Birkhoff sums as distributions II: Applications to deformations of dynamical systems

The paper proves that, for piecewise expanding interval maps f in B^k_exp(C) (k≥3) that are locally Lasota–Yorke stable, the topological class T(f) is a C^[k/2] Banach submanifold modeled on the horizontal space Eh(f), of codimension 2n−2. It does so by characterizing infinitesimal deformations via the twisted cohomological equation, encoding obstructions with functionals J(f,·,·), constructing a finite set of transverse directions {w_c} with J(f0,w_c,d)=δ_cd, and applying the Banach implicit function theorem (Theorem 10.2) , leveraging the equivalences in Theorems 8.1 and 8.3 . The candidate solution follows the same strategy (TCE, horizontality via J, obstruction map, IFT), differing mostly in presentation and minor normalizations (e.g., a small sign/scale slip in relating J to boundary values of the infinitesimal conjugacy).