$\omega$-limit set; fixed points
Let $M$ and $K$ be closed subsets of [0,1] with $K$ a subset of the limit points of $M$. Necessary and sufficient conditions are found for the existence of a continuous function $f:[0,1]\rightarrow [0,1]$ such that $M$ is an $\omega$-limit set for $f$ and $K$ is the set of fixed points of $f$ in $M$.
 S. J. Agronsky A. M. Bruckner J. G. Ceder, T. L. Pearson: The structure of $\omega$-limit sets for continuous functions
. Real Analysis Exchange 15 (1989-90), 483-510. DOI 10.2307/44152033
| MR 1059418
 A. M. Bruckner J. Smítal: The structure of $\omega$-limit sets for continuous maps of an interval
. to appear in Časopis pro Pěstování Mat. MR 1154053
 M. J. Evans P. D. Humke C. M. Lee, R. J. O'Malley: Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets. to appear.