Anosov flow

A nonsingular smooth flow on a compact smooth manifold is Anosov if there is a continuous invariant splitting $TN=E^s\oplus \mathbb{R}F\oplus E^u$ and constants $C,\lambda >0$ with $\Vert D{\varphi }_t{v}_s\Vert \le C{e}^{-\lambda t}\Vert v_s\Vert$ and $\Vert D{\varphi }_{-t}{v}_u\Vert \le C{e}^{-\lambda t}\Vert v_u\Vert$ for $t\ge 0$. Uniform contraction and expansion are transverse to the neutral flow direction, which distinguishes the splitting from that of an Anosov diffeomorphism.

Let $F$ generate a nonsingular smooth flow on a closed manifold. Suppose a continuous nondegenerate indefinite quadratic form on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative along that flow. Then the flow is an Anosov flow. Compactness makes the positivity uniform; the positive and negative cones give the unstable and stable transverse directions. The flow direction must be quotiented out: its derivative cannot satisfy strict positivity.

Invariance gives $dh(v_s)=dh(D{\varphi }_t{v}_s)$ for a stable direction. Boundedness of $dh$ and exponential contraction make this zero; negative time does the same for unstable directions. Also $dh(F)=0$. The Anosov splitting spans the tangent bundle, so $dh=0$ and the function is constant on each connected component. This proof does not need an ergodicity hypothesis.

For a topologically transitive Anosov flow, a Hölder function has a Hölder flow coboundary precisely when its integral around every periodic orbit vanishes. The coboundary identity is $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. For smooth flow and smooth $h$, the smooth regularity theorem gives smooth $u$. Necessity follows by setting $t$ equal to a period. The solution is unique up to a constant by density of an orbit.

The unstable bundle consists of transverse directions exponentially contracted in negative time. Together with the stable bundle and the flow direction it gives the invariant splitting of an Anosov flow. Under a positive time change its graph coefficient is obtained from the convergent past integral $-\rho (x{)}^{-1}{\int }_{-\infty }^{0}d\rho (D{\varphi }_{r}v)dr$.

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.

The weak stable bundle adjoins the flow direction to the strong stable bundle. It is tangent to the weak stable foliation. For a geodesic flow on a hyperbolic surface, the canonical frame gives $E^{ws}=\mathrm{span}\{X,H-V\}$.