Hyperbolic characteristic flow for an inverted oscillator ($A_t=\left(\begin{array}{cc}\cosh t& \sinh t\\ \sinh t& \cosh t\end{array}\right)$)

The Hamiltonian $(v^2-x^2)/2$ gives $\dot{x}=v$, $\dot{v}=x$. Its characteristic flow map is multiplication by $A_t$, with $A_t{A}_s=A_{t+s}$ and determinant one. The backward map is $A_{-t}$. This expanding-contracting linear flow preserves phase-space Lebesgue measure and every finite-$p$ integral of a transported density, despite stretching its level sets.