Smooth flow (${\varphi }_t$)

A smooth flow on a smooth manifold is a smooth map $(t,x)\mapsto {\Phi }_t(x)$ withThe identity implies ${\Phi }_{-t}={\Phi }_t^{-1}$, so each time map is a diffeomorphism. Its generating vector field is $F(x)={\partial }_t{\Phi }_t(x){|}_{t=0}$; trajectories solve the autonomous differential equation $\dot{x}=F(x)$.

For example, ${\Phi }_t(x)=x+t$ is a smooth flow on the real line, while ${\Phi }_t(\theta )=\theta +t(\mathrm{mod}2\pi )$ is a rotation flow on a circle. The second has periodic trajectories, even though the same additive time parameter acts in both examples.

A smooth vector field initially gives only a local flow. The equation $\dot{x}=x^2$ has $x(t)=x_0/(1-x_{0}t)$, which blows up at $t=1/x_0$ for $x_0>0$. On a closed manifold, compactness prevents escape in finite time and local existence can be continued for every real time. Thus every smooth vector field there generates a global smooth flow. On a noncompact manifold, global existence is a separate property.

An observable $h$ is a flow coboundary if it is the derivative of a function $u$ along a flow map. An integrated formulation isWhen $u$ is differentiable along the flow, this is $Fu=h$, where $F$ is the generating vector field. Spatial regularity must be specified separately: $u$ may be smooth or merely Hölder continuous while still being differentiable along every flow line.

The function $c(t,x)=u({\Phi }_{t}x)-u(x)$ is a cocycle over a dynamical system. Its identity follows by adding the two differences from $x$ to ${\Phi }_{t}x$ and from ${\Phi }_{t}x$ to ${\Phi }_{s+t}x$. Being a coboundary says that the accumulated observable can be recovered from endpoint values.

For a periodic orbit of period $T$, the endpoint values coincide, so its integral of $h$ is zero. This is the periodic orbit obstruction. If $u$ is bounded, then the time average of $h$ also vanishes:For a transitive Anosov flow, the Livšic theorem identifies when the periodic-orbit obstruction is sufficient. This is stronger than checking just one average or finitely many periodic trajectories.

A periodic orbit obstruction is a necessary compatibility condition for solving a cohomological equation. If an dynamical observable is a flow coboundary, $h=Xu$, where $X$ differentiates along the flow, then on a periodic orbit of period $T$,A nonzero integral therefore rules out a single-valued transfer function $u$. Under suitable transitive Anosov flow and Hölder regularity hypotheses, the Livšic theorem provides a converse. Without those dynamical and regularity assumptions, vanishing periodic integrals alone is not a general existence theorem.^[ref]

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 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.

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.

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.

The Anosov closing lemma turns a sufficiently close return of an Anosov flow into a nearby periodic orbit. On a compact manifold, uniform hyperbolicity ensures that an orbit segment returning close to its starting point is shadowed by a genuine closed orbit, with a small adjustment of the period and of the time parametrization.

Stable directions contract toward the shadowing orbit, while unstable directions are matched by choosing the appropriate nearby initial point. This makes periodic trajectories a tool for studying nonperiodic recurrence, for example when proving that periodic-orbit data control a cohomological equation. The time adjustment distinguishes the flow statement from its discrete-map counterpart.^[ref]

Let a nonsingular smooth flow on a closed manifold $N$ be generated by $F$. Suppose a continuous nondegenerate indefinite quadratic form $Q$ on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative there. Then the flow is an Anosov flow.^[ref]

Here differentiating the form means differentiating $Q_{{\Phi }_{t}x}(D{\Phi }_{t}v)$. Compactness gives constants $M,c>0$ such that $|Q(v)|\le M\Vert v{\Vert }^2$ and $\dot{Q}(v)\ge c\Vert v{\Vert }^2$. On the positive cone,so the form increases at least exponentially. Negative cones supply the analogous backward-time estimate. Strict cone invariance produces uniformly expanding and contracting transverse bundles; the full splitting also contains the neutral flow direction.

For a geodesic flow on a compact negatively curved surface, a transverse Jacobi field has scalar coordinates satisfying $\dot{y}=z$ and $\dot{z}=-K(t)y$, with Gaussian curvature $K\le -\kappa <0$. The indefinite form $Q(y,z)=yz$ obeysfor every nonzero transverse vector. This directly verifies the criterion. The quotient by $F$ is essential: $D{\Phi }_{t}F=F\circ {\Phi }_t$ gives the neutral direction that cannot satisfy this strict transverse positivity. The estimates concern the form; constructing the invariant bundles is the additional cone argument, rather than an assertion that every positive-cone vector has monotonically growing norm.

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.

The Livšic theorem turns periodic-orbit data into a global derivative along a hyperbolic flow. Let ${\Phi }_t$ be a transitive Anosov flow on a compact smooth manifold, with generating vector field $F$, and let $h$ be a real Hölder continuous observable. There is a Hölder continuous function $u$, differentiable along flow lines, satisfyingif and only if ${\int }_0^{T}h({\Phi }_{t}p)dt=0$ for every periodic orbit $p$ of period $T$. The solution preserves the Hölder exponent.^[ref]

Necessity is immediate by putting ${\Phi }_{T}p=p$ in the identity. Two solutions differ by a continuous invariant function. Transitivity supplies a dense orbit; their difference is constant on that orbit, and continuity makes it constant everywhere. Thus the solution is unique up to an additive constant.

The harder sufficiency argument defines $u({\Phi }_t{x}_0)$ by integrating $h$ along a dense orbit. The Anosov closing lemma converts a sufficiently close return into a nearby periodic trajectory. Its vanishing integral and exponential shadowing bound the discrepancy between the endpoint values in terms of their distance. This gives Hölder control on the dense orbit and permits extension to the whole manifold.

When both the flow and $h$ are smooth, a separate regularity theorem gives smooth $u$. This does not follow merely from the preceding continuity argument.^[ref] The theorem is named for A. N. Livšic.

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