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Sylvain Crovisier; Raphael Krikorian; Carlos Matheus; Samuel Senti. (Ed.)We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be cal- culated 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 unimodal 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 topo- logical conjugacy. Another consequence is for the thermodynamic formalism: A non- degenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic.more » « less
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We study the essential minimum of the (stable) Faltings' height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.more » « less
