expansive continuous flow

Question

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 $t_0 \in \mathbb{R}.$

Answer 1

Bowen-Walters definition implies KH. The converse is fase in general but true on certain spaces (e.g. surfaces). This is clarified in artigue's thesis where the HK type are called geometric separating flows

Answer 2

I think that the definition in KH book should be this:

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 $t_0\in\mathbb{R}$.

As far as I know, this definition appears first in a 1988's paper by O. Ruggiero.