We study the asymptotic behaviour of contractive operators and strongly
continuous semigroups on separable Hilbert spaces using the notion of rigidity.
In particular, we show that a "typical" contraction $T$ contains the unit
circle times the identity operator in the strong limit set of its powers, while
$T^{n_j}$ converges weakly to zero along a sequence $\{n_j\}$ with density one.
The continuous analogue is presented for isometric ang unitary
$C_0$-(semi)groups.