Peixoto graphs of Morse-Smale foliations on surfaces

Bronstein, I U; Nikolaev, Igor

doi:10.1016/s0166-8641(96)00101-0

Cited by 13 publications

(16 citation statements)

References 3 publications

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“…To analyze a line field, it is common to study the behavior of its orbits [10,14]. We take a more topological approach, consisting of grouping orbits with similar behavior [16,17]. To construct such a decomposition, we observe that the dynamics of a line field is usually trivial almost everywhere except near the points for which the field is not defined or vanishes, the so called critical points [18] (Figure 1.1).…”

Section: Introductionmentioning

confidence: 99%

Discrete line fields on surfaces

Novello

Paixão²,

Tomei

et al. 2021

Topology and its Applications

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I would like to express my sincere gratitude to my advisers: João Paixão, for the trust, dedication, and genuine friendship, to whom I will be eternally grateful; Thomas Lewiner, for believing and trusting in my work; Carlos Tomei, for the valuable teachings. To them, my infinite respect and friendship. I wish to o er my thanks to the Departamento de Matemática of PUC-Rio and its professors for helping me to be a better professional. In particular, Creuza, Kátia, and Orlando, who have always been very e cient and helpful.

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Section: Introductionmentioning

confidence: 99%

Discrete line fields on surfaces

Novello

Paixão²,

Tomei

et al. 2021

Topology and its Applications

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“…‡ And foliations, see [3]. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds [9].…”

Section: Introductionmentioning

confidence: 99%

Graphs and flows on surfaces

Nikolaev

1998

Ergod. Th. Dynam. Sys.

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In 1971, M. M. Peixoto [15] introduced an important topological invariant of Morse–Smale flows on surfaces, which he called a distinguished graph $X^*$ associated with a given flow. Here we show how the Peixoto invariant can be essentially simplified and revised by adopting a purely topological point of view connected with the embeddings of arbitrary graphs into compact surfaces. The newly obtained invariant, $X^R$, is a rotation of a graph $X$ generated by a Morse–Smale flow. (A rotation $R$ is a cyclic order of edges given in every vertex of $X$.) The invariant $X^R$ ‘reads-off’ the topological information carried by a flow, being in a one-to-one correspondence with the topological equivalence classes of Morse–Smale flowsAnd foliations, see [3]. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds [9].. As a counterpart to the equivalence result we prove a realization theorem for an ‘abstractly given’ $X^R$. (Our methods are completely different from those of Peixoto and they clarify the connections between graphs and flows on surfaces.) The idea of ‘rotation systems’ on graphs can be further exploited in the study of recurrent flows (and foliations) with several disjoint quasiminimal sets on surfaces [10].

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“…For the investigation of one-dimensional foliations with nonorientable singularities Guti6rrez [39], Kadyrov [18], Bronstein and Nikolaev [16,17,31,32,50,51] suggested the use of the concept of the field of line elements. In these works, they understood line elements as the pair {v,-v} of mutually opposite tangent vectors.…”

Section: Introductionmentioning

confidence: 99%

Qualitative theory of foliations on closed surfaces

Aranson¹,

Bronstein²,

Zhuzhoma³

et al. 1998

J Math Sci

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PrefaceFoliations on surfaces are natural generalizations of flows (continuous-time dynamical systems) on surfaces. According to the classical theory, in the neighborhood of a nonsingular point (i.e., a point different from the equilibrium state), the trajectories of a flow are a family of parallel straight lines. This is what served as a starting point for defining a foliation.Let M be a closed surface (a closed two-dimensional manifold). The foliation F on M with a set of singularities Sing (F) is the partitioning M -Sing (F) into nonintersecting curves (called fibers) which are locally homeomorphic to the family of straight lines. As to the set of singularities Sing (F), this set must be described separately for every class of fibers. In the sequel, unless otherwise specified, we shall consider the set Sing (F) to be finite.Foliations occupy an intermediate place between flows and arbitrary families of curves on surfaces. Foliations which can be embedded into a flow are said to be orientable; otherwise they are nonorientable.The origination of a qualitative theory of foliations goes back to the works by H. Kneser, G. Reeb, A. Haefliger, and S. P. Novikov. The theory of foliations attracted special attention early in the sixties in connection with the study of Y-flows and Y-diffeomorphisms introduced by D. V. Anosov. The technique of foliations allowed G. Franks to classify Y-diffeomorphisms of codimension one the nonwandering set of whose points coincides with the whole manifold. The application of a "surgical operation" to Y-diffeomorphisms of codimension one leads to nontrivial base sets of codimension one (attractors and repellers) the profound results in the study of whose geometry and topology belong to P. V. Plykin and V. Z. Grines.A more general approach was suggested by Ya. G. Sinai, who introduced a class of dynamical systems with two invariant transversal fibers. When we use this approach, we neglect such properties of the systems as the estimates of the contraction and extension of the fibers of invariant foliations, the everywhere density of the periodic trajectories in a nonwandering set, and others, and leave only those properties which are closely connected with the topology' of the manifolds.A new impetus to the study of foliations and homeomorphisms with invariant foliations was given by the works of W. Thurston, in which he complemented the homotopic classification of homeomorphisms of surfaces obtained by d. Nielson in 1920-30 and gave a new interpretation to it. The introduction by Thurston of the concept of pseudo-Anosov homeomorphism, which generalized the concept of the Anosov diffeomorphism, stimulated further investigations in this direction based on the study of the action of homeomorphisms in a fundamental group.Beginning in the 1980s, the geometry and topology of foliations with saddle-point singularities were studied by H. Rosenberg (the construction of labyrinths), G. Levitt (equivalence in the sense of Whitehead), G. Papandopoulos, K. Dantoni, and others.At the same time, many qu...

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Peixoto graphs of Morse-Smale foliations on surfaces

Cited by 13 publications

References 3 publications

Discrete line fields on surfaces

Discrete line fields on surfaces

Graphs and flows on surfaces

Qualitative theory of foliations on closed surfaces

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