Invariant set

D.Lazard (talk) 13:24, 9 February 2024 (UTC)

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In mathematics, an invariant set is a subset which does not change under the action of a group or other dynamical system. It often has the interpretation of a "place that one can never leave according to the given dynamical system".

Depending on the subject and on the author, "invariant set" may denote a variant of one of these two related, but distinct notions:

1. A subset $S\subseteq X$ such that every point of $S$ is mapped again to $S$. Points from outside of $S$ may still be mapped to $S$. This is the notion mostly considered in differential geometry and related fields. This variant is described below at one-sided definition;
2. A subset $S\subseteq X$ such that every point of $X$ is mapped to $S$ if and only if it is already in $S$. This is the notion mostly considered in probability theory and related fields, sometimes up to null sets. This variant is described below at two-sided definition.

The second variant is a special case of the first one, and for the case of group actions, the two variants coincide.

One-sided definition

Invariant sets in their one-sided definition have the property of being stable under the action, in the sense that their points will not leave the set. We give the definition for single functions, possibly with extra properties (such as being continuous or measurable), then for group actions, and finally for general monoid actions.

Definition for single functions

Let $f:X\to X$ be a function. A subset $S\subseteq X$ is $f$-invariant if for every $x\in X$,

$x\in S⟹f(x)\in S.$

We can restate the condition equivalently in terms of preimages:

$f^{-1}(S)\subseteq S.$

Definition for group actions

Let $G$ be a monoid, let $\alpha :G\times X\to X$ be a group action, and denote the action of $g\in G$ on $X$ by ${\alpha }_g:X\to X$. A subset $S\subseteq X$ is $\alpha$-invariant if for every $g\in G$ and every $x\in X$,

$x\in S⟹{\alpha }_g(x)\in S.$

Equivalently, in terms of preimages: for every $g\in G$,

${\alpha }_g^{-1}(S)\subseteq S.$

Note that since $g\in G$ is invertible, the inclusion can be replaced by an equality, and so for groups the notion coincides with the two-sided definition given below.

General definition

More generally, let $M$ be a monoid, let $\alpha :M\times X\to X$ be a monoid action, and denote the action of $m\in M$ on $X$ by ${\alpha }_m:X\to X$. A subset $S\subseteq X$ is $\alpha$-invariant if for every $m\in M$ and every $x\in X$,

$x\in S⟹{\alpha }_m(x)\in S.$

Equivalently, in terms of preimages: for every $m\in M$,

${\alpha }_m^{-1}(S)\subseteq S.$

This generalizes the notion for groups, since every group is a monoid (but in this case it does not coincide with the two-sided version). It also generalizes the notion for functions, since every function $f:X\to X$ induces a unique action of the monoid $(\mathbb{\mathbb{N}},+)$ by ${\alpha }_n=f^n$, and every action of $(\mathbb{\mathbb{N}},+)$ arises in this way.

Examples

• In linear algebra, an invariant subspace of a vector space is an invariant subset under a linear map.
• In dynamical systems, an invariant manifold is a particular invariant subset under the flow of a differential equation.

One can construct more examples by replacing the set $X$ and the function $f$ with objects and morphisms of a more general category.

Two-sided definition

Invariant sets in their two-sided definition are mostly used in probability theory and related fields such as information theory and ergodic theory. They can have the interpretation of being "indifferent" to the action.

Definition

Let $f:X\to X$ be a function. A subset $S\subseteq X$ is $f$-invariant if for every $x\in X$,

$x\in S⟺f(x)\in S.$

Equivalently, in terms of preimages:

$f^{-1}(S)=S.$

More generally, let $M$ be a monoid, let $\alpha :M\times X\to X$ be a monoid action, and denote the action of $m\in M$ on $X$ by ${\alpha }_m:X\to X$. A subset $S\subseteq X$ is $\alpha$-invariant if for every $m\in M$ and every $x\in X$,

$x\in S⟺{\alpha }_m(x)\in S.$

Equivalently, in terms of preimages: for every $m\in M$,

${\alpha }_m^{-1}(S)=S.$

Properties

• Every invariant set in the two-sided sense is invariant in the one-sided sense.

• For group actions, the one-sided and two-sided versions of invariant set coincide.

• The complement of an invariant set in the two-sided definition is also invariant (in the two-sided definition).

In what follows, given $f:X\to X$, we call a function $g:X\to Y$ invariant if and only if $g\circ f=g$, i.e. if $g(f(x))=g(x)$ for all $x\in X$.

• A subset $S\subseteq X$ is invariant (in the two-sided definition) if and only if its indicator function $1_S:X\to \{0,1\}$ is invariant.

• Somewhat conversely, a function $g:X\to Y$ is invariant if and only if for every $T\subseteq Y$, the preimage $g^{-1}(T)$ is invariant (in the two-sided definition).

In measure and probability theory

When $X$ is a measure or measurable space and the action is given by measurable functions, one is interested in measurable invariant sets (in the two-sided definition).

It is also common to consider invariance only up to null sets: Given a probability space $(X,{ℱ},p)$ and a measure-preserving function $f:(X,{ℱ},p)\to (X,{ℱ},p)$, a measurable subset (event) $S\in {ℱ}$ is called almost surely invariant if and only if its indicator function satisfies

$1_S(x)=1_S(f(x))$

for almost all $x\in X$, i.e. the sets $S$ and $f^{-1}(S)$ only differ by a null set.

Similarly, given a measure-preserving Markov kernel $k:(X,{ℱ},p)\to (X,{ℱ},p)$, we call a set $S\in {ℱ}$ almost surely invariant if and only if

$1_S(x)=k(S\mid x)$

for almost all $x\in X$.

When the action is given by measurable functions or by Markov kernels, invariant measurable subsets (in the two-sided definition) form a sigma-algebra, the invariant sigma-algebra. This is true both for almost surely invariant sets as well as for the invariant sets in the strict sense.

See also

• Invariant (mathematics)
• Invariant subspace
• Invariant manifold
• Invariant sigma-algebra
• Group action
• Monoid action

References

• Viana, Marcelo; Oliveira, Krerley (2016). Foundations of Ergodic Theory. Cambridge University Press. ISBN 978-1-107-12696-1. Search this book on

• Billingsley, Patrick (1995). Probability and Measure. John Wiley & Sons. ISBN 0-471-00710-2. Search this book on

• Durrett, Rick (2010). Probability: theory and examples. Cambridge University Press. ISBN 978-0-521-76539-8. Search this book on

• Douc, Randal; Moulines, Eric; Priouret, Pierre; Soulier, Philippe (2018). Markov Chains. Springer. doi:10.1007/978-3-319-97704-1. ISBN 978-3-319-97703-4. Search this book on

• Klenke, Achim (2020). Probability Theory: A comprehensive course. Universitext. Springer. doi:10.1007/978-1-4471-5361-0. ISBN 978-3-030-56401-8. Search this book on

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