scholarly journals Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

2021 ◽  
Vol 17 (3) ◽  
pp. 321-334
Author(s):  
M. K. Barinova ◽  
◽  
E. Y. Gogulina ◽  
O. V. Pochinka ◽  
◽  
...  
Keyword(s):  
Isomorphism Class ◽  
Sufficient Conditions ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Connected Sum ◽  
Hasse Diagrams ◽  
Surface Diffeomorphisms ◽  
Necessary And Sufficient

The present paper gives a partial answer to Smale’s question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}\subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.

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 Classification of periodic transformations of an orientable surface of genus two

2021 ◽  
Vol 23 (2) ◽  
pp. 147-158
Author(s):  
Denis A. Baranov ◽  
Olga V. Pochinka
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Modular Surface ◽  
Periodic Surface ◽  
Genus 2 ◽  
Genus Two ◽  
Necessary And Sufficient ◽  
Genus One

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

Homoclinic and Big Bang Bifurcations of an Embedding of 1D Allee’s Functions into a 2D Diffeomorphism

2017 ◽  
Vol 27 (09) ◽  
pp. 1730030 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Big Bang ◽  
Complete Classification ◽  
Two Dimensional ◽  
Contour Lines ◽  
Dimension One ◽  
Necessary And Sufficient ◽  
Homoclinic Tangencies

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

The λ-classification of continuous-time birth-and-death processes

2003 ◽  
Vol 35 (04) ◽  
pp. 1111-1130 ◽  
Author(s):  
Andrew G. Hart ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Birth And Death Processes ◽  
Positive Recurrent ◽  
Necessary And Sufficient

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

scholarly journals An Application of Spherical Geometry to Hyperkähler Slices

2020 ◽  
pp. 1-30
Author(s):  
Peter Crooks ◽  
Maarten van Pruijssen
Keyword(s):  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Spherical Geometry ◽  
Spherical Subgroup ◽  
Cotangent Bundles ◽  
Complex Semisimple ◽  
Reductive Subgroup ◽  
Necessary And Sufficient

Abstract This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group  $G$ , a reductive subgroup $H\subseteq G$ , and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$ , defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of  $G$ . This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$ , $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ -regularity of $(G,H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

scholarly journals The Q0-matrix completion problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Kalyan Sinha
Keyword(s):  
Matrix Completion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
Necessary And Sufficient ◽  
The Relationship

A matrix is a Q0-matrix if for every k∈{1,2,…,n}, the sum of all k×k principal minors is nonnegative. In this paper, we study some necessary and sufficient conditions for a digraph to have Q0-completion. Later on we discuss the relationship between Q and Q0-matrix completion problem. Finally, a classification of the digraphs of order up to four is done based on Q0-completion.

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