Titlebook: Linear Chaos; Karl-G. Grosse-Erdmann,Alfred Peris Manguillot Textbook 2011 Springer-Verlag London Limited 2011 Chaos.Dynamical systems.Hyp

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Hypercyclic and chaotic operatorsis shown that every hypercyclic operator possesses a dense subspace all of whose nonzero vectors are hypercyclic (the Herrero–Bourdon theorem), and that linear dynamics can be as complicated as nonlinear dynamics. We begin the chapter with an introduction to Fréchet spaces since they provide the setting for some important chaotic operators.

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Connectedness arguments in linear dynamics that every multi-hypercyclic operator is hypercyclic, the León–Müller theorem that any unimodular multiple of a hypercyclic operator is hypercyclic, and the Conejero–Müller–Peris theorem that every operator in a hypercyclic semigroup is hypercyclic.

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Existence of hypercyclic operatorse set of hypercyclic operators in two ways: it forms a dense set in the space of all operators when endowed with the strong operator topology; and it is shown that any linearly independent sequence of vectors appears as the orbit under a hypercyclic operator.

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Hypercyclic subspacese existence of hypercyclic subspaces. The first proof provides an explicit construction via basic sequences, the second one relies on the study of left-multiplication operators. We also obtain conditions that prevent the existence of hypercyclic subspaces; as an application we show that Rolewicz’s operators do not have hypercyclic subspaces.

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