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Orbits of homogeneous polynomials on Banach spaces

Part of: Holomorphic mappings and correspondences General theory of linear operators Complex dynamical systems Nonlinear operators and their properties

Published online by Cambridge University Press:  13 April 2020

RODRIGO CARDECCIA  and
SANTIAGO MURO
RODRIGO CARDECCIA
Affiliation:
Departamento de Matemática – PAB I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina email rcardeccia@dm.uba.ar IMAS-CONICET, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
SANTIAGO MURO
Affiliation:
Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, Argentina CIFASIS-CONICET, Universidad Nacional de Rosario, Argentina email muro@cifasis-conicet.gov.ar
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Abstract

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

Keywords

Polynomical dynamicshypercyclic operatorsJulia sets

MSC classification

Primary: 47H60: Multilinear and polynomial operators 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 32H50: Iteration problems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1627 - 1655
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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