Invariant set

Invariant set

A subsetS of the domain U of a mapping T: U → U is an invariant set under the mapping when ${\displaystyle x\in S\iff T(x)\in S.}$ The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, 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 homothety of space.