Single compatible metric ensuring finite equicontinuity from per-iterate equicontinuity

Determine whether, for every metrizable space X and family F of self-maps on X, if for each n ≥ 1 there exists a compatible metric d_n on X such that the family F^n of all n-fold compositions of functions from F is equicontinuous with respect to d_n, then there exists a single compatible metric d on X such that F is finitely equicontinuous (i.e., F^n is equicontinuous for all n ≥ 1 with respect to d).

Background

The paper studies remetrization techniques for dynamical systems to control distances over time, focusing on Lipschitz bounds and moduli of continuity for iterates and compositions. A key notion is finite equicontinuity: a family F ⊂ X^X is finitely equicontinuous if, for every n ≥ 1, the family F^n of all n-fold compositions is equicontinuous.

Through Theorem 2.7, the author characterizes when one can remetrize a space so that all F^n admit prescribed moduli of continuity, under the assumption that F is finitely equicontinuous. The paper then provides Example 2.11, demonstrating subtle behavior in the noncompact setting: a family can be equicontinuous in some metric while a closely related family is not equicontinuous in any compatible metric.

Motivated by this example, the authors pose the problem of whether equicontinuity of each iterate family F^n, potentially with different compatible metrics d_n for each n, guarantees the existence of a single compatible metric d under which F is finitely equicontinuous.