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

TL;DR

The paper analyzes gradient-like diffeomorphisms of the 2-torus whose action on the fundamental group matches $A_2$ and shows that the isotopy class splits into four stable components, distinguished by the number of fixed sinks. It develops a framework connecting gradient-like dynamics to periodic homeomorphisms, proving that each map in the class is isotopic to a periodic map of order three, with a characteristic three-fixed-point structure and all other points of period three. By constructing canonical representatives (the simplest diffeomorphisms) $g_i$ in each component and using invariants such as a three-color graph and knot-type data on the torus, the authors prove a stable-arc classification: two diffeomorphisms are stably connected if and only if they have the same number of fixed sinks. Consequently, the set $G_2$ comprises exactly four stable components corresponding to $i\in\{0,1,2,3\}$, providing a complete stable decomposition for these gradient-like toral diffeomorphisms. The results illuminate how gradient-like surface dynamics interact with periodic torus symmetries and offer explicit combinatorial and topological tools for stable classification.

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.

Figures (16)

• Figure 1: Stable arc in the space of diffeomorphisms
• Figure 2: Phase portrait of the diffeomorphism $g_1$
• Figure 3: Three-color graph of diffeomorphism $g_1$
• Figure 4: Phase portrait of the diffeomorphism $g_0$
• Figure 5: Three-color graph of diffeomorphism $g_0$

Theorems & Definitions (18)

• Lemma 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof