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An Anosov flow is a nonsingular smooth flow on a compact smooth manifold with a continuous derivative-invariant splittingHere $F$ is the generating vector field, the flow direction is one-dimensional, and in the usual nontrivial convention both $E^s$ and $E^u$ have positive dimension. Constants $C,\lambda >0$ must work uniformly over all points, all $t\ge 0$ and the corresponding tangent vectors:The stable bundle of an Anosov flow contracts forward; the unstable bundle of an Anosov flow contracts backward. The neutral direction along $F$ is essential: changing the starting point slightly along an orbit mainly shifts its clock, so that direction cannot be included in either uniform contraction estimate.

A suspension flow of the cat map gives a three-dimensional example. A positive time change of a smooth flow preserves the Anosov property on a compact manifold, although its stable and unstable bundles can tilt along the flow direction. These are uniform structural conditions, stronger than a sign observed in one finite-time Lyapunov exponent calculation.

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.

A suspension flow turns a discrete-time map into continuous motion by moving along an extra time coordinate. For a diffeomorphism $f:M\to M$ and constant roof one, form the mapping torusThe flow moves vertically at unit speed; each crossing of the top applies $f$ and re-enters at height zero. If $M$ has dimension $d$, the suspension has dimension $d+1$.

Suspending a hyperbolic toral automorphism gives an Anosov flow. After $k$ roof crossings, its transverse derivative is $A^k$, so the base map's stable and unstable exponential bounds become the transverse flow bounds. A remaining fractional crossing has a uniformly bounded effect on a compact mapping torus. The vertical direction supplies the neutral flow direction. In particular, the two-dimensional cat map gives a three-dimensional suspension.

A constant roof also shows why this flow need not be mixing. The well-defined height function $H([x,s])=e^{2\pi is}$ obeys $H({\Phi }_t([x,s]))=e^{2\pi it}H([x,s])$. Under the product invariant probability measure its mean is zero, but its autocorrelation has modulus one for all $t$. Thus even a mixing base map can produce a suspension with persistent clock correlations.