A remark on uniform expansion

Abstract

For every U ⊂ Diffvol(T) there is a measure of finite support contained in U which is uniformly expanding. 0. Introduction Let μ be a probability measure in Diff(M) where M is a closed manifold of dimension d := dim(M). We denote μ(1) = μ and μ(n) = μ∗μ(n−1). Note that μ(n) is the pushforward by the composition of the product measure μ in (Diff(M))n. Definition 0.1 ([8, 4]). A probability measure μ in Diff(M) is uniformly expanding if there exists N > 0 such that for every (x, v) ∈ T 1M one has that ∫ log ‖Dxfv‖ dμ(N)(f) > 2. This is a robust1 condition on μ. This notion as well as similar ones have been studied extensively recently, as it allows one to describe quite precisely the stationary measures for random walks with μ as law (see below for more discussion). Here we will make a remark (which can be related to some results, e.g. in [3, 6, 14]) that points in the direction of the abundance of uniform expansion. Theorem 0.2. For every open set U in Diffvol(T) there is a finitely supported probability measure μ whose support is contained in U and μ is uniformly expanding. As a consequence of the results of [5, 13, 6] one deduces that: Corollary 0.3. For every U ⊂ Diffvol(T) there is a probability measure μ finitely supported in U such that the orbit of every point under the random walk on T2 produced by μ equidistributes in T2. Moreover, for every μ′ close to μ in the weak-∗ 2020 Mathematics Subject Classification. 37H15. Rafael Potrie was partially supported by CSIC 618, FCE-1-2017-1-135352. This work was started while the author was a Von Neumann fellow at IAS, funded by the Minerva Research Foundation Membership Fund and NSF DMS-1638352. 1To be precise, if μ has compact support, then there is a neighborhood U of its support such that any measure μ′ which has support in U and is weak-∗-close to μ, is also uniformly expanding (see (3.1) below).