# Orbit of a point and flow

Let $$\varphi:X\times \mathbb{R}\to X$$ be a continuous flow on compact metric space $$X$$ without singularity. For $$\delta>0$$ and $$x\in X$$, take $$$$\Gamma_\delta(x)=\bigcup_{h\in\mathcal{C}}\bigcap_{t\in\mathbb{R}}\varphi_{-h(t)}(B(\varphi_t(x), \delta))$$$$ where $$\mathcal{C}$$ is the set of continuous functions $$h:\mathbb{R}\to \mathbb{R}$$ with $$h(0)=0$$ and $$B(a, \delta)=\{b:d(a,b)<\delta\}$$.

In the following, I give a proof of the following statement:

There is $$\alpha>0$$ such that $$\varphi_{(-\alpha, \alpha)}(x)\subseteq \Gamma_\delta(x)$$ for all $$x\in X$$.

If it is not true, then for every $$n\in\mathbb{N}$$, there is $$x_n\in X$$ such that

$$$$\varphi_{(-\frac{1}{n}, \frac{1}{n})}(x_n)\nsubseteq \Gamma_\delta(x_n)$$$$ Hence for every $$n\in\mathbb{N}$$, there is $$y_n\in\varphi_{(-\frac{1}{n}, \frac{1}{n})}(x_n)$$ such that $$y_n\notin \Gamma_\delta(x_n)$$. This implies that for every $$h\in\mathcal{C}$$, there is $$t_n\in\mathbb{R}$$ with $$d(\varphi_{h(t_n)}(y_n), \varphi_{t_n}(x_n))>\delta$$. By $$y_n\in\varphi_{(-\frac{1}{n}, \frac{1}{n})}(x_n)$$, we can say that $$lim_{n\to\infty}y_n=lim_{n\to\infty}x_n=x$$ that is a contradiction with $$d(\varphi_{h(t_n)}(y_n), \varphi_{t_n}(x_n))>\delta$$ if $$h(t)=t$$.