# On equicontinuous factors of flows on locally path-connected compact spaces

## TL;DR

This paper investigates the structure of equicontinuous factors of flows on locally path-connected compact spaces with finite first Betti number, showing they are isomorphic to flows on low-dimensional compact abelian Lie groups.

## Contribution

It provides a new proof that the quotient map to the maximal equicontinuous factor is monotone, using a novel characterization of local connectedness via Banach lattices.

## Key findings

- Every equicontinuous factor is isomorphic to a flow on a compact abelian Lie group of dimension less than b_1(K)
- A new proof of the monotonicity of the quotient map onto the maximal equicontinuous factor
- Characterization of local connectedness in terms of the Banach lattice C(K)

## Abstract

We consider a locally path-connected compact metric space $K$ with finite first Betti number $b_1(K)$ and a flow $(K, G)$ on $K$ such that $G$ is abelian and all $G$-invariant functions $f\in\mathrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than $b_1(K)$. For this purpose, we use and provide a new proof for [HJop, Theorem 2.12] which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces $K$ and $L$ that we obtain by characterizing the local connectedness of $K$ in terms of the Banach lattice $\mathrm{C}(K)$.