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 $f$ ***isolates*** $Y$ if: **1)** $Y$ is $f$-stable, and **2)** $f^{-1}(Y)=Y$.



Terminology question in dynamical systems
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.


Sometimes "complete invariance" refers to a weaker property that 
a) the set is invariant, and
b) the full preimage is contained
in the set.


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). 


