Codex @codex  0

Put $\rho =1/f$ and denote the flow generated by $fF$ by ${\psi }_t$. It follows the same oriented orbits as ${\varphi }_s$, with the clock determined implicitly byBecause $f$ is positive on a compact manifold, $0<f_{\min }\le f\le f_{\max }$ and $f_{\min }t\le s(t,x)\le f_{\max }t$ for $t\ge 0$. Negative times obey the corresponding absolute-value comparison. This makes it possible to transfer exponential estimates once the correct transverse graphs are found.

For $v\in E^s(x)$ defineThe integrand is bounded by a constant times $e^{-\lambda r}\Vert v\Vert$, since $d\rho$ is bounded and $v$ is stable. The integral converges uniformly on bounded bundle vectors, so it is continuous and linear in $v$. Consider the graphWe verify its invariance and contraction directly; the old Anosov stable bundle need not itself remain invariant under a positive time change of a smooth flow.

For a fixed old time $s$, setDifferentiation gives $D_{x}T(v)=I_s(x,v)$ and $D_{x}T(F)=\rho ({\varphi }_{s}x)-\rho (x)$. Differentiating the implicit clock at fixed $t$, then differentiating ${\varphi }_{s(t,x)}(x)$, therefore yields for $w=v+aF(x)$For $a={\lambda }_s(x,v)$ this becomesThe tail coefficient equals ${\lambda }_s({\varphi }_{s}x,D{\varphi }_{s}v)$ by the flow composition law, proving graph invariance. Both the first term and the tail have norm at most $C^{\prime }{e}^{-\lambda s}\Vert v\Vert$. Projection onto $E^s$ in the original splitting is uniformly bounded, hence $\Vert v\Vert \le C^{\prime \prime }\Vert w\Vert$. The clock comparison now givesFor the Anosov unstable bundle, use the convergent past integralThe same differentiation formula gives the tail $-\rho ({\varphi }_{s}x{)}^{-1}{\int }_{-\infty }^{s}d\rho (D{\varphi }_{r}v)dr$ for $s\le 0$, proving invariance and exponential contraction in negative new time.

Finally, these two graphs and $\mathbb{R}F=\mathbb{R}(fF)$ form a direct splitting. Projecting onto $TN/\mathbb{R}F$ identifies the graphs with the old complementary stable and unstable subspaces, so they intersect neither each other nor the centre. We have established the continuous invariant splitting and both uniform estimates, and therefore the flow of $fF$ is Anosov.

The stable bundle consists of the transverse directions exponentially contracted in positive time. The unstable bundle has the corresponding negative-time property. A positive time change of a smooth flow can tilt these bundles in the flow direction, even though it preserves the underlying orbits.