Smooth deformations of piecewise expanding unimodal maps

TL;DR

This paper characterizes smooth deformations of piecewise expanding unimodal maps that preserve topological dynamics, identifying specific tangent directions and providing conditions for their existence.

Contribution

It introduces a precise description of smooth deformations in the space of unimodal maps and defines the horizontal directions as kernels of explicit linear functionals.

Findings

Characterization of C^1 smooth families with unchanged topological dynamics.

Identification of codimension-one subspaces as kernels of linear functionals.

Existence of C^{k-1+Lip} deformations tangent to any given horizontal direction for k>=2.

Abstract

In the space of C^k piecewise expanding unimodal maps, k>=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 an explicit 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>=2.