# expansive continuous flow

I encounter with two definitions for expansive continuous flows and their equivalence is unclear for me. Could anyone can explain for me please? Thanks in advance. I cite below these two definitions.

The first one in the paper "expansive one-parameter flow" of Bowen and Walters (1972): A flow $\varphi^t$ acting on a compact metric space $X$ is expansive if $\forall \epsilon, \exists \delta$ with the property that if $d(\varphi^t x, \varphi^{s(t)}y) \le \delta$ for all $t\in \mathbb{R}$ a pair of points $x,y \in X$ and a continuous function $s: \mathbb{R} \rightarrow \mathbb{R}$ with $s(0)=0$ then $y= \varphi^{t_0}x$ for some $|t_0| \le \epsilon.$

The second one in the book "Introduction to the modern theory of dynamical system" of Katok and Hasselblatt, page 125: A flow $\varphi^t$ acting on a compact metric space $X$ is expansive if there exists a positive $\delta$ such that any given continuous function $s: \mathbb{R} \rightarrow \mathbb{R}$ with $s(0)=0$ satisfying $d(\varphi^t x, \varphi^{s(t)}x) \le \delta$ for all $t\in \mathbb{R}$ then the property $d(\varphi^t x, \varphi^{s(t)}y) \le \delta$ implies $y= \varphi^{t_0}x$ for some $t_0 \in \mathbb{R}.$

• There seem to be some quantifiers missing here: do the inequalities depending on $t$ hold for all $t$, or only for certain $t$? If only for $t$ in a certain range, then on what does the range depend? – Ian Morris Jan 9 '14 at 21:17
• Is the bad English really from the sources that are cited here? – Michael Renardy Jan 10 '14 at 2:08
• It might be possible that the definition are not equivalent, but agree on classical examples. – Benoît Kloeckner Jan 28 '14 at 21:53

A flow $\phi^t$ is expansive if there is $\delta>0$ such that if $x,y$ satisfy $d(\phi^t(x),\phi^{s(t)}(y))<\delta$ for every $t\in\mathbb{R}$, for some continuous map $s:\mathbb{R}\to\mathbb{R}$ with $s(0)=0$, then $y=\phi^{t_0}(x)$ for some $t_0\in\mathbb{R}$.