DOMINATED PESIN THEORY: CONVEX SUM OF HYPERBOLIC MEASURES CHRISTIAN BONATTI AND KATRIN GELFERT Abstract. In the uniformly hyperbolic setting it is well known that the mea- sure supported on periodic orbits is dense in the convex space of all invariant measure. In this paper we consider the reverse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a con- vex sum of hyperbolic ergodic measures, what can we deduce on the initial measures? To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a convex sum of finitely many ergodic hyperbolic measures of the same index is accumulated by ergodic measures if, and only if, they share the same intersection class. This result also holds if the measures fail to be ergodic but are supported on hyperbolic sets. We provide examples which indicate the importance of the domination assumption. 1. Introduction 1.1. Quick presentation of the results. The space M(f ) of invariant measures by a homeomorphism f of a compact metric space is a compact metric space (for the the weak∗ topology) and is convex. The ergodic measures are the extremal points of this convex set and any invariant measure can be written as a convex sum of ergodic measures which is unique (up to 0-measure) and called its disintegration in ergodic measures or ergodic decomposition. Nevertheless, a typical picture of hyperbolic dynamics it that the ergodic measure associated to periodic orbits may be dense in M(f ). More precisely, if you consider a shift or a subshift of finite type, there are periodic orbits following an arbitrary itinerary. Hence, given any ergodic measures µ 1 ,...,µ k , there are periodic orbits which follow a given proportion of time a typical point of the measure µ 1 in an orbit segment long enough for approaching µ 1 and then follow µ 2 and so on, so that the measure obtained at the period is arbitrarily close to a given convex combination of the µ i . Now, if f is a diffeomorphism on a manifold and if Λ is an invariant (uniformly) hyperbolic basic set, the existence of a Markov partitions allows us to transfer this property to the set M(Λ) of invariant measures supported on Λ. 2010 Mathematics Subject Classification. 37C29, 37C40, 37C50, 37D25, 37D30, 28A33. Key words and phrases. dominated splitting, C 1 -Pesin theory, hyperbolic measures, Poulsen simplex, homoclinic relation. This paper was supported by Ciˆ encia Sem Fronteiras CAPES and CNPq (Brazil). The authors acknowledge the kind hospitality of Institut de Math´ ematiques de Bourgogne, Dijon (France) and of Departamento de Matem´atica, PUC-Rio (Brazil). 1 arXiv:1503.05901v1 [math.DS] 19 Mar 2015