Abstract
This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology. We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.
Highlights
This paper is a sequel of [FH13] in which the authors described a surgery construction adapted to Reeb flows (“contact flows” to dynamicists; see Subsection 2.2)
We show that much of the complexity of the resulting flow is reflected in the cylindrical contact homology and is realized in any Reeb flow associated to the contact structure resulting from the surgery (Theorems 3.9, 3.13, and 3.14), and
The Bernoulli property and the Ornstein Isomorphism Theorem [Orn74] imply that the flows we obtain from our surgery are measure-theoretically isomorphic to the original contact Anosov flow up to a constant rescaling of time, the constant being the ratio of the Liouville entropies. (This answers a question of Vershik.)
Summary
The unstable foliation of the flow is contained in the positive cone Q+ and the stable foliation in the positive cone of Q−. If q > 0, the image of the first and third quadrant (i.e., the trace of C+) is a subcone of the first and third quadrant that shares the horizontal axis (see Figure 4.3) Speaking, this implies that the cone field C+ is preserved by the surgery and one can define a new cone field on the surgered manifold by. The return map sends it into the half-cone 0 a Kb, which is the other half of the cone in which we started This is incompatible with the existence of a continuous invariant cone field that extends to points that miss the surgery region, and with the Anosov property.
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