A "'pattern "'is the repeatable components of an image, defined as the S-invariant subset of an image.
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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 ).
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The topological dynamical system \ textstyle ( X, T ) is called "'minimal "'if it has no proper non-empty closed \ textstyle T-invariant subsets.
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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 ).
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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.
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