Reflection symmetries of almost periodic functions

We study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit pe riodic case, the criterion of Jitomirskaya and Simon ensuring absence of ei genvalues for almost periodic Schrodinger operators is only applicable on a set of zero Haar measure. We complement this by giving examples of limit p eriodic functions where the Jitomirskaya Simon criterion can be applied to every element of the hull. (C) 2000 Academic Press.