  An operation that maps one natural number to another can be considered as a
dynamical system in $\mathbb{N}^+$. Some of such systems, e.g. the mapping in
the so-called 3x+1 problem proposed by Collatz, is conjectured to have a single
global attractor, whereas other systems, e.g. linear congruence, could be
ergodic. Here we demonstrate that an operation that is based on digital
reversal, has a spectrum of dynamical behaviors, including 2-cycle, 12-cycle,
periodic attractors with other cycle lengths, and diverging limiting dynamics
that escape to infinity. This dynamical system has infinite number of cyclic
attractors, and may have unlimited number of cycle lengths. It also has
potentially infinite number of diverging trajectories with a recurrent pattern
repeating every 8 steps. Although the transient time before settling on a
limiting dynamics is relatively short, we speculate that transient times may
not have an upper bound.
