Editing Invariant (mathematics) (section)

==Invariant set==

A [[subset]] ''S'' of the domain ''U'' of a mapping ''T'': ''U'' → ''U''  is an '''invariant set''' under the mapping when <math>x \in S \iff T(x) \in S.</math> The [[element (mathematics)|elements]] of ''S'' are not necessarily [[Fixed point (mathematics)|fixed]], even though the set ''S'' is fixed in the [[power set]] of ''U''. (Some authors use the terminology ''setwise invariant,''<ref name="Simon">{{cite book|author=Barry Simon|title=Representations of Finite and Compact Groups|publisher=American Mathematical Soc.|isbn=978-0-8218-7196-6|page=16|url=https://books.google.com/books?id=SFlDLVqVZJgC}}</ref> vs. ''pointwise invariant,''<ref>{{cite book|author=Judith Cederberg|title=A Course in Modern Geometries|url=https://archive.org/details/courseinmodernge0000cede|url-access=registration|year=1989|publisher=Springer |isbn=978-1-4757-3831-5|page=[https://archive.org/details/courseinmodernge0000cede/page/174 174]}}</ref> to distinguish between these cases.)
For example, a circle is an invariant subset of the plane under a [[rotation]] about the circle's center. Further, a [[conical surface]] is invariant as a set under a [[Homothetic transformation|homothety]] of space.

An invariant set of an operation ''T'' is also said to be '''stable under''' ''T''. For example, the [[normal subgroup]]s that are so important in [[group theory]] are those [[subgroup]]s that are stable under the [[inner automorphism]]s of the ambient [[group (mathematics)|group]].<ref>{{harvtxt|Fraleigh|1976|p=103}}</ref><ref>{{harvtxt|Herstein|1964|p=42}}</ref><ref>{{harvtxt|McCoy|1968|p=183}}</ref>
In [[linear algebra]], if a [[linear transformation]] ''T'' has an [[eigenvector]] '''v''', then the line through '''0''' and '''v''' is an invariant set under ''T'', in which case the eigenvectors span an [[invariant subspace]] which is stable under ''T''.

When ''T'' is a [[screw displacement]], the [[screw axis]] is an invariant line, though if the [[pitch (screw)|pitch]] is non-zero, ''T'' has no fixed points.

In [[probability theory]] and [[ergodic theory]], invariant sets are usually defined via the stronger property <math>x \in S \Leftrightarrow T(x) \in S.</math><ref name=billingsley>{{harvp|Billingsley|1995|pages=313-314}}</ref><ref name="douc">{{harvp|Douc|Moulines|Priouret|Soulier|2018|page=99}}</ref><ref name="klenke">{{harvp|Klenke|2020|page=494-495}}</ref> When the map <math>T</math> is measurable, invariant sets form a [[sigma-algebra]], the [[invariant sigma-algebra]].