Orbit growth of contact structures after surgery
Annales Henri Lebesgue, Volume 4 (2021), pp. 1103-1141.

KeywordsAnosov flow, 3-manifold, contact structure, contact flow, Reeb flow, surgery, contact homology

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 3-manifolds 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.

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