I'm reading one of the classical theorems presented in Bowen's lecture notes, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms." I'm trying to figure out a very short line of reasoning that's eluding me in the proof of the following:

Let $f$ be a transitive $C^2$ Anosov diffeomorphism. If $\mbox{det}~ (Df^n: T_xM \to T_xM) = 1$ for each $n$-periodic point, then $f$ admits an invariant measure of the form $d\mu = h dm$, with $h$ positive and Holder and $m$ the natural volume form.

The sketch of the proof is to use the periodicity condition to construct a function $h(x) = e^{u(x)}$, where the Holder function $u : M\to \mathbb{R}$ is related to the log of the Jacobian $\phi(x) = \mbox{log}~\mbox{Jac} (Df: T_xM \to T_xM)$ via the cohomology relation $\phi(x) = u(fx) - u(x)$. One easily derives the fact $h(fx) = h(x) e^{\phi(x)}$ from this relation. I understand the method of proof up to here. It remains to show that the measure $hdm$ is $f$-invariant.

Bowen proceeds as follows:

Thinking of $\mu = h dm$ as the absolute value of a form,

$\begin{eqnarray*}f^* (hdm)(fx) &=& h(x) e^{\phi(x)} \\ &=& dm(fx) \\ &=& h(fx) dm(fx) \\ &=& (hdm)(fx),\end{eqnarray*}$

hence $\mu$ is $f$-invariant.

The first equality is what's giving me trouble (the second and third follow from our aforementioned fact and the definition of the form hdm). If $f^*$ is the usual pullback of forms, things seem not to transform as I am expecting (maybe I forgot all my calculus...) can someone with more experience here please enlighten me as to what he's doing in this string?