A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 (Z)\PSL 2 (R). The complement of any finite number of orbits is a hyperbolic 3-manifold, which thus has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.
The fact that the modular template coincides with the Lorenz template,
discovered by Ghys, implies modular knots have very peculiar properties. We
obtain a generalization of these results to other Hecke triangle groups. In
this context, the geodesic flow can never be seen as a flow on a subset of
$S^3$, and one is led to consider embeddings into lens spaces. We will
geometrically construct homeomorphisms from the unit tangent bundles of the
orbifolds into the lens spaces, elliminating the need for elliptic functions.
Finally we will use these homeomorphisms to compute templates for the geodesic
flows. This offers a tool for topologically investigating their otherwise well
studied periodic orbits.Comment: 29 pages, 21 figure
We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow on the modular surface, once compactifying the latter.
We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i) for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.
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