A review of some recent work on hypercyclicity

Abstract

Even linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinite-dimensional property, when iterations of the operator generate a dense subspace. A Frechet space admits a hypercyclic operator if and only if it is separable and infinite-dimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. This article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet spaces.

Invited paper, Workshop celebrating the 65 birthday of L. A. Cordero, Santiago de Compostela, June 27-29, 2012. 15 pages 116 references

References in corpus (11)

• Recurrent Linear Operators
• Second order tangent bundles of infinite dimensional manifolds
• Weighted diffeomorphism groups of Banach spaces and weighted mapping groups
• Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds
• Hypercyclic subspaces on Fréchet spaces without continuous norm
• Orbits of coanalytic Toeplitz operators and weak hypercyclicity
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• Some recent work in Frechet geometry
• On numerically hypercyclic operators
• Hypercyclic Abelian Affine Groups
• Growth of frequently Birkhoff-universal functions of exponential type on rays