ON THE DISCONTINUITIES OF HAUSDORFF DIMENSION IN GENERIC DYNAMICAL LAGRANGE SPECTRUM

The uploaded paper proves Theorem 1.1: for a residual set of diffeomorphisms φ in a small C^2-neighborhood and a Cr-residual set of observables f, one has max L_{φ,f} = HD(L_{φ,f}) = min{1, HD(Λ)}, c_{φ,f} equals the smallest accumulation point of L_{φ,f}, and the map t ↦ L_{φ,f}(t) has only finitely many discontinuities on closed subintervals that avoid the endpoints specified by the Hausdorff-dimension regime. This is stated explicitly and proved via a detailed construction using subhorseshoes, a technical proposition producing large-dimension subhorseshoes within level sets, and a connection criterion that controls discontinuities, culminating in the finiteness statements (Theorem 1.1 and its corollaries) . By contrast, the candidate solution reproduces the dimension equality using known results on images/sums of Cantor sets and genericity, but its arguments for the novel discontinuity structure are circular: it invokes precisely the main result of the uploaded paper to conclude that c_{φ,f} is the smallest accumulation point and that discontinuities can only accumulate at c_{φ,f} (and possibly c̃_{φ,f}). It omits the paper’s core combinatorial-dynamical machinery (e.g., the construction/connection of subhorseshoes and the technical propositions underpinning the discontinuity analysis) that are essential to establish the new claims .