Moduli of 3-dimensional diffeomorphisms with saddle-foci

TL;DR

This paper studies the classification invariants (moduli) of 3D diffeomorphisms with saddle-focus fixed points, revealing generic properties of eigenvalues and conjugacy maps in such systems.

Contribution

It characterizes the moduli space for a class of 3D diffeomorphisms with saddle-focus points, showing generic eigenvalue properties and conjugacy restrictions.

Findings

Eigenvalues of $Df(p)$ are moduli for generic systems.

Conjugacy homeomorphisms are linear conformal on unstable manifolds.

The study identifies invariants that classify these diffeomorphisms.

Abstract

We consider a space $\mathcal{U}$ of 3-dimensional diffeomorphisms $f$ with hyperbolic fixed points $p$ the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that $Df(p)$ has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in $\mathcal{U}$. We show that, for a generic element $f$ of $\mathcal{U}$, all the eigenvalues of $Df(p)$ are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal map.