  In this work we treat a famous topic in Ergodic Theory and Dynamical Systems:
uniformly expanding maps. We relate regularity of expanding maps and
conjugacies with Lyapunov exponents, metric and topological entropies for
expanding maps of the circle. In Theorem C, we present a result of rigidity of
expanding maps on the torus $\mathbb{T}^2,$ after we naturally generalize it in
higher dimensions. Here, we present techniques involving ordinary differential
equations to study rigidity problems of expanding maps. Along of this work we
provide another proof of a Shub-Sullivan's Theorem, concerning the regularity
of the conjugacy between expanding endomorphism of the circle.
