Invariantsete

Invariantsete is a concept in mathematics used to describe a subset of a given space that remains within itself under a specified transformation or action. Formally, if f is a map from a space X to itself, a subset S ⊆ X is forward invariant if f(S) ⊆ S. If the transformation is invertible, a stronger notion is S = f(S). In the context of a semigroup or group of transformations, S is invariant if g(S) ⊆ S for every transformation g in the family.

Common types and terminology

- Forward (or simply) invariant: under the action of f, points in S stay inside S after one or more applications.

- Backward invariant: S is closed under preimages, meaning f^{-1}(S) ⊆ S.

- Invariant under a family: S is invariant if it is mapped into itself by every transformation in a given set of maps.

Examples

- The unit circle in the plane is invariant under any rotation about the origin, since rotating a point on the circle yields another point on the circle.

- For the map f(x) = x^2 on the real line, the set [0, ∞) is forward invariant, because squaring any nonnegative number yields another nonnegative number.

- In dynamical systems, invariant sets include fixed points, periodic orbits, and invariant manifolds, which describe regions where trajectories remain for all future (and sometimes past) times.

Properties

- The intersection of invariant sets for the same transformation is invariant.

- The union of invariant sets is not necessarily invariant.

- Invariance concepts extend to group actions and to invariant measures, which describe preserved distributions, not just preserved sets.

Applications

Invariantsete concepts appear in dynamical systems, control theory (invariant sets for safety and stability), differential equations, and ergodic theory, where they help describe long-term behavior and structural stability.