Classification of Morse–Smale systems and topological structure of the underlying manifolds

Гринес, Вячеслав Зигмундович; Gurevich, E. Ya.; Zhuzhoma, E. V.; Pochinka, O. V.

doi:10.1070/rm9855

Cited by 30 publications

(13 citation statements)

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“…Without trying to be exhaustive, see for instance [5,19,[22][23][24]. This interest continues during this first part of the twenty-first century, see for example [2,3,9,10,[15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

Cufí-Cabré

Llibre

2021

J. Fixed Point Theory Appl.

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We study the set of periods of the Morse-Smale diffeomorphisms on the n-dimensional sphere S n , on products of two spheres of arbitrary dimension S m × S n with m = n, on the n-dimensional complex projective space CP n and on the n-dimensional quaternion projective space HP n . We classify the minimal sets of Lefschetz periods for such Morse-Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.

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Section: Introductionmentioning

confidence: 99%

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

Cufí-Cabré

Llibre

2021

J. Fixed Point Theory Appl.

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“…Therefore, one can glue a disk d j to each circle c j extending f to d j with a sink inside of d j . If K is without an isolated saddles, then K ∪ j d − J is a 2-sphere with a unique source and a unique sink [14]. Therefore if K is without an isolated saddles, then K is a disk with a unique source.…”

Section: Some Applicationsmentioning

confidence: 99%

“…Anosov [3] and Smale [44] were first who realize the fundamental role of hyperbolicity for dynamical systems. Numerous topological invariants were constructed for special classes of A-diffeomorphisms including Anosov systems [12,29,34] and Morse-Smale systems, see the books [6,13] and the surveys [14,30].…”

Section: Introductionmentioning

confidence: 99%

Smale regular and chaotic A-homeomorphisms and A-diffeomorphisms

Medvedev¹,

Zhuzhoma²

2021

Preprint

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We introduce Smale A-homeomorphisms that includes regular, semi-chaotic, chaotic, and super chaotic homeomorphisms of topological n-manifold M n , n ≥ 2. Smale A-homeomorphisms contain A-diffeomorphisms (in particular, structurally stable diffeomorphisms) provided M n admits a smooth structure. Regular A-homeomorphisms contain all Morse-Smale diffeomorphisms, while semi-chaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Super chaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. We describe invariant sets that determine completely dynamics of regular, semi-chaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms. We apply this necessary and sufficient conditions for structurally stable surface diffeomorphisms with arbitrary number of one-dimensional expanding attractors. We also use this conditions to get the complete classification of Morse-Smale diffeomorphisms on projective-like n-manifolds for n = 2, 8, 16.

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“…In class T 2 there is a classification of structurally stable Morse -Smale diffeomorphisms with orientable heteroclinic intersection in [5][6][7][8]. The paper [9] contains a review of the results obtained so far in the classification of Morse -Smale diffeomorphisms in general.…”

Section: Introductionmentioning

confidence: 99%

On a Classification of Periodic Maps on the 2-Torus

Baranov¹,

Grines²,

Pochinka³

et al. 2023

Nelin. Dinam.

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In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, 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. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

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Classification of Morse–Smale systems and topological structure of the underlying manifolds

Cited by 30 publications

References 0 publications

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

Smale regular and chaotic A-homeomorphisms and A-diffeomorphisms

On a Classification of Periodic Maps on the 2-Torus

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