On Periodic Points in Covering Systems

We study a system of intervals $I_1,\ldots,I_k$ on the real line and a continuous map $f$ with $f(I_1 \cup I_2 \cup \ldots \cup I_k)\supseteq I_1 \cup I_2 \cup \ldots \cup I_k$. It's conjectured that there exists a periodic point of period $\le k$ in $I_1\cup \ldots \cup I_k$. In this paper, we prove the conjecture by a discretization method and reduce the initial problem to an interesting combinatorial lemma concerning cyclic permutations. We also obtain a non-concentration property of periodic points of small periods in intervals.