Non-stationary normal forms for contracting extensions

TL;DR

This paper develops a comprehensive theory of non-stationary normal forms for uniformly contracting smooth extensions with narrow Mather spectrum, providing explicit descriptions and new results in resonance normal forms.

Contribution

It introduces the first detailed treatment of resonance normal forms in the narrow spectrum setting, including explicit non-uniqueness descriptions and regularity results.

Findings

Explicit description of non-uniqueness in normal forms

Results applicable in any regularity above the critical level

Extension of normal form theory to resonance cases in narrow spectrum

Abstract

We present the theory of non-stationary normal forms for uniformly contracting smooth extensions with sufficiently narrow Mather spectrum. We give coherent proofs of existence, (non)uniqueness, and a description of the centralizer results. As a corollary, we obtain corresponding results for normal forms along an invariant contracting foliation. The main improvements over the previous results in the narrow spectrum setting include explicit description of non-uniqueness and obtaining results in any regularity above the precise critical level, which is especially useful for the centralizer. In addition to sub-resonance normal form, we also prove corresponding results for resonance normal form, which is new in the narrow spectrum setting.