Let $(X, \{T_g\}_{g \in G}), (Y, \{S_g\}_{g \in G})$ be topological dynamical systems, with $G$ a group, $(X, d_X), (Y, d_Y)$ compact metric spaces and $\{T_g\}_{g \in G}, \{S_g\}_{g \in G}$ groups of homeomorphisms acting on $X$ and $Y$ respectively. Let $p \colon X \to Y$ be a topological factor, i.e., a continuous surjective map such that $p(T_g(x)) = S_g(p(x))$ for every $x \in X$.
- Is it true that $(Y, \{S_g\}_{g \in G})$ must be an equicontinuous system if $(X, \{T_g\}_{g \in G})$ is equicontinuous? If this isn't true, is there a simple counterexample?
- I know that if $(X, \{T_g\}_{g \in G})$ is distal, then $(Y, \{S_g\}_{g \in G})$ is also distal, but the proof I found derives this fact from more general facts using properties of minimality and almost periodicity (Corollary 2.7.7 of Brin and Stuck - Introduction to Dynamical Systems). Is there a way to prove this directly from the definition of a distal system?
Many thanks in advance.