Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 14 / 1

For a nonsingular smooth flow generated by $F$ on a closed manifold, the Anosov flow condition is a continuous $D{\varphi }_t$-invariant splittingand constants $C,\lambda >0$ such that, for all $t\ge 0$,The norm can come from any auxiliary Riemannian metric; compactness makes the resulting definitions equivalent. The centre direction is the flow direction, with $F$ nowhere zero.

One construction is a suspension flow. The hyperbolic toral automorphism of ${\mathbb{T}}^2$ defined byhas eigenvalues ${\lambda }_{\pm }=(3\pm \sqrt{5})/2$, with $0<{\lambda }_{-}<1<{\lambda }_{+}$. Form the closed three-dimensional mapping torus $({\mathbb{T}}^2\times [0,1])/((x,1)\sim (Ax,0))$ and move in the second coordinate at unit speed. Each crossing of the section applies $A$. The two one-dimensional eigenspaces give invariant transverse bundles, and after $k$ crossings their derivatives multiply by ${\lambda }_{-}^k$ and ${\lambda }_{+}^k$. The difference between $k$ and elapsed time is bounded, so any smooth Riemannian metric gives the required uniform exponential estimates. The vertical direction supplies the centre line.

A different construction is the geodesic flow on the unit tangent bundle of a closed oriented hyperbolic surface. This bundle is a closed three-manifold. In curvature $-1$, a normal Jacobi field along a unit-speed geodesic satisfies $j^{\prime \prime }-j=0$. The initial data $(j,j^{\prime })=(a,-a)$ give $j(t)=a{e}^{-t}$, while $(a,a)$ give $j(t)=a{e}^t$. The linearized geodesic flow transports these initial data as $(j(t),j^{\prime }(t))$, so they give one stable and one unstable direction with precisely those exponential rates. The remaining tangent direction is the geodesic vector field. Thus this also constructs an Anosov flow, using geometric rather than return-map hyperbolicity.

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.