on periodic orbits, and moreover the quality of this approximation is superpolynomial with respect to the period: see \[t.super\_pol\]. This extends a result of Morris [@Morris_rapidly], who gave a quantitative version of the celebrated theorem of Berger–Wang [@BWang].

We also introduce *Mather sets* in our context; these sets are the habitat of Lyapunov maximizing measures. We prove an important structural result on the existence of *dominated splittings* on the Mather sets, namely \[t.dom\], which is an essential ingredient in the proof of the aforementioned \[t.super\_pol\].

Organization of the paper
-------------------------

In \[s.setting\] we introduce the setting for our results, providing the definitions and properties of fiber-bunched automorphisms and related concepts. In \[s.subbundles\] we study irreducibility and related concepts.

In \[s.bounded\] we provide sufficient conditions for *relative product boundedness*, an intermediate property which is required for the existence of extremal norms.

The construction of extremal norms is given in \[s.norms\], together with the construction of Barabanov-like norms for shifts and an application to the regularity of $\beta (\mathord{\cdot})$.

In \[s.Mather\] we introduce Mather sets in a very general setting and, under the assumption of existence of an extremal norm, establish fine properties about them.

In \[s.app\] we collect several applications of 