Behaviour of leaves of codimension-one foliations

Nishimori, Toshiyuki

doi:10.2748/tmj/1178240656

Cited by 14 publications

(18 citation statements)

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“…In this section, inspired by ideas of Cantwell and Conlon [10,11], and also of Hector [17], Nishimori [27,28] and Tsuchiya [34] for codimension 1 transversally C 2 -differentiable foliations, we introduce a quantifier of dynamical complexity of a set in (S n , ) which, following [10], we call level. We repeat now Definition 1.6.…”

Section: Theory Of Levels For Graph Matchbox Manifoldsmentioning

confidence: 99%

“…Cantwell and Conlon [10] introduced the notion of a level of a leaf or a transitive subset, which can be seen as a quantifier of dynamical complexity. A similar, but not the same notion of 'depth' of a leaf was considered by Nishimori [27,28]. The theory of levels for C 2foliations relies heavily on the Kopell lemma [10,8], which does not apply in our setting.…”

Section: Introductionmentioning

confidence: 98%

“…In the rest of the paper we make precise the hierarchy of graph matchbox manifolds, suggested by Theorem 1.5, associating to them a quantifier of dynamical complexity, called level, inspired by the ideas of Cantwell and Conlon [10,11], Hector [17,18,19], Nishimori [27,28] and Tsuchiya [34]. Our main tool is the fusion technique of Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…The consideration of the extension of the Poincaré Recurrence Theorem for flows to the closures of leaves of foliations began in the 1950's. In foliation theory, the strongest results have been obtained for codimension 1 transversally C 2 -differentiable foliations by Cantwell and Conlon [10,11], Hector [17,18], Nishimori [27,28] and Tsuchiya [34]. Later, the extent to which these ideas carry on to the codimension 1 C 0 case was studied by Salhi [30,31,32], and, for foliations of higher codimensions satisfying certain additional conditions, by Marzougui and Salhi [24].…”

Section: Introductionmentioning

confidence: 99%

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Hierarchy of graph matchbox manifolds

Lukina

2012

Topology and its Applications

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We study a class of graph foliated spaces, or graph matchbox manifolds, initially constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier of dynamical complexity which we call its level. We develop the fusion construction, which allows us to associate to every two graph foliated spaces a third one which contains the former two in its closure. Although the underlying idea of the fusion is simple, it gives us a powerful tool to study graph foliated spaces. Using fusion, we prove that there is a hierarchy of graph foliated spaces at infinite levels. We also construct examples of graph foliated spaces with various dynamical and geometric properties.Comment: New examples added at the end of Section 2; Introduction is rewritten. To appear in Topology and its Application

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Section: Theory Of Levels For Graph Matchbox Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hierarchy of graph matchbox manifolds

Lukina

2012

Topology and its Applications

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“…In particular, in [22,Theorem 1.3] the second author studied a partial order on M G given by inclusions, which is equivalent to the study of the orbit structure of the pseudogroup dynamical system (X, G). Using the notion of a level of a leaf, initially introduced for codimension 1 foliations by J. Cantwell and L. Conlon [8] (see also [9,14,15,23,24]) she constructed hierarchies of graph matchbox manifolds at infinite levels. The results of [22] also show that the orbit structure of (X, G) is reminiscent of the orbit structure of Bernoulli shifts, for example there is a meager subset of points with finite orbits, and a residual subset of points with dense orbits.…”

Section: Introductionmentioning

confidence: 99%

Suspensions of Bernoulli shifts

Rojo

Lukina

2013

Dynamical Systems

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We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov-Hausdorff metric and natural partial action of a free group. Since the latter can be realised as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations.

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Foliations

Fuks¹

1982

J Math Sci

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The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970-1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.). I. General TheoryThe theory of foliations has existed as an independent mathematical discipline for about 30 years, The first major work in this area is the dissertation of Reeb [302] which contains a number of deep theorems together with the basic definitions. During the fifties and sixties the number of works on the theory of foliations was relatively small, but some of these works were very important, for example, the works of Haefliger [137], Novikov [19], and Wood [388]. In the seventies interest in foliations rose considerably; the principal stimuli were the theory of characteristic classes of foliations which arose in 1971 and the works of Thurston of 1974-1976 on the existence and classification of foliations. Beginning in 1970 about 400 works on the theory of foliations were reviewed in RZhMatmatika; the present survey is written on the basis of these works~ Among these works specially devoted to foliations there are the book of Tamura[355] reeently published in Russian translation, the survey paper [211], more restricted survey papers [24,27, 47, 120,173,187, 311, 320, 3541, and the popular papers [69, 341, 385]. Among other works of general character we mention the story of the sources and the basic problems of foliations written by the founder [303] and the list of problems compiled by Schweitzer [329].i. Basic Concepts. Let X be a smooth n-dimensional manifold without boundary (smoothness everywhere means C =~ unless otherwise specified), and let 0 -< p -< n. We say that on X there is given a smooth foliation of dimension p if X is equipped with a partition into (linearly) connected subsets F~ possessing the following property: Each point x 6 X is covered by a chart ~: U ~ IR n from the structural atlas of the manifold X such that the components (relative to the linear connectedness) of the intersections UflF= are mapped by ~ onto pdimensional planes of the space R n parallel to the plane of the first p coordinate axes. The sets Fc~ are called leaves of the foliation, and X itself is called the total manifold. The number n-p is called the codimension of the foliation; the eodimension of the foliation ~ is denoted by codim ~-.These definitions (as the majority of definitions below) have obvious complex versions. They can also be extended to manifolds with boundary, while here there are two concurrent possibilities: it is possible to require that the leaves be transverse to the boundary, and it is possible to require that the components of the boundary be leaves. In the first case the foliation is called transversal to the boundary, while in the second case (in which the codimension of the foliation must be equal to i) the foliation is trivial on the boundary. Exampl...

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Behaviour of leaves of codimension-one foliations

Cited by 14 publications

References 2 publications

Hierarchy of graph matchbox manifolds

Hierarchy of graph matchbox manifolds

Suspensions of Bernoulli shifts

Foliations

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