On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Let φ₀ be a C²-conservative diffeomorphism of a compact surface S and let Λ₀ be a mixing horseshoe of φ₀. Given a smooth real function f defined in S and some diffeomorphism φ, close to φ₀, let Lφ, f be the Lagrange spectrum associated to the hyperbolic continuation Λ(φ) of the horseshoe Λ₀ and f. We show that, for generic choices of φ and f, if Lφ, f is the map that gives the Hausdorff dimension of the set Lφ, f∩ (-∞, t) for t∈ R, then there are at most two points that can be limit of a infinite sequence of discontinuities of Lφ, f.