  In this paper, we introduce the notion of recurrence entropy in the context
of nonlinear control systems. A set is said to be ($\tau$-)recurrent if every
trajectory that starts in the set returns to it (within at most $\tau$ units of
time). Recurrence entropy quantifies the complexity of making a set
$\tau$-recurrent measured by the average rate of growth, as time increases, of
the number of control signals required to achieve this goal. Our analysis
reveals that, compared to invariance, recurrence is quantitatively less
complex, meaning that the recurrence entropy of a set is no larger than, and
often strictly smaller than, the invariance entropy. Our results further offer
insights into the minimum data rate required for achieving recurrence. We also
present an algorithm for achieving recurrence asymptotically.
