= Anosov flow
{c}
An <Anosov flow> is a nonsingular <smooth flow> on a compact <smooth manifold> with a continuous derivative-invariant splitting
$$
TN=E^s\oplus\mathbb RF\oplus E^u.
$$
Here $F$ 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 $t\geq0$ and the corresponding tangent vectors:
$$
\|D\Phi_t v_s\|\leq Ce^{-\lambda t}\|v_s\|,\qquad
\|D\Phi_{-t}v_u\|\leq Ce^{-\lambda t}\|v_u\|.
$$
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.
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