On Periodic Points in Covering Systems

2023/06/18 by Yihan Wang, Wang, Yihan
Mathematics · #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2306.10523

openalex publication_date 2023/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a system of intervals I₁,…,I_k on the real line and a continuous map f with f(I₁ ∪ I₂ ∪ … ∪ I_k)⊇ I₁ ∪ I₂ ∪ … ∪ I_k. It's conjectured that there exists a periodic point of period ≤ k in I₁∪ … ∪ 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.

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