invariant subset in a sentence

1. A "'pattern "'is the repeatable components of an image, defined as the S-invariant subset of an image.
2. Every recurrent point is a nonwandering point, hence if f is a homeomorphism and X is invariant subset of the non-wandering set of f ( and may be a proper subset ).
3. The topological dynamical system \ textstyle ( X, T ) is called "'minimal "'if it has no proper non-empty closed \ textstyle T-invariant subsets.
4. A diffeomorphism " f " is omega stable if and only if it satisfies axiom A and the "'no-cycle condition "'( that an orbit, once having left an invariant subset, does not return ).
5. Unlike the theory of smooth dynamical systems, where the main object of study is a smooth manifold with a diffeomorphism or a smooth flow, phase spaces considered in topological dynamics are general metric spaces ( usually, invariant subsets of a manifold are frequently very complicated topologically ( cf limit cycle, strange attractor ); additionally, shift spaces arising via symbolic representations can be considered on an equal footing with more geometric actions.
6. It's difficult to find invariant subset in a sentence.