  We study different pointwise recurrence notions for linear dynamical systems
from the Ergodic Theory point of view. We show that from any reiteratively
recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach
space $X$, one can construct a $T$-invariant probability measure which contains
$x_0$ in its support. This allows us to establish some equivalences, for these
operators, between some strong pointwise recurrence notions which in general
are completely distinguished. In particular, we show that (in our framework)
reiterative recurrence coincides with frequent recurrence; for complex Hilbert
spaces uniform recurrence coincides with the property of having a spanning
family of unimodular eigenvectors; and the same happens for power-bounded
operators on complex reflexive Banach spaces. These (surprising) properties are
easily generalized to product and inverse dynamical systems, which implies some
relations with the respective hypercyclicity notions. Finally we study how
typical is an operator with a non-zero reiteratively recurrent vector in the
sense of Baire category.
