Are there $\delta_n<\alpha$ and $x_n\in X$ such that $\{\varphi_{(-\delta_n, \delta_n)}(x_n):n\in\mathbb{N}\}$ is a cover for $\varphi_\mathbb{R}(x)$.

Question

Let $\varphi:X\times \mathbb{R}\to X$ be a continuous action on compact metric space $X$. Fix $\alpha>0$ and $x\in X$.

Are there $\delta_n<\alpha$ and $x_n\in X$ such that $\{\varphi_{(-\delta_n, \delta_n)}(x_n):n\in\mathbb{N}\}$ is a cover for $\varphi_\mathbb{R}(x)$.

Please help me to know it.

Answer 1

If I understood your notations properly, $X=[0,1], \phi (x,t)=t$ for $0 \leq t \leq 1$, $1$ for $t >1$, $0$ for $t <0$ and $\alpha =1/2$ gives counterexample.