On components of stable connectivity of gradient-like diffeomorphisms of the 2-torus

TL;DR

This paper investigates the stable connectivity components of gradient-like diffeomorphisms on the 2-torus, showing that non-isotopic classes form finitely many such components distinguished by periodic data.

Contribution

It establishes that gradient-like diffeomorphisms of the 2-torus not isotopic to the identity split into finitely many stable connectivity components characterized by periodic data.

Findings

Non-isotopic gradient-like diffeomorphisms of the 2-torus form finitely many stable components.

Periodic data uniquely determines the stable connectivity component within each isotopy class.

Contrasts with the 2-sphere case, which has countably many components.

Abstract

Gradient-like diffeomorphisms of a closed surface $M^2$ are characterized by a finite hyperbolic limit set and the absence of intersections of invariant manifolds of distinct saddle points. In the case where such diffeomorphisms $f_0, f_1:M^2\to M^2$ are isotopic, they are connected by some arc $\{f_t:M^2\to M^2, t\in [0,1]\}$ in the space of diffeomorphisms. If every diffeomorphism of the arc has a finite limit set and the arc is stable (does not change its qualitative properties under small perturbations) in the space of diffeomorphisms, then $f_0,f_1$ are said to be {\it stably connected}. Thus, the set of isotopic diffeomorphisms splits into components of stable connectivity, of which there may, in general, be infinitely many. For instance, it is known that gradient-like diffeomorphisms of the 2-sphere (both orientation-preserving and orientation-reversing) consist of a countable number of stable connectivity components. Moreover, belonging to a particular component is uniquely determined by the periodic data of the diffeomorphism. In the present paper, we consider gradient-like diffeomorphisms of the 2-torus that are not isotopic to the identity. We establish that the set of such diffeomorphisms splits into a finite number of stable connectivity components. For each isotopy class, we define the periodic data of the diffeomorphism, which uniquely determine membership in a given component.