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 Fr´echet 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 Fr´echet spaces.