A criterion for the topological conjugacy of Hamiltonian flows on two-dimensional compact surfaces

Bifurcations of two-dimensional diffeomorphisms with a homoclinic tangency are studied in one-and two-parameter families. Due to the well-known impossibility of a complete study of such bifurcations, the problem is restricted to the study of the bifurcations of the so-called low-round periodic orbits. In this connection, the idea of taking Ω-moduli (continuous invariants of the topological conjugacy on the nonwandering set) as the main control parameters (together with the standard splitting parameter) is proposed. In this way, new bifurcational effects are found which do not occur at a one-parameter analysis. In particular, the density of cusp-bifurcations is revealed.

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Kneading sequences for toy models of Hénon maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.117 ◽

2020 ◽

pp. 1-20

Author(s):

ERMERSON ARAUJO

Keyword(s):

Conjugacy Class ◽

Topological Conjugacy ◽

Two Dimensional ◽

One Dimensional ◽

Henon Maps ◽

Hénon Maps ◽

Critical Lines ◽

Model Family

Abstract The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer’s theorem for the toy model family.

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The representation of an arbitrary, two-dimensional completely integrable system as the common action of two commuting one-dimensional Hamiltonian flows

Lecture Notes in Mathematics - The Riemann Problem, Complete Integrability and Arithmetic Applications ◽

10.1007/bfb0093501 ◽

1982 ◽

pp. 85-94

Author(s):

D. V. Chudnovsky

Keyword(s):

Integrable System ◽

Two Dimensional ◽

One Dimensional ◽

Completely Integrable ◽

Completely Integrable System ◽

Hamiltonian Flows ◽

The Common ◽

Common Action

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The projective fundamental group of a ℤ2-shift

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700009810 ◽

1995 ◽

Vol 15 (6) ◽

pp. 1091-1118 ◽

Cited By ~ 4

Author(s):

William Geller ◽

James Propp

Keyword(s):

Topological Space ◽

Fundamental Group ◽

Topological Conjugacy ◽

Fundamental Groups ◽

Two Dimensional ◽

Limit Group ◽

Long Distance ◽

Mixing Properties ◽

Point Data ◽

Factor Maps

AbstractWe define a new invariant for symbolic ℤ2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the ℤ2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the ℤ2-action that π1; of a topological space bears to π0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.

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

Journal of Mathematical Sciences ◽

10.1007/bf02365197 ◽

1999 ◽

Vol 94 (4) ◽

pp. 1457-1476

Author(s):

A. V. Bolsinov ◽

A. T. Fomenko

Keyword(s):

Topological Classification ◽

Two Dimensional ◽

Hamiltonian Flows

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

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.23.202103.295-307 ◽

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.

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