Foliations 2005 2006
DOI: 10.1142/9789812772640_0005
|View full text |Cite
|
Sign up to set email alerts
|

Hirsch Foliations in Codimension Greater Than One

Abstract: We generalize the Hirsch construction of a smooth foliation on a 3-manifold with a unique exceptional minimal set, to obtain a method for constructing smooth foliations of arbitrary codimension with exotic minimal sets. The method also yields a procedure to realize a given system ofétale correspondences as the holonomy of a smooth foliation of a compact manifold. This generalizes the well-known group suspension construction.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
2
1

Citation Types

0
20
0

Year Published

2008
2008
2021
2021

Publication Types

Select...
4
1
1

Relationship

5
1

Authors

Journals

citations
Cited by 10 publications
(20 citation statements)
references
References 43 publications
0
20
0
Order By: Relevance
“…Hirsch constructed in [148] an analytic foliation F of codimension-one with an exceptional minimal set on a closed 3-manifold M , starting from a familiar method in dynamical systems to construct diffeomorphisms of compact manifolds with expanding isolated invariant sets which are solenoids [325,326]. The construction of the Hirsch foliation was generalized in [23] to codimension q > 1.…”
Section: Parabolic Foliationsmentioning
confidence: 99%
See 1 more Smart Citation
“…Hirsch constructed in [148] an analytic foliation F of codimension-one with an exceptional minimal set on a closed 3-manifold M , starting from a familiar method in dynamical systems to construct diffeomorphisms of compact manifolds with expanding isolated invariant sets which are solenoids [325,326]. The construction of the Hirsch foliation was generalized in [23] to codimension q > 1.…”
Section: Parabolic Foliationsmentioning
confidence: 99%
“…Suppose there exists a collection of smooth maps F = {f i : N → N | 1 ≤ i ≤ k} such that each f i : N → N is a covering map. We call this a system of etale correspondences in [23]. Then there exists a codimension-q foliation F F of a closed manifold M , such that its holonomy pseudogroup R F F is equivalent to that generated by the collection of maps F. The foliation F F is constructed using the generalized suspension construction, as described in §5, [23].…”
Section: Hyperbolic Foliationsmentioning
confidence: 99%
“…Another realization of a Markov pseudogroup, obtained from a hyperbolic Markov system, as the holonomy pseudogroup of a codimension one foliation on a compact 3-manifold, was provided by Biś, Hurder and Shive [2] in their construction of generalized Hirsch foliations. Several months after that paper was written, similar results were obtained in [7].…”
Section: Definition 2 (Following [18]) a Finite Systemmentioning
confidence: 99%
“…Choose an integer n > 1 and an analytic embedding ϕ : S 1 → S 1 × D so that its homotopy class is equal to ng, where g is a generator of the fundamental group of the solid torus. Now we recall the construction of a generalized Hirsch foliation in codimension one, which was presented in detail in Section 2 of [2], in the following way. Choose a nonzero interior point z 0 ∈ D (such that 0 < |z 0 | < 1) and ε > 0 such that 0 < 2ε < min{|z 0 |, 1 − |z 0 |}.…”
Section: Theoremmentioning
confidence: 99%
See 1 more Smart Citation