This mainly expository article is devoted to recent advances in the study of
dynamical aspects of the Cuntz algebras O_n, with n finite, via their
automorphisms and, more generally, endomorphisms. A combinatorial description
of permutative automorphisms of O_n in terms of labeled, rooted trees is
presented. This in turn gives rise to an algebraic characterization of the
restricted Weyl group of O_n. It is shown how this group is related to certain
classical dynamical systems on the Cantor set. An identification of the image
in Out(O_n) of the restricted Weyl group with the group of automorphisms of the
full two-sided n-shift is given, for prime n, providing an answer to a question
raised by Cuntz in 1980. Furthermore, we discuss proper endomorphisms of O_n
which preserve either the canonical UHF-subalgebra or the diagonal MASA, and
present methods for constructing exotic examples of such endomorphisms.Comment: 2 figures, uses pictex, to appear in the Proceedings of the Workshop
on Noncommutative Harmonic Analysis, Bedlewo 201
