arXiv:2011.12264v3 [math.DS] 10 Aug 2021 HAUSDORFF DIMENSION OF CANTOR INTERSECTIONS FOR COUPLED HORSESHOE MAPS YOSHITAKA SAIKI, HIROKI TAKAHASI, JAMES A. YORKE Abstract. As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a C 2 -open set of diffeomorphisms of R 3 having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly 2 whose cross sections are Cantor sets; the intersec- tion of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly 1. Our proof employs the thicknesses of Cantor sets. 1. Introduction For a diffeomorphism ϕ of a Riemannian manifold M equipped with a distance d, the unstable and stable sets of a point p ∈ M are given by W u (p)= {q ∈ M : d(ϕ n (p),ϕ n (q )) → 0 as n → −∞}, W s (p)= {q ∈ M : d(ϕ n (p),ϕ n (q )) → 0 as n → ∞}, respectively. If these sets are submanifolds of M , they are called unstable and stable manifolds of p. An incredibly rich array of complicated dynamical phenomena is unleashed by a non-transverse intersection between unstable and stable manifolds, see for example [12]. The existence of an intersection of Cantor sets is a fundamental tool to con- struct examples of open sets of diffeomorphisms which are not structurally stable. Newhouse [9] defined a non-negative quantity called the “thickness” of a Cantor set on the real line, in order to formulate conditions which guarantee that two Cantor sets intersect each other. These conditions have been applied to surface diffeomorphisms to show the robustness of tangencies between unstable and sta- ble manifolds whose cross sections are Cantor sets [9, 10, 11, 12, 14]. In higher dimension, the thickness is still useful for the construction of robust tangencies [5, 8, 13, 15]. Newhouse’s result [9] asserts that two Cantor sets on the real line intersect each other if the product of their thicknesses is greater than one, and neither set lies in a gap of the other. His result does not imply any lower bound of the Hausdorff dimension of the intersection of the two Cantor sets. Indeed, Williams [17] observed that two interleaved Cantor sets can have thicknesses well above 1 and still only intersect at a single point. Therefore, the results [5, 8, 9, 10, 11, 12, 13, 14, 15] mentioned above do not imply any lower bound of the Hausdorff dimension of the sets of tangencies. 2020 Mathematics Subject Classification. 37C29, 37C45, 37G25. Keywords: horseshoe map, Cantor sets, thickness, Hausdorff dimension. 1