Lie vector fields

One starts by noting that ${\displaystyle T^{1}H}$ is isomorphic to the Lie group PSL(2,R). This group is the group of orientation-preserving isometries of the upper half-plane. The Lie algebra of PSL(2,R) is sl(2,R), and is represented by the matrices

${\displaystyle J={\begin{pmatrix}1/2&0\\0&-1/2\\\end{pmatrix}}\qquad X={\begin{pmatrix}0&1\\0&0\\\end{pmatrix}}\qquad Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}}$

which have the algebra

${\displaystyle [J,X]=X\qquad [J,Y]=-Y\qquad [X,Y]=2J}$

The exponential maps

${\displaystyle g_{t}=\exp(tJ)={\begin{pmatrix}e^{t/2}&0\\0&e^{-t/2}\\\end{pmatrix}}\qquad h_{t}^{*}=\exp(tX)={\begin{pmatrix}1&t\\0&1\\\end{pmatrix}}\qquad h_{t}=\exp(tY)={\begin{pmatrix}1&0\\t&1\\\end{pmatrix}}}$

define right-invariant flows on the manifold of ${\displaystyle T^{1}H=\operatorname {PSL} (2,\mathbb {R} )}$, and likewise on ${\displaystyle T^{1}M}$. Defining ${\displaystyle P=T^{1}H}$ and ${\displaystyle Q=T^{1}M}$, these flows define vector fields on P and Q, whose vectors lie in TP and TQ. These are just the standard, ordinary Lie vector fields on the manifold of a Lie group, and the presentation above is a standard exposition of a Lie vector field.

Anosov flow

The connection to the Anosov flow comes from the realization that ${\displaystyle g_{t}}$ is the geodesic flow on P and Q. Lie vector fields being (by definition) left invariant under the action of a group element, one has that these fields are left invariant under the specific elements ${\displaystyle g_{t}}$ of the geodesic flow. In other words, the spaces TP and TQ are split into three one-dimensional spaces, or subbundles, each of which are invariant under the geodesic flow. The final step is to notice that vector fields in one subbundle expand (and expand exponentially), those in another are unchanged, and those in a third shrink (and do so exponentially).

More precisely, the tangent bundle TQ may be written as the direct sum

${\displaystyle TQ=E^{+}\oplus E^{0}\oplus E^{-}}$

or, at a point ${\displaystyle g\cdot e=q\in Q}$, the direct sum

${\displaystyle T_{q}Q=E_{q}^{+}\oplus E_{q}^{0}\oplus E_{q}^{-}}$

corresponding to the Lie algebra generators Y, J and X, respectively, carried, by the left action of group element g, from the origin e to the point q. That is, one has ${\displaystyle E_{e}^{+}=Y,E_{e}^{0}=J}$ and ${\displaystyle E_{e}^{-}=X}$. These spaces are each subbundles, and are preserved (are invariant) under the action of the geodesic flow; that is, under the action of group elements ${\displaystyle g=g_{t}}$.

To compare the lengths of vectors in ${\displaystyle T_{q}Q}$ at different points q, one needs a metric. Any inner product at ${\displaystyle T_{e}P=sl(2,\mathbb {R} )}$ extends to a left-invariant Riemannian metric on P, and thus to a Riemannian metric on Q. The length of a vector ${\displaystyle v\in E_{q}^{+}}$ expands exponentially as exp(t) under the action of ${\displaystyle g_{t}}$. The length of a vector ${\displaystyle v\in E_{q}^{-}}$ shrinks exponentially as exp(-t) under the action of ${\displaystyle g_{t}}$. Vectors in ${\displaystyle E_{q}^{0}}$ are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant,

${\displaystyle g_{s}g_{t}=g_{t}g_{s}=g_{s+t}}$

but the other two shrink and expand:

${\displaystyle g_{s}h_{t}^{*}=h_{t\exp(-s)}^{*}g_{s}}$

and

${\displaystyle g_{s}h_{t}=h_{t\exp(s)}g_{s}}$

where we recall that a tangent vector in ${\displaystyle E_{q}^{+}}$ is given by the derivative, with respect to t, of the curve ${\displaystyle h_{t}}$, the setting ${\displaystyle t=0}$.

Geometric interpretation of the Anosov flow

When acting on the point ${\displaystyle z=i}$ of the upper half-plane, ${\displaystyle g_{t}}$ corresponds to a geodesic on the upper half plane, passing through the point ${\displaystyle z=i}$. The action is the standard Möbius transformation action of SL(2,R) on the upper half-plane, so that

${\displaystyle g_{t}\cdot i={\begin{pmatrix}\exp(t/2)&0\\0&\exp(-t/2)\end{pmatrix}}\cdot i=i\exp(t)}$

A general geodesic is given by

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot i\exp(t)={\frac {ai\exp(t)+b}{ci\exp(t)+d}}}$

with a, b, c and d real, with ${\displaystyle ad-bc=1}$. The curves ${\displaystyle h_{t}^{*}}$ and ${\displaystyle h_{t}}$ are called horocycles. Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane.