TITLE: Fixed Set Property?
QUESTION [6 upvotes]: As far as I know, there are fixed-point-like results for continuous functions from a convex compact subset $K$ of an Euclidean space to itself. I have one question in mind: Does there exist a set $A\subseteq X$ for which $f(A)=A$? Let's say $X$ is a compact metric space and $f$ is continuous.

REPLY [2 votes]: Additional Reference: Theorem 1.8 from Dynamical Systems and Ergodic Theory by M. Pollicott and M. Yuri partially answers your question.

Let $T:X\to X$ be a homeomorphism of a compact metric space $X$. Then there exists a non-empty closed set $Y\subset X$ with $TY=Y$ and $T:Y\to Y$ is minimal. 

Minimality means that $\{T^ny:n\in\mathbb{Z}\}$ is dense in $Y$ for every $y\in Y$.
This result is proved using Zorn's Lemma, much like in Henno Brandsma's answer.