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Hypercyclicity and linear chaos is a relatively recent area of work which, over the last two decades, has
developed into a very active research area. In the past few years several open problems, some
of which long-standing, have been solved, and a number of landmark results have been
obtained. While chaos is commonly believed to be intrinsically linked to non-linearity, the
investigations into linear dynamics have thoroughly refuted this assumption. Many, even quite
natural, linear dynamical systems exhibit chaos. This effect, however, is only visible when one
studies infinite-dimensional phase spaces.
From its very beginning, linear dynamics has been at the crossroads of several areas of
mathematics (Operator theory, Dynamical systems, Complex analysis, Semigroups and
applications to partial differential equations, Ergodic theory). An operator T on a Banach space
X is called hypercyclic if there is a vector x in X whose orbit {x, Tx, T^2x, ...} is dense in X.
The interest of operator theorists in this notion was motivated by the invariant subspace problem.
A more dynamical point of view then suggested to call an operator chaotic if it is hypercyclic
and it has a dense set of periodic points. Further extensions, like hypercyclic semigroups and
supercyclic operators, followed, leading to what may now be regarded as the Theory of Linear
Dynamical Systems.