If you would like to continue working on the submission, click on the "Edit" tab at the top of the window.
If you have not resolved the issues listed above, your draft will be declined again and potentially deleted.
If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors.
Please do not remove reviewer comments or this notice until the submission is accepted.
Where to get help
If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews.
If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed.
To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags.
Please note that if the issues are not fixed, the draft will be declined again.
Comment: We have already Invariant set, which covers the same subject. This is a section of Invariant (mathematics). It is possible that this section deserves to be expanded, and eventually to be split into an independent article, but this requires a WP:consensus at Talk:Invariant (mathematics). For the moment, this draft is a WP:REDUNDANTFORK. Also, the terminology ("one-sided invariant set", "two-sided invariant set", "a dynamics") of this article seems WP:OR.
Comment: We have already Invariant set, which covers the same subject. This is a section of Invariant (mathematics). It is possible that this section deserves to be expanded, and eventually to be split into an independent article, but this requires a WP:consensus at Talk:Invariant (mathematics). For the moment, this draft is a WP:REDUNDANTFORK. Also, the terminology ("one-sided invariant set", "two-sided invariant set", "a dynamics") of this article seems WP:OR. D.Lazard (talk) 13:24, 9 February 2024 (UTC)
Notions of "invariant set" in dynamical systems and ergodic theory.
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:
A subset such that every point of is mapped again to . Points from outside of may still be mapped to . This the notion mostly considered in differential geometry and related fields. This variant is described below at one-sided definition;
A subset such that every point of is mapped to if and only if it is already in . This is the notion mostly considered in probability theory and related fields[1][2][3], sometimes up to to null sets.[4][5][3] 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 be a function. A subset is -invariant if for every ,
We can restate the condition equivalently in terms of preimages:
Definition for group actions
Let be a monoid, let be a group action, and denote the action of on by .
A subset is -invariant if for every and every ,
Note that since is invertible, the inclusion can be replace by an equality, and so for groups the notion coincides with the two-sided definition given below.
General definition
More generally, let be a monoid, let be a monoid action, and denote the action of on by .
A subset is -invariant if for every and every ,
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 induces a unique action of the monoid by , and every action of arises in this way.
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 be a function. A subset is -invariant if for every ,
It is also common to consider invariance only up to null sets:[4][5][3]
Given a probability space and a measure-preserving function , a measurable subset (event) is called almost surely invariant if and only if its indicator function satisfies
for almost all , i.e. the sets and only differ by a null set.
Similarly, given a measure-preserving Markov kernel, we call a set almost surely invariant if and only if
for almost all .
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.