BRAIDS AND THE NIELSEN-THURSTON CLASSIFICATION

1995 ◽  
Vol 04 (04) ◽  
pp. 549-618 ◽  
Author(s):  
DIEGO BERNARDETE ◽  
ZBIGNIEW NITECKI ◽  
MAURICIO GUTIERREZ
Keyword(s):  
Isotopy Class ◽  
Mapping Classes

The well-known connection between braids and mapping classes on the n-punctured disc can be exploited to decide, using braid-theoretic techniques, the place of a given isotopy class in the Nielsen-Thurston classification as reducible or irreducible, and in the latter case, as periodic or pseudo-Anosov.

scholarly journals Conjugacy invariants for Brouwer mapping classes

2016 ◽  
Vol 37 (6) ◽  
pp. 1765-1814
Author(s):  
JULIETTE BAVARD
Keyword(s):  
Isotopy Class ◽  
Closed Curves ◽  
Mapping Classes

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

PL Link Isotopy, Essential Knotting and Quotients of Polynomials

1991 ◽  
Vol 34 (4) ◽  
pp. 536-541 ◽  
Author(s):  
Dale Rolfsen
Keyword(s):  
Knot Theory ◽  
Piecewise Linear ◽  
Rational Functions ◽  
Isotopy Class ◽  
Link Invariants ◽  
Link Theory ◽  
The Individual

AbstractPiecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

scholarly journals Commuting conjugates of finite-order mapping classes

2020 ◽  
Vol 209 (1) ◽  
pp. 69-93
Author(s):  
Neeraj K. Dhanwani ◽  
Kashyap Rajeevsarathy
Keyword(s):  
Finite Order ◽  
Mapping Classes

scholarly journals The stratum of random mapping classes

2017 ◽  
Vol 38 (7) ◽  
pp. 2666-2682 ◽  
Author(s):  
VAIBHAV GADRE ◽  
JOSEPH MAHER
Keyword(s):  
Random Walks ◽  
Mapping Class Group ◽  
Class Group ◽  
Sample Path ◽  
Mapping Class ◽  
Elementary Subgroup ◽  
Random Mapping ◽  
Mapping Classes ◽  
First Moment

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

scholarly journals Examples of Non-trivial Contact Mapping Classes in all Dimensions

2016 ◽  
Vol 2016 (15) ◽  
pp. 4784-4806 ◽  
Author(s):  
Patrick Massot ◽  
Niederkrüger Klaus
Keyword(s):  
Contact Mapping ◽  
Mapping Classes
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