Asymptotic expansion of smooth interval maps
  We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various different ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unicritical maps. This also solves a conjecture of Luzzatto in dimension 1.   Combined with a result of Nowicki and Przytycki, these considerations imply that several natural nonuniform hyperbolicity conditions are invariant under topological conjugacy. Another consequence is for the thermodynamic formalism of non-degenerate smooth interval maps: A non-degenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic. 
Juan Rivera-Letelier
math.DS