Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings

Abstract

In order to study the perturbations of a family of mappings with a hyperbolic mixing attractor, an apparatus of foliated functions is developed. Foliated functions are analogues of distributions based on smooth measures on leaves (traces), which are embedded manifolds in a neighborhood of the attractor. The dimension of such manifolds must coincide with the dimension of the expanding foliation, and the values of a foliated function on a trace must vary smoothly under smooth transverse deformations of the trace (which include deformations of the measure itself).