  Although recurrence for dynamical systems has been studied since the end of
the nineteenth century, the study of recurrence for linear operators started
with papers by Costakis, Manoussos and Parissis in 2012 and 2014. We explore
recurrence in Banach algebras, in the space of continuous linear operators on
$\omega=\mathbb{C}^{\mathbb{N}},$ and for composition operators whose symbols
are linear fractional transformations, acting on weighted Dirichlet spaces. In
particular, we show that composition operators with a parabolic non
automorphism symbol are never recurrent. Our results relate to work by
Gallardo-Guti\'errez and Montes-Rodr\'iguez in their 2004 AMS Memoir in which
they show, among other things, that composition operators induced by parabolic
non automorphisms are never hypercyclic in weighted Dirichlet spaces. Our work
also relates to a more recent paper by Karim, Benchiheb, and Amouch.
