  The periodic (ordinal) patterns of a map are the permutations realized by the
relative order of the points in its periodic orbits. We give a combinatorial
characterization of the periodic patterns of an arbitrary signed shift, in
terms of the structure of the descent set of a certain cyclic permutation
associated to the pattern. Signed shifts are an important family of
one-dimensional dynamical systems that includes shift maps and the tent map as
particular cases. Defined as a function on the set of infinite words on a
finite alphabet, a signed shift deletes the first letter and, depending on its
value, possibly applies the complementation operation on the remaining word.
For shift maps, reverse shift maps, and the tent map, we give exact formulas
for their number of periodic patterns. As a byproduct of our work, we recover
results of Gessel--Reutenauer and Weiss--Rogers and obtain new enumeration
formulas for pattern-avoiding cycles.
