time-shift in partial differential equation of density function

Question

I am reading the following paper:

http://www.nacad.ufrj.br/~amit/lyapdual.pdf (see Appendix A. Lemma A.1. proof)

My understanding of a few notations are:

1. $\rho_t(z)$: the density function around initial point $z$ at time $t$
2. $\phi_t(z)$: the solution $x(t)$ of $\dot{x}(t)=f(x(t))$ with $x(0)=z$. This is also called phase flow.

Note that $$\rho_t(z)=\rho(\phi_t(z))|(\partial\phi(t)/\partial z)(z)|$$

this is coming from the evolution of density with time. The term $|(\partial\phi(t)/\partial z)(z)|$ is actually the Jacobian. Also note that $$\dot{\phi}_t(z) = f(z) = f$$

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My question is

1. How can we change $t=\tau$ to $h=0$.
2. Why did the author add $\rho_h$, the $h$ parameter?