Smooth deformations of piecewise expanding unimodal maps

Abstract

In the space of $C^k$ piecewise expanding unimodal maps, $k\geq 1$, we characterize the $C^1$ smooth families of maps where the topological dynamics does not change (the 'smooth deformations') as the families tangent to a continuous distribution of codimension-one subspaces (the 'horizontal' directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of anexplicit class of linear functionals. As a consequence we show the existence of $C^{k-1+Lip}$ deformations tangent to every given $C^k$ horizontal direction, for $k\ge 2$.