Abstract:
We study a system of intervals $I_1,\dots,I_k$ on the real line and a continuous map $f$ with $$ f(I_1 \cup I_2 \cup \dots \cup I_k)\supseteq I_1 \cup I_2 \cup \dots \cup I_k. $$ It's conjectured that there exists a periodic point of period $\le k$ in $I_1\cup \dots \cup I_k$. This generalization extends a fundamental fixed point theorem in one-dimensional dynamics. In this paper, we prove the conjecture by a discretization method and reduce the initial problem to an interesting combinatorial lemma, which exhibits a close connection to the adjacency matrix of a periodic point modulo 2. And we show that the explicit form of the characteristic polynomial depends only on the size of the matrix, which turns out to be an own property of a cyclic permutation in the symmetric group. Another consequence of the combinatorial lemma is a non-concentration property of periodic points of small periods in intervals, making it possible to find new periodic points of lower periods inside the gaps of a given periodic orbit.
Keywords:periodic point, interval, discretization, adjacency matrix, cyclic permutation.
