spaces are each subbundles, and are preserved (are invariant) under the action of the geodesic flow; that is, under the action of group elements . To compare the lengths of vectors in at different points q, one needs a metric. Any inner product at extends to a left-invariant Riemannian metric on P, and thus to a Riemannian metric on Q. The length of a vector expands exponentially as exp(t) under the action of . The length of a vector shrinks exponentially as exp(-t) under the action of . Vectors in are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant, but the other two shrink and expand: and where we recall that a tangent vector in is given by the derivative, with respect to t, of the curve , the setting . Geometric interpretation of the Anosov flow When acting on the point of the upper half-plane, corresponds to a geodesic on the upper half plane, passing through the point . The action is the standard Möbius transformation action of SL(2,R) on the upper half-plane, so that A general geodesic is given by with a, b, c and d real, with . The curves and are called horocycles. Horocycles correspond to the motion of the normal vectors of a horosphere on the upper half-plane. See also Ergodic flow Morse–Smale system Pseudo-Anosov map Notes References Anthony Manning, Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature, (1991), appearing as Chapter 3 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). (Provides an expository introduction to the Anosov flow on SL(2,R).) Toshikazu Sunada, Magnetic flows on a Riemann surface, Proc. KAIST Math. Workshop (1993), 93–108. Diffeomorphisms Dynamical systems Hyperbolic geometry