I'm reading "Expansive one-parameter flows" by Bowen-Walters.
Let $(X,d)$ be a compact metric space and $\Phi:X\times \mathbb{R}\to X$ be a continuous flow on $X$.
They consider the following property: $\forall \varepsilon>0\,\exists \delta>0$ so that $d(\phi_tx,\phi_ty)<\delta$ for all $t\in\mathbb{R}$ implies $y=\phi_sx$ with $|s|<\epsilon$
Then they state that this property is not a conjugacy invariant and there are flows with this property which have uncountably many periodic orbits; for example if $X$ is the annulus in the plane bounded by the circles of radius 1 and 2 and the flow is an anticlockwise rotation with constant speed".
I would be very grateful if someone can give me some details that will allow me to understand that statement.