On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Let ₀ be a C²-conservative diffeomorphism of a compact surface S and let ₀ be a mixing horseshoe of ₀. Given a smooth real function f defined in S and some diffeomorphism, close to ₀, let L, ₅ be the Lagrange spectrum associated to the hyperbolic continuation () of the horseshoe ₀ and f. We show that, for generic choices of and f, if L, ₅ is the map that gives the Hausdorff dimension of the set L, ₅ (-, t) for t R, then there are at most two points that can be limit of a infinite sequence of discontinuities of L, ₅.