  Let $E$ be a homogeneous compact set, for instance a Cantor set of positive
length. Further let $\sigma$ be a positive measure with
$\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous
part of $\sigma$ satisfies a Szeg\"o--type condition we give an asymptotic
representation, on and off the support, for the polynomials orthonormal with
respect to $\sigma$. For the special case that $E$ consists of a finite number
of intervals and that $\sigma$ has no singular component this is a nowaday well
known result of Widom. If $E=[a,b]$ it becomes a classical result due to
Szeg\"o and in case that there appears in addition a singular component, it is
due to Kolmogorov--Krein. In fact the results are presented for the more
general case that the orthogonality measure may have a denumerable set of
mass--points outside of $E$ which are supposed to accumulate on $E$ only and to
satisfy (together with the zeros of the associated Stieltjes function) the
free--interpolation Carleson--type condition. Up to the case of a finite number
of mass points this is even new for the single interval case. Furthermore, as a
byproduct of our representations, we obtain that the recurrence coefficients of
the orthonormal polynomials behave asymptotically almost periodic. Or in other
words the Jacobi matrices associated with the above discussed orthonormal
polynomials are compact perturbations of a one--sided restriction of almost
periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory
of Hardy spaces of character--automorphic functions and forms on Riemann
surfaces of Widom type, we use also some ideas of scattering theory for
one--dimensional Schr\"odinger equations.
