A review of some recent work on hypercyclicity

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 Fréchet 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. The main part of this article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Fréchet spaces. M.S.C. 2010: 58B25 58A05 47A16, 47B37.