and Strang [@RS].

If, for example, $N=1$ and $A_0 = \left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$, then no extremal norm exists. However, if the set $\{A_0,\dots,A_{N-1}\}$ is *irreducible*, in the sense that there is no common invariant non-trivial subspace, then extremal norms ${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {\cdot}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}$ do exist, and can be taken so that ${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {u}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}_x$ is independent of $x \in X$. Actually, Barabanov [@Barabanov] proved that there exists a norm ${\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {\cdot}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}}$ on ${\mathbb{R}}^d$ with the following stronger property: $$\label{e.Barabanov}
\forall u \in {\mathbb{R}}^d, \quad
\max_{i \in \{0,\dots,N-1\}} {\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {A_i u}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}} = e^{\beta(\Phi)} {\mathopen{| {\kern -1.5pt} | {\kern -1.5pt} |} {u}
\mathclose{| {\kern -1.5pt} | {\kern -1.5pt} |}} \, .$$ For more information on the joint spectral radius and Barabanov norms, see [@Wirth; @Jungers]. Further applications of extremal norms were obtained by Morris [@Morris_rapidly; @Morris_Mather].

Still in the setting of one-step cocycles, a modification 