Reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ and exceptional hyperbolic CR-singularities

TL;DR

This paper classifies holomorphic and anti-holomorphic parabolic diffeomorphisms of 2,0) with certain symmetries and integrals, and applies these results to classify real analytic surfaces with hyperbolic CR singularities.

Contribution

It provides a canonical formal normal form and a complete analytic classification for these diffeomorphisms, extending to anti-holomorphic cases and real surfaces with hyperbolic CR singularities.

Findings

Complete analytic classification in formal generic cases.

Reduction to Birkhoff-c9calle-Voronin modulus for restricted germs.

Classification results for real analytic surfaces with hyperbolic CR singularities.

Abstract

The aim of this article is twofold: First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants. Their restriction to an irreductible component of the zero locus of the first integral reduces to the Birkhoff--\'Ecalle--Voronin modulus of the 1-dimensional restricted parabolic germ. We then generalize this classification also to germs of anti-holomorphic diffeomorphisms of $(\mathbb{C}^2,0)$ whose square iterate is of the above form. Related to it, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $\mathbb{C}^2$ with non-degenerate CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hyperplane of $\mathbb{C}^2$.