Thurston’s classification of surface diffeomorphisms

E518460

classification theorem mathematical theorem result in low-dimensional topology

Thurston’s classification of surface diffeomorphisms is a foundational theorem in low-dimensional topology that categorizes self-maps of surfaces into periodic, reducible, or pseudo-Anosov types, profoundly influencing the study of 3-manifolds and dynamical systems.

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Observed surface forms (2)

Surface form	Occurrences
Nielsen–Thurston classification	1
On the geometry and dynamics of diffeomorphisms of surfaces	1

Statements (46)

Predicate	Object
instanceOf	classification theorem ⓘ mathematical theorem ⓘ result in low-dimensional topology ⓘ
alsoKnownAs	Nielsen–Thurston classification NERFINISHED ⓘ Thurston–Nielsen classification NERFINISHED ⓘ
appliesTo	compact surfaces ⓘ oriented surfaces ⓘ surface diffeomorphisms ⓘ surface homeomorphisms ⓘ
author	William P. Thurston NERFINISHED ⓘ
buildsOn	Jakob Nielsen’s work on surface homeomorphisms ⓘ
characterizes	periodic mapping classes as having some power isotopic to the identity ⓘ pseudo-Anosov mapping classes as having invariant transverse measured foliations ⓘ pseudo-Anosov mapping classes as stretching along one measured foliation and contracting along the transverse one by a dilatation factor ⓘ reducible mapping classes as preserving a multicurve up to isotopy ⓘ
classifiesAs	periodic mapping classes ⓘ pseudo-Anosov mapping classes ⓘ reducible mapping classes ⓘ
concerns	mapping class group of a surface ⓘ
field	dynamical systems ⓘ low-dimensional topology ⓘ topology ⓘ
formalizes	trichotomy of surface mapping classes into periodic, reducible, and pseudo-Anosov types ⓘ
hasConsequence	pseudo-Anosov maps are generic in mapping class groups ⓘ surface automorphisms decompose into simple dynamical types ⓘ
historicalPeriod	late 20th century ⓘ
implies	existence of a canonical representative in each mapping class ⓘ existence of a pseudo-Anosov representative for irreducible non-periodic mapping classes ⓘ
influenced	development of Thurston’s geometrization ideas for 3-manifolds ⓘ study of entropy of surface diffeomorphisms ⓘ
involvesConcept	invariant geodesic laminations ⓘ isotopy class of homeomorphisms ⓘ mapping class ⓘ measured foliations ⓘ stretch factor of a pseudo-Anosov map ⓘ train tracks ⓘ
relatedTo	JSJ decomposition of 3-manifolds NERFINISHED ⓘ geometrization of Haken 3-manifolds ⓘ prime decomposition of 3-manifolds ⓘ
statesThat	every mapping class of a compact surface is periodic, reducible, or pseudo-Anosov ⓘ
usedIn	Teichmüller theory NERFINISHED ⓘ geometric group theory ⓘ study of 3-manifolds ⓘ study of fibered 3-manifolds ⓘ study of hyperbolic 3-manifolds ⓘ topological dynamics on surfaces ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

William Thurston → knownFor → Thurston’s classification of surface diffeomorphisms ⓘ

William Thurston → notableWork → Thurston’s classification of surface diffeomorphisms ⓘ

this entity surface form: On the geometry and dynamics of diffeomorphisms of surfaces

Dehn twist → playsRoleIn → Thurston’s classification of surface diffeomorphisms ⓘ

this entity surface form: Nielsen–Thurston classification