$\Gamma$-compact maps on an interval and fixed points

Abstract:

We characterize the $\Gamma$-compact continuous functions $f:X \to X$ where $X$ is a possibly-noncompact interval. The map $f$ is called $\Gamma$-compact if the closed topological semigroup $\Gamma (f)$ generated by $f$ is compact, or equivalently, if every sequence of iterates of $f$ under functional composition $(\ast )$ has a subsequence which converges uniformly on compact subsets of $X$. For compact $X$ the characterization is that the set of fixed points of $f\ast f$ is connected. If $X$ is noncompact an additional technical condition is necessary. We also characterize those maps $f$ for which iterates of distinct orders agree ($\Gamma (f)$ finite) and state a result on common fixed points of commuting functions when one of the functions is $\Gamma$-compact.