On the ergodicity of the geodesic flow on manifolds of non-positive curvature
This text consists in lecture notes from a course given by the author at University of Tours, in 2010. It is a survey of dynamical properties of the geodesic flow on the unit tangent bundle of a nonpositively curved manifold, from a topological point of view, and from the point of view of ergodic theory, with emphasis on the similarities and differences between negatively curved manifolds, and nonpositively curved manifolds. Not all proofs are given, but the main arguments are given or sketched in a short and elegant style. In the first two parts, the author recalls convexity properties in nonpositive curvature, and investigates differences between negative and nonpositive curvature. He also shows how to prove transitivity of the geodesic flow, using recurrent vectors or hyperbolic periodic orbits. He describes the relations and differences between strong (un)stable manifolds of the geodesic flow and (un)stable horospheres. The third part contains definitions and classical results about rank-one manifolds, i.e., nonpositively curved manifolds which admit at least one hyperbolic orbit. The last and main part deals with the ergodicity of the geodesic flow. First, the author focuses on the ergodicity of the Liouville measure. He explains how this property follows from the Hopf argument and the absolute continuity of the strong (un)stable foliation. He cites \textit{D. V. Anosov}'s theorem [Tr. Mat. Inst. Steklova 90, 210 p. (1967; Zbl 0163.43604)] on ergodicity of the Liouville measure on negatively curved manifolds, and recalls the work of \textit{Ja. B. Pesin} [Russ. Math. Surv. 32, No. 4, 55--114 (1977; Zbl 0383.58011); Math. USSR, Izv. 11, 1195--1228 (1977; Zbl 0399.58010)] for nonpositively curved manifolds. Second, the author adresses the ergodicity of the measure of maximal entropy. In the same way, he begins with the works of \textit{R. Bowen} [Math. Syst. Theory 7(1973), 300--303 (1974; Zbl 0303.58014)] and \textit{G. A. Margulis} [Funct. Anal. Appl. 4, 55--67 (1970); translation from Funkts. Anal. Prilozh. 4, No.1, 62--76 (1970; Zbl 0245.58003)] in negative curvature, and afterwards explains the main results of \textit{G. Knieper} [Ann. Math. (2) 148, No. 1, 291--314 (1998; Zbl 0946.53045)] on rank-one manifolds. At last, he explains a result about the genericity of ergodicity, obtained in [the author and the reviewer, Isr. J. Math. 179, 157--172 (2010; Zbl 1229.53078)].
scientific article; zbMATH DE number 5976735