Generalized dichotomies via time rescaling

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $ \mu $, we show that the notion of $ \mu $ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain certain nonautonomous topological and smooth linearization results.