Smooth flow (${\varphi }_t$)

A smooth flow is a smooth one-parameter family of diffeomorphisms satisfying ${\varphi }_0=\mathrm{id}$ and ${\varphi }_{t+s}={\varphi }_t\circ {\varphi }_s$. Its generating vector field is $F(x)={\partial }_t{\varphi }_t(x){|}_{t=0}$. On a closed manifold, every smooth vector field has a flow defined for all real times.

A smooth function $h$ is a flow coboundary if $h=Fu$ for some smooth function $u$, where $F$ is the flow generator. Equivalently $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. Thus its integral over every periodic orbit is zero. For a transitive Anosov flow, the Livsic theorem supplies the converse and smooth regularity.

For a volume form $\Omega$, define divergence by ${\mathcal{L}}_F\Omega =({\mathrm{div}}_{\Omega }F)\Omega$. The product rule for the Lie derivative gives ${\mathrm{div}}_{e^h\Omega }F={\mathrm{div}}_{\Omega }F+Fh$. Consequently the flow preserves a smooth positive volume precisely when its divergence in any reference volume is a smooth flow coboundary. The displayed density has the minus sign dictated by this product rule.

A suspension flow moves vertically in the mapping torus of a diffeomorphism, applying that diffeomorphism whenever an orbit crosses the identifying section. A hyperbolic toral automorphism with constant roof gives a three-dimensional Anosov flow: after $k$ crossings its transverse derivative is $A^k$, while the remaining fractional time contributes a uniformly bounded factor.

Write $\rho =1/f$ and $t={\int }_0^s\rho ({\varphi }_{r}x)dr$. For an Anosov stable bundle, the new stable directions are $v+{\lambda }_s(x,v)F(x)$, where ${\lambda }_s=\rho (x{)}^{-1}{\int }_0^{\infty }d{\rho }_{{\varphi }_{r}x}(D{\varphi }_{r}v)dr$. Exponential contraction makes this integral uniformly convergent. Differentiating the clock equation shows that its flow-direction coefficient after time $t$ is $\rho ({\varphi }_{s}x{)}^{-1}{\int }_s^{\infty }d\rho (D{\varphi }_{r}v)dr$, which decays exponentially and has the same graph form at the new point. Negative-time integrals give the unstable graph. Compactness and positivity of $f$ compare old and new times uniformly, preserving the Anosov flow property.

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 $E^{ws}=\mathrm{span}\{X,H-V\}$.