Definition:Invariant Set

Definition

Let $S$ be a set.

Let $f: S \to S$ be a self-map on $S$.

Let $T \subseteq S$ be a subset of $S$.

Then $T$ is an invariant set of $f$ if and only if:

$f \sqbrk Y = Y$

Positively Invariant Set

Let $S$ be a set.

Let $f: S \to S$ be a self-map on $S$.

Let $T \subseteq S$ be a subset of $S$ such that:

$f \sqbrk T \subseteq T$

Then $T$ is a positively invariant set of $f$.

Negatively Invariant Set

Let $S$ be a set.

Let $f: S \to S$ be a self-map on $S$.

Let $T \subseteq S$ be a subset of $S$ such that:

$f^{-1} \sqbrk T \subseteq T$

Then $T$ is a negatively invariant set of $f$.

Also known as

An invariant set of a mapping is also known as an invariant subset of a set under a mapping if it is important to attach more emphasis to the set than to the mapping.

Also see

• Results about invariant sets can be found here.

Sources

• 1991: Gertrude Ehrlich: Fundamental Concepts of Abstract Algebra: $\S 2$: Definition $2.11.2$
• 1999: Clark Robinson: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos ... (previous) ... (next): Chapter $\text {II}$: $\S 2.3$