Which cycles force uncountably many orbit-types? (Q6140867)

Two orbits of a closed-interval map are said to be of the same type if their respective elements are connected via an order-preserving bijection. Orbits of the same type form equivalence classes, referred to as orbit-types of the map. In this paper, the authors prove that the existence of a cycle for a closed-interval map implies the existence of either only finite orbit-types or uncountably many orbit-types. The authors also determine which types of cycles belong to each of these two cases.