Geometry and Iteration of Dianalytic Transformation

TL;DR

This paper studies the dynamics of dianalytic transformations on nonorientable Klein surfaces, focusing on the real projective plane, revealing that automorphisms are projections of rotations and some transformations relate to Blaschke products.

Contribution

It characterizes the automorphisms and certain dianalytic transformations of the real projective plane as projections of well-known complex functions, extending previous work on Klein surfaces.

Findings

Automorphisms of P^2 are projections of rotations of the Riemann sphere.

Some dianalytic transformations are projections of Blaschke products.

The paper advances understanding of the structure of dianalytic transformations on nonorientable surfaces.

Abstract

This paper is a continuation of the paper [5] dealing with dynamics of dianalytic transformations of nonorientable Klein surfaces. We are examining mainly the transformations of the real projective plane $P^{2}, $ whose orientable double cover is the Riemann sphere $\overline{C}$ . It is shown that the automorphisms of $P^{2}$ are projections of the rotations of \ $\overline{C}$ and some of the other dianalytic transformations of $P^{2}$ are projections of Blaschke products.