Parametric linearization of skew products

TL;DR

This paper develops a criterion for linearizing skew products of contractions across dimensions, ensuring smooth or holomorphic parameter dependence, and applies it to expanding Cantor sets.

Contribution

It introduces a new linearization criterion for skew products of contractions and demonstrates their smooth or holomorphic dependence on parameters.

Findings

Established a linearization criterion for skew products of contractions.

Proved smooth and holomorphic parameter dependence.

Applied results to linearize expanding Cantor sets.

Abstract

We establish a linearization criterion for skew products of contractions in any dimension. We prove their smooth or holomorphic parameter dependence. In the smooth setting, we use the language of tame Fr\'echet spaces. We apply our result to the linearization of totally projectively expanding Cantor sets.