On components of stable connectivity of gradient-like diffeomorphisms of the 2-torus
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916020160036864 |
|---|---|
| author | Baranov, D. A. Nozdrinova, E. V. Pochinka, O. V. |
| author_facet | Baranov, D. A. Nozdrinova, E. V. Pochinka, O. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17130 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On components of stable connectivity of gradient-like diffeomorphisms of the 2-torus Baranov, D. A. Nozdrinova, E. V. Pochinka, O. V. Dynamical Systems 37E30, 58D05 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. |
| title | On components of stable connectivity of gradient-like diffeomorphisms of the 2-torus |
| topic | Dynamical Systems 37E30, 58D05 |
| url | https://arxiv.org/abs/2605.17130 |