  Chaotic linear dynamics deals primarily with various topological ergodic
properties of semigroups of continuous linear operators acting on a topological
vector space. We treat questions of characterizing which of the spaces from a
given class support a semigroup of prescribed shape satisfying a given
topological ergodic property.
  In particular, we characterize countable inductive limits of separable Banach
spaces that admit a hypercyclic operator, show that there is a non-mixing
hypercyclic operator on a separable infinite dimensional complex Fr\'echet
space $X$ if and only if $X$ is non-isomorphic to the space $\omega$ of all
sequences with coordinatewise convergence topology. It is also shown for any
$k\in\N$, any separable infinite dimensional Fr\'echet space $X$ non-isomorphic
to $\omega$ admits a mixing uniformly continuous group $\{T_t\}_{t\in C^n}$ of
continuous linear operators and that there is no supercyclic strongly
continuous operator semigroup $\{T_t\}_{t\geq 0}$ on $\omega$. We specify a
wide class of Fr\'echet spaces $X$, including all infinite dimensional Banach
spaces with separable dual, such that there is a hypercyclic operator $T$ on
$X$ for which the dual operator $T'$ is also hypercyclic. An extension of the
Salas theorem on hypercyclicity of a perturbation of the identity by adding a
backward weighted shift is presented and its various applications are outlined.
