Let $T$ be a topological semigroup and $(T, X)$ be a semiflow on topological space $X$. This means that for every $t\in T$, $t:X\to X$ is continuous map and for $t_0, t_1\in T$, we have $t_1(t_0(x))= (t_1t_0)(x)$.
Question. Take an open cover $\mathcal{A}$ for $X$ and $g\in T$. In my research, I need to know that does there exist an open cover $\mathcal{B}$ such that for every open set $B\in\mathcal{B}$, there is open set $A\in\mathcal{A}$ with $g^{-1}(B)\subseteq A$?
I know that above question does not hold in general. For example, assume that $X=\mathbb{R}\times \{0\}\cup \mathbb{R}\times \{1\}$ and $g:X\to X$ defined by $g(x, 0)=(x, 1)$ while $g(x, 1)=(x, 1)$. Then for open cover $\bigcup_{n\in\mathbb{N}}\{(-n, n)\times \{1\}, (-n, n)\times \{0\}\}$, there is no any open cover $\mathcal{B}$ such that for every open set $B\in\mathcal{B}$, there is open set $A\in\mathcal{A}$ with $g^{-1}(B)\subseteq A$?
What condition on $g\in T$ implies that the above question does hold?