On Closed Invariant Sets in Local Dynamics
Cinzia Bisi2007, arXiv: Complex Variables
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Abstract
We investigate the dynamical behaviour of a holomorphic map on an f-invariant subset C of U , where f : U → C k. We study two cases: when U is an open, connected and polynomially convex subset of C k and C U , closed in U , and when ∂U has a p.s.h. barrier at each of its points and C is not relatively compact in U. In the second part of the paper, we prove a Birkhoff's type theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f , defined in a neighborhood of U , with U star-shaped and f (U) a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of U which touches ∂U .
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