Lyapunov Exponents Everywhere and Rigidity

We study the possibility that Anosov or expanding maps have Lyapunov exponents defined everywhere. We discover that, in low dimensions, we have that the maps with exponents defined everywhere are smoothly conjugate to linear maps. In higher dimensions, we present somewhat weaker results ($C^{1 +\varepsilon}$ conjugacy with extra hypothesis on the spectrum of the homology or proximity to linear) and we exhibit examples of $C^{\infty}$ maps which have Lyapunov exponents everywhere, but are are not $C^1$ conjugate to linear.