Stable components for gradient-like diffeomorphisms of torus inducing matrix (−1 −1 1 0)

Working papers by Cornell University. Series math "arxiv.org". 2025. P. 1–22.

Pochinka O., Baranov D.

An isotopy between two diffeomorphisms f0, f1 : M → M means the existence of an arc {ft : M → M, t ∈ [0, 1]} 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 T2 whose induced isomorphism f∗ : π1(T2) → π1(T2) given by a matrix f⋆ =−1 −1 1 0. 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. 

Keywords: stable arc, устойчивая дуга