A relation between ergodicity and regularity in pseudo-almost periodic equations is established. For this purpose the following operator is considered
It is shown that the following three statements are equivalent:
(1) The operator is regular.
(2) An solution to the homogeneous equation exhibits an exponential dichotomy.
(3) For every , the inhomogeneous equation has a unique solution in .
Here, and are the th powers of , respectively, where is the space of the bounded continuous functions on the real line supplied with the norm and is the space of differentiable functions with (with the norm ). is the space of pseudo-almost periodic functions and consists of those for which .
The main result states that, if the matrix is such that for all and , , are ergodic, then the operator is regular if and only if , . Furthermore, if and are in then the unique solution to is again in .