Stable components for gradient-like diffeomorphisms of torus inducing matrix $\begin{pmatrix} -1 & -1\cr 1& 0\end{pmatrix}$

TL;DR

This paper studies gradient-like diffeomorphisms of the 2-torus induced by a specific matrix, proving they form four stable components distinguished by the number of fixed sinks, and analyzes their stability under perturbations.

Contribution

It identifies four stable components of such diffeomorphisms and characterizes stable connectivity based on fixed sinks, advancing understanding of their structural stability.

Findings

The set of these diffeomorphisms decomposes into four stable components.

Two diffeomorphisms are stably connected if and only if they have the same number of fixed sinks.

The stability persists under small perturbations within each component.

Abstract

An isotopy between two diffeomorphisms means the existence of an arc connecting them in the space of diffeomorphisms. Among such arcs there are so-called stable arcs, which do not qualitatively change under small perturbations. In the present paper we consider a set of gradient-like diffeomorphisms f of 2-torus whose induced isomorphism given by a matrix $\begin{pmatrix} -1 & -1\cr 1& 0\end{pmatrix}$. We prove that the set of such diffeomorphisms is decomposed into four stable components. Moreover, we establish that two diffeomorphisms under consideration are stably connected if and only if they have the same number of fixed sinks.