Some Remarks on Expanding Maps

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.