The geodesic flow on a compact locally CAT(-1) metric space, first
studied by Gromov, is a far-reaching generalization of the geodesic
flow on a closed negative curvature Riemannian manifold. While one
expects these flows to exhibit similar behaviour to the classical
case, the lack of smooth structure has been a major obstacle to
extending many of the finer aspects of the dynamical theory to this
setting. Our new approach to this problem is to show that such
geodesic flows are Smale flows.
A Smale flow is a topological flow equipped with a continuous bracket
operation which is an abstraction of the local product structure from
uniform hyperbolicity. In 1987, Pollicott showed that a version of Bowens
construction of symbolic dynamics for Axiom A flows can be extended to this
setting. By symbolic dynamics, we mean there exists a suspension flow over
a shift of finite type which describes the original dynamics. By taking
additional care in the construction, we are able to verify that the roof
function can be taken to be Lipschitz in our setting. This is achieved by
using carefully chosen geometric rectangles as the building blocks for the
construction.
With this additional ingredient, the symbolic dynamics machine switches on
and ergodic-theoretic results which are true for Axiom A flows are extended
to this setting. For example, we obtain that the Bowen-Margulis measure for
the geodesic flow is Bernoulli and satisfies the Central Limit Theorem.
This is joint work with Dave Constantine and Jean-Francois Lafont.
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