There are no structural stable Axiom A 3-diffeomorphisms with dynamics "one-dimensional surfaced attractor-repeller"

TL;DR

This paper investigates the structural stability of three-dimensional diffeomorphisms with one-dimensional attractors and repellers, concluding that such systems are not structurally stable when embedded in a surface.

Contribution

It proves that three-dimensional diffeomorphisms with one-dimensional surface-embedded attractors and repellers are not structurally stable, extending known results to this specific setting.

Findings

Diffeomorphisms with surface-embedded attractors and repellers lack structural stability.

Structural stability fails for these systems when attractors and repellers are canonical surface embeddings.

Results apply to systems with hyperbolic repellers and attractors in three dimensions.

Abstract

In this paper, we study the structural stability of three-dimensional diffeomorphisms with source-sink dynamics. Here the role of source and sink is played by one-dimensional hyperbolic repeller and attractor. It is well known that in the case when the repeller and the attractor are solenoids (not embedded in the surface), the diffeomorphism is not structurally stable. The author proves that in the case when the attractor and the repeller are canonically embedded in a surface, the diffeomorphism is also not structurally stable.