Terminology question in dynamical systems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:38:36Z http://mathoverflow.net/feeds/question/117339 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117339/terminology-question-in-dynamical-systems Terminology question in dynamical systems Mahdi Majidi-Zolbanin 2012-12-27T17:11:40Z 2012-12-28T17:19:24Z <p>Let $X$ be a topological space and let $f:X\rightarrow X$ be a continuous self-morphism of topological spaces. Let $Y$ be a closed $f$-stable subset of $X$, that is, suppose $f(Y)\subseteq Y$. Consider the additional condition that $f^{-1}(Y)=Y$. Is there a terminology for this situation in topological dynamics? I am not sure if there exists a terminology for this, but I am tempted to say $f$ <strong><em>isolates</em></strong> $Y$ if: <strong>1)</strong> $Y$ is $f$-stable, and <strong>2)</strong> $f^{-1}(Y)=Y$.</p> http://mathoverflow.net/questions/117339/terminology-question-in-dynamical-systems/117353#117353 Answer by Alexandre Eremenko for Terminology question in dynamical systems Alexandre Eremenko 2012-12-27T20:27:24Z 2012-12-28T17:19:24Z <p>The commonly accepted term is "completely invariant". A set which is mapped to itself is called simply "invariant" and a stronger property to coincide with its preimage is called complete invariance.</p> <p>Sometimes "complete invariance" refers to a weaker property that a) the set is invariant, and b) the full preimage is contained in the set.</p> <p>EDIT. On your further questions: For a reference, see for example the survey "Dynamics of analytic transformations", Leningrad Math. J. (1990). (It is avalable on my web site). </p>