Construction of minimal non-invertible skew-product maps on 2-manifolds

TL;DR

This paper constructs minimal non-invertible skew-product maps on 2-manifolds, specifically on the Klein bottle and the torus, using blow-up techniques and simplified methods.

Contribution

It provides the first known examples of non-invertible minimal maps on these 2-manifolds, expanding understanding of dynamical systems on surfaces.

Findings

Existence of non-invertible minimal circle-fibered maps on the Klein bottle.

Simpler construction of a non-invertible minimal self-map on the 2D torus.

Application of Hric-Jäger blow-up technique to surface dynamics.

Abstract

Applying the Hric-J\"ager blow up technique, we give an affirmative answer to the question about the existence of non-invertible minimal circle-fibered self-maps of the Klein bottle. In addition, we present a simpler construction of a non-invertible minimal self-map of two dimensional torus.