  This series of two papers is devoted to the study of the principal spectral
theory of nonlocal dispersal operators with almost periodic dependence and the
study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with
almost periodic dependence. In this first part of the series, we investigate
the principal spectral theory of nonlocal dispersal operators from two aspects:
top Lyapunov exponents and generalized principal eigenvalues. Among others, we
provide various characterizations of the top Lyapunov exponents and generalized
principal eigenvalues, establish the relations between them, and study the
effect of time and space variations on them. In the second part of the series,
we will study the asymptotic dynamics of nonlinear nonlocal dispersal equations
with almost periodic dependence applying the principal spectral theory to be
developed in this part.
