Exact topological classification of Hamiltonian flows on smooth two-dimensional surfaces

1999 ◽  
Vol 94 (4) ◽  
pp. 1457-1476
Author(s):  
A. V. Bolsinov ◽  
A. T. Fomenko
Keyword(s):  
Topological Classification ◽  
Two Dimensional ◽  
Hamiltonian Flows

scholarly journals On periodic mapping data of a two-dimensional torus with one saddle orbit

2019 ◽  
Vol 21 (2) ◽  
pp. 164-174
Author(s):  
Anna A. Bosova ◽  
Olga V. Pochinka
Keyword(s):  
Periodic Orbit ◽  
Nodal Point ◽  
Zeta Functions ◽  
Topological Classification ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Periodic Surface ◽  
Periodic Data ◽  
Surface Diffeomorphism

Periodic data of diffeomorphisms with regular dynamics on surfaces were studied using zeta functions in a series of already classical works by such authors as P. Blanchard, J. Franks, S. Narasimhan, S. Batterson and others. The description of periodic data for gradient-like diffeomorphisms of surfaces were given in the work of A. Bezdenezhnykh and V. Grines by means of the classification of periodic surface transformations obtained by J. Nielsen. V. Grines, O. Pochinka, S. Van Strien showed that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. Namely, the construction of filtering for Morse-Smale diffeomorphisms makes it possible to reduce the problem of studying periodic surface diffeomorphism data to the problem of calculating periodic diffeomorphism data with a single saddle periodic orbit. T. Medvedev, E. Nozdrinova, O. Pochinka solved this problem in a general formulation, that is, the periods of source orbits are calculated from a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. In this paper we establish a complete list of periodic data of diffeomorphisms of a two-dimensional torus with one saddle orbit, provided that at least one nodal point of the map is fixed.

scholarly journals On non-hyperbolic algebraic automorphisms of a two-dimensional torus

2021 ◽  
Vol 23 (3) ◽  
pp. 295-307
Author(s):  
Sergey V. Sidorov ◽  
Ekaterina E. Chilina
Keyword(s):  
Sufficient Conditions ◽  
Complete Classification ◽  
Orientable Surface ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Algebraic Automorphism ◽  
Necessary And Sufficient

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

scholarly journals Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds

2017 ◽  
Vol 39 (9) ◽  
pp. 2403-2432 ◽  
Author(s):  
CH. BONATTI ◽  
V. GRINES ◽  
F. LAUDENBACH ◽  
O. POCHINKA
Keyword(s):  
Equivalence Class ◽  
Topological Classification ◽  
Two Dimensional ◽  
Characteristic Space ◽  
Topological Conjugation

We show that, up to topological conjugation, the equivalence class of a Morse–Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

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