  We introduce the Szeg\"o class, Sz(E), for an arbitrary Parreau-Widom set E
in R and study the dynamics of its elements under the left shift. When the
direct Cauchy theorem holds on C\E, we show that to each J in Sz(E) there is a
unique element J' in the isospectral torus, T_E, so that the left-shifts of J
are asymptotic to the orbit {J'_m} on T_E. Moreover, we show that the ratio of
the associated orthogonal polynomials has a limit, expressible in terms of Jost
functions, as the degree n tends to infinity. This enables us to describe the
large n behaviour of the orthogonal polynomials for every J in the Szeg\"o
class.
