h-dichotomies via noncritical uniformity and expansiveness for noninvertible evolution families

Abstract

We establish characterizations of h-dichotomies for noninvertible evolution families acting on Banach spaces, in terms of appropriate notions of expansiveness and uniform noncriticality. Our results extend the corresponding characterizations previously obtained for invertible evolution families. To the best of our knowledge, no such characterizations have previously been established for general noninvertible evolution families. A key difference is that noninvertibility makes the structure of the stable spaces more delicate: in our positive-half-line result, finite codimension must be imposed at every time, rather than only at the initial time. We show by an example that the latter, weaker condition is insufficient. We also study evolution families on the negative half-line, using duality to obtain corresponding characterizations in terms of dual expansiveness and dual uniform noncriticality. General growth rates are treated by a time-rescaling argument.