  Often topological classes of one-dimensional dynamical systems are finite
codimension smooth manifolds. We describe a method to prove this sort of
statement that we believe can be applied in many settings. In this work we will
implement it for piecewise expanding maps. The most important step will be the
identification of infinitesimal deformations with primitives of Birkhoff sums
(up to addition of a Lipschitz function), that allows us to use the ergodic
properties of piecewise expanding maps to study the regularity of infinitesimal
deformations.
