Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$.
Let $d^{1,f}$ be the Bowen-Walters metric in $Y^{1,f}$, then $Y^{1,f}$ is a compact metric space.
If $\mu$ is a Borel probability measure in $Y^{1,f}$, then I wonder if you can define a Borel probability measure in $X$ restricting the first coordinate:
these are my attempts:
1) $\nu={\pi_{1}}_*\mu$ where $\pi_1:Y^{1,f}\rightarrow X$ is the first projection (the problem here is that there should be an identification as $x\sim f(x)$)
2) $\nu(M)=\mu((M\times[0,1])\cap Y^{1,f})$ (but I think this makes no sense )
I appreciate if you could give me some suggestions to define the probability $\nu$ in $X$.
I presume that you are assuming that $\mu$ is invariant under the flow. Otherwise, there is really no canonical way of defining a measure on the base.
If $\mu$ is invariant, then basically both of your definitions, properly adjusted, define a unique invariant measure on the base such that $\mu$ is locally the product of $\nu$ and Lebesgue measure (along the flow direction).
Simply define:
$\nu={\pi_{1}}_*\mu$ restricted to the $\sigma$-algebra of $X\times(0,1)$ and then notice that this $\sigma$-algebra generates the one of $Y$; or
$\nu(M)=\mu((M\times[0,1])\cap Y^{1,f})$ for sets $M\subset X\times(0,1)$ and then use the same obervation as in 1).
The outcome is the same.