How should the "measure theoretic" Jacobians of a dynamical map be understood in Lai-Sang Young's "Recurrence Times and Rates of Mixing"

Question

In Young's article: Recurrence Times and Rates of Mixing, she uses multiple times the notation $JF, JF^k, JF^R$ to mean the Jacobian of a dynamical map $F:\Delta\to\Delta$ w.r.t. a given reference measure $m$. Unfortunately, details of this Jacobian are left out, and I have not found any source or book which discusses the meaning of such a Jacobian in detail. So how should $JF, JF^k$ or $JF^R$ be understood, and what properties do they have? I suppose an easy result to see the details in practice is the Sublemma 2 of the same article where, for $$\mu = F_*^k(\lambda\mid \Omega) = \left(\lambda\mid\Omega\right)\circ F^{-k}$$, it is shown that

$$\left|\frac{d\mu}{dm}(x)/\frac{d\mu}{dm}(y) - 1\right|\leq C_0$$

for all $x,y\in \Delta_0$, with $\Delta_0$ an arbitrary (a priori chosen) subset of $\Delta$. By writing $\varphi = \frac{d\lambda}{dm}$ and taking such $x_0,y_0$ that $F^kx_0 = x, F^ky_0 = y$, Young concludes the proof by estimating

$$\left|\frac{\varphi x_0}{JF^kx_0}/\frac{\varphi y_0}{JF^ky_0} - 1\right|$$

above. Right now I don't really see how the quantity

$$\left|\frac{\varphi x_0}{JF^kx_0}/\frac{\varphi y_0}{JF^ky_0} - 1\right|$$

connects to $$\left|\frac{d\mu}{dm}(x)/\frac{d\mu}{dm}(y) - 1\right|\leq C_0$$

Answer 1

To motivate the terminology, let $m$ be Lebesgue measure on $\mathbb{R}^d$, let $U\subset \mathbb{R}^d$ be open, and let $F\colon U \to \mathbb{R}^d$ be a diffeomorphism onto its image. Recall from calculus that $$ \label{mf}\tag{1} m(F(U)) = \int_U JF(x) \,dm(x), $$ where $JF(x) = |\det(DF)|$ and $DF$ is the matrix of partial derivatives. We may say that $JF$ is the Jacobian with respect to Lebesgue measure; it can also be described by $$ \label{RN}\tag{2} \frac{d(F^* m)}{dm}(x) = JF(x), $$ where $F^*m(E) = m(F(E))$, or equivalently, $$ \label{push}\tag{3} \frac{d(F_* m)}{dm}(F(x)) = (JF(x))^{-1}. $$ The general definition of "Jacobian of $F$ with respect to a reference measure $m$" is identical, just replacing Euclidean space and Lebesgue measure by an arbitrary measure space $(\Delta,m)$. The formulations \eqref{mf}, \eqref{RN}, and \eqref{push} are equivalent, provided one understands them in the correct way: in \eqref{mf}, $U$ must be a set on which $F$ is injective; in \eqref{RN}, $F$ must be injective on a neighborhood of $x$, and $F^*m$ is the pullback of $m$ by the restriction of $F$ to this neighborhood; in \eqref{push}, $F$ must be injective on a neighborhood of $x$.

Answer 2

I believe the book Nonuniformly Hyperbolic Attractors by Jose F. Alves covers the measure theoretic Jacobian in detail. Namely section 2.2, on page 17.