=f^{n}(U)

{\ displaystyle f ^ {m + n} (U) = f ^ {n} (U)}

.

Literature

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.
Milnor, J. Holomorphic dynamics. Introductory lectures. = Dynamics in One Complex Variable. Introductory Lectures. - Izhevsk: Research Center "Regular and chaotic dynamics", 2000. - 320 p. - ISBN 5-93972-006-4 .

Sullivan's theorem on the absence of wandering components

Wording

Literature

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