Leaves of Markov local minimal sets in foliations of codimension one
type: Item
instance of: scholarly article
title: Leaves of Markov local minimal sets in foliations of codimension one
zbMATH Open document ID: 0703.57016
DOI: 10.5565/PUBLMAT_33389_07
author: John Cantwell
author: Lawrence Conlon
published in: Publicacions Matemàtiques
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41085
review text: Let M be a compact orientable manifold, and \({\mathcal F}\) a codimension one, transversely orientable \(C^ 2\) foliation of M. An exceptional local minimal set X is said to be Markov if the holonomy pseudogroup of the foliated set (X,\({\mathcal F}| X)\) is generated by a subshift of finite type. The authors prove that if L is an arbitrary leaf of a Markov local minimal set X, then the space of ends of L which are asymptotic to L is homeomorphic to the Cantor set. In the paper there is a proof of the following theorem: Let \(X\subset M\) be a Markov local minimal set of \({\mathcal F}\). Then X contains exactly a countable infinity of resilient leaves. Furthermore, for M 3-dimensional, either \(genus(L)=1\), for each resilient leaf \(L\subset X\), the remaining leaves of \({\mathcal F}| X\) having genus 0, or each leaf \(L\subset X\) has infinite genus and each end asymptotic to L is a cluster point of handles.
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4153491
zbMATH Keywords: codimension one
zbMATH Keywords: transversely orientable
zbMATH Keywords: foliation
zbMATH Keywords: exceptional local minimal set
zbMATH Keywords: subshift of finite type
zbMATH Keywords: Markov local minimal set
zbMATH Keywords: space of ends
zbMATH Keywords: resilient leaves
MaRDI profile type: Publication
OpenAlex ID: W2095229618
description: scientific article; zbMATH DE number 4153491