Hasse diagrams and orbit class spaces

Bonatti, Christian; Hattab, Hawete; Salhi, Ezzeddine; Vago, Gioia

doi:10.1016/j.topol.2010.12.010

Cited by 7 publications

(7 citation statements)

References 6 publications

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“…The spaces of leaves Y of foliations often appear as spaces of orbits of flows and more generally of group actions and play an important role in the understanding the dynamics of that actions, e.g. [5,15,4,7,10,2,3] and others. The usual difficulty arising at once when we pass from the manifold X to the space of leaves Y is that Y is usually non-Hausdorff.…”

Section: Introductionmentioning

confidence: 99%

Actions of groups of foliated homeomorphisms on spaces of leaves

Maksymenko¹,

Polulyakh²

2020

Preprint

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Let ∆ be a foliation on a topological manifold X, Y be the space of leaves, and p : X → Y be the natural projection. Endow Y with the factor topology with respect to p. Then the group H(X, ∆) of foliated (i.e. mapping leaves onto leaves) homeomorphisms of X naturally acts on the space of leaves Y , which gives a homomorphism ψ : H(X, ∆) → H(Y ). We present sufficient conditions when ψ is continuous with respect to the corresponding compact open topologies.In fact similar results hold not only for foliations but for a more general class of partitions ∆ of locally compact Hausdorff spaces X.

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Section: Introductionmentioning

confidence: 99%

Actions of groups of foliated homeomorphisms on spaces of leaves

Maksymenko¹,

Polulyakh²

2020

Preprint

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“…In fact, the Conley-Morse graphs are implemented as a computer software CHomP (Computational Homology Project software) [2], and there are several relative works for analyzing dynamical systems using algorithms [14,20,22]. The finite realizability of the orbit class spaces is studied [7,8]. Therefore the question of how to reduce dynamical systems into finite topological invariants is essential from a theoretical and application point of view.…”

Section: Introductionmentioning

confidence: 99%

Quotient spaces and topological invariants of flows

Yokoyama¹

2020

Preprint

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We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on topological spaces are generalizations of both Morse graphs of flows on compact metric spaces and Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces. Moreover, we show that the abstract weak orbit spaces are complete and finite for several kinds of flows on manifolds, and we state several examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons. In addition, we consider when the time-one map reconstructs the topology of the original flow. Therefore we show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization, and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. Furthermore, the abstract weak orbit space of a Morse flow on a closed manifold is a refinement of the CW decomposition which consists of the unstable manifolds of singular points. Though the CW decomposition of a Morse flow on a closed manifold is finite, the intersection of the unstable manifold and the stable manifold of saddles of a Morse-Smale flow on a closed manifold need not consist of finitely many connected components (or equivalently need not consist of finitely many abstract weak orbits). Therefore we study the finiteness of abstract weak orbit spaces of Morse(-Smale) flow on compact manifolds.

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“…By [3], any connected finite poset can be realized by the singular part of the leaf class space of a transversally oriented codimension one C 1 -foliation on a closed oriented three manifold. In [4], Bonatti et al showed that any finite connected poset can be realized by the orbit class space of a finitely generated group of homeomorphisms of a compact connected topological space. They also state the question of whether any finite Hasse diagram can be realized on a compact manifold.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of surface flows

Yokoyama

2017

Preprint

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We construct a complete invariant for surface flows of finite type. In fact, although the set of topological equivalence classes of minimal flows (resp. Denjoy flows) on a torus is uncountable, we enumerate the set of topological equivalence classes of flows with at most finitely many limit cycles but without Q-sets and non-degenerate singular points on a compact surface using finite labelled graphs. To enumerate such flows, we describe properties of border points of a flow without degenerate singular points on a compact surface. In particular, we show that each connected component of the complement of the "saddle connection diagram" is either an open disk, an open annulus, a torus, a Klein bottle, an open Möbius band, or an open essential subset. Moreover, we generalize the Poincaré-Bendixson theorem for a flow with arbitrarily many singular points on a compact surface. In fact, the ω-limit set of any nonclosed orbit is either a nowhere dense subset of singular points, a limit cycle, a limit "quasi-circuit", a locally dense Q-set, or a "quasi-Q-set". In addition, for such a surface flow, we characterize the necessary and sufficient conditions for the closure of the union of closed orbits corresponding to the non-wandering set.

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Hasse diagrams and orbit class spaces

Cited by 7 publications

References 6 publications

Actions of groups of foliated homeomorphisms on spaces of leaves

Actions of groups of foliated homeomorphisms on spaces of leaves

Quotient spaces and topological invariants of flows

Decompositions of surface flows

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