Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

Bonatti, Christian; Grines, V. Z.; Pochinka, O. V.

doi:10.1215/00127094-2019-0019

Cited by 25 publications

(16 citation statements)

References 42 publications

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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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“…The solution to Palis problem in case n ≥ 3 was possible due to the significant progress in solving the topological classification problem for Morse-Smale diffeomorphisms. In papers [3][4][5][6][7][8]44] by Bonatti, Grines, Pochinka, Pécou, Medvedev and Laudenbach these was introduced a new complete topological invariant for Morse-Smale diffeomorphisms on 3-manifolds called a scheme of a diffeomorphism and the problem of realization of all classes of topological conjugation was solved. Due to this it was possible to state necessary and sufficient conditions for Morse-Smale diffeomorphism to be embedded in the topological flow.…”

Section: A Scheme Of a Diffeomorphismmentioning

confidence: 99%

On Embedding of the Morse–Smale Diffeomorphisms in a Topological Flow

2022

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This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to be embedded in topological flows. The progress achieved in solving of Palis's problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on threedimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

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“…In 1993, Langevin [23] proposed to consider the orbit space of the basin of the sink and project to this closed surface the unstable separatries of the saddle points. This approach was generalized and successfully applied by Bonatti, Grines, Medvedev, Pecou and Pochinka in [9], [10] for the topological classification of Morse-Smale diffeomorphisms f with beh(f ) ≤ 3 on 3-manifolds. In 2010, Mitryakova and Pochinka [30] applied this method to the topological classification of Morse-Smale diffeomorphisms f with beh(f ) ≤ 3 on orientable surfaces.…”

Section: Classification Of Non-gradient-like Diffeomorphisms On Close...mentioning

confidence: 99%

“…One of the main features of this classification is that, given this abstract data, we will always be able construct a diffeomorphism that realises this data. More precisely, in the present paper we consider class M S(M 2 ) preserving orientation Morse-Smale diffeomorphisms of an orientable surface M 2 and give a complete topological classification within this class, and solve the corresponding realization problem, by means of topological invariants similar to those used in [9,10,30,31]. To make this precise we need to introduce some notions.…”

Section: Classification Of Non-gradient-like Diffeomorphisms On Close...mentioning

confidence: 99%

A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

Grines¹,

Pochinka²,

Strien³

2017

Preprint

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In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a 'scheme'. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conjugacy class of MS orientation preserving diffeomorphisms (on some orientation preserving surface). As a corollary we obtain a different proof that nearby MS diffeomorphisms are topologically conjugate.

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Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

Cited by 25 publications

References 42 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 $${\mathbb {H}}{\mathbf{P }}^n$$

On Embedding of the Morse–Smale Diffeomorphisms in a Topological Flow

A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

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