Let $X$ be a compact metric space. A flow $\Phi=\{\phi_t:-\infty<t<+\infty\}$ on $X$ which is continuous in $(t,x)$. Given $\Phi$ and $\Psi$ flows on $X$ and $Y$ respectively, we say that $\Phi$ and $\Psi$ are weakly equivalent if there exists a homeomorphism $\pi$ of $X$ to $Y$ such that $\Phi$ and
$$\hat{\Phi}=\{\hat{\phi}_t=\pi^{-1}\circ \psi_t\circ\pi;-\infty<t<+\infty\}$$
have the same orbits i.e. $\{\phi_t(x):-\infty<t<+\infty\}=\{\hat{\phi}_t(x):-\infty<t<+\infty\}$ for all $x\in X$.
Assume that $\Phi$ and $\hat{\Phi}$ have the same orbits with the same directions. Let $X_0$ denote the set of all $\Phi$-fixed points. Define $\theta(t,x),-\infty<t<+\infty,x\in X\setminus X_0$ an additive functional with the following properties:
- $\phi_t(x)=\pi^{-1}\circ \psi_{\theta(t,x)}\circ\pi=\hat{\phi}_{\theta(t,x)}(x)$
- $\theta(t+s,x)=\theta(t,\phi_s(x))+\theta(s,x)$
- $\theta(0,x)=0$ and $\theta(t,x)$ is strictly increasing in $t$
- $\theta(t,x)$ is continuos in $(t,x)$
Let $\epsilon(\Phi)$ denote the family of all $\Phi$-invariant ergodic borel probability measures in $X$. Then, for each $m\in \epsilon(\Phi)$ define
$$E_{\hat{m}}(f)=\frac{1}{E_m(\theta(1,x))}E_m\left(\int_0^{\theta(1,x)}f(\hat{\phi}_t(x))dt\right),$$
where $E_m(f)=\int f(x)dm(x)$.
I'm trying to show that $\hat{m}$ is a $\hat{\Phi}$-invariant probability measure.