On an existence problem of periodic points in intervals whose images cover themselves

TL;DR

This paper proves a conjecture about the existence of periodic points within certain intervals under continuous maps, specifically confirming the case for five intervals, and introduces a discretization method to approach the problem.

Contribution

The paper establishes the conjecture for five intervals and proposes a discretization approach to address the existence of periodic points in such settings.

Findings

Confirmed the conjecture for k=5 intervals

Introduced a discretization method for the problem

Provided insights into periodic points in self-covering intervals

Abstract

We consider $k$ intervals on the real line whose images under a continuous map $f$ contain themselves. It's conjectured that there exists a periodic point of period not bigger than $k$ in these intervals. We prove the conjecture for $k=5$ in this paper. We also propose a discretization method in attempt to solve the problem.