Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows

We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d \lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro–Gardiner, Hutchings, Ramos, and inspired by the argument of Marques–Neves–Song, who proved a similar equidistribution result for minimal hypersurfaces.We also discuss a question about generic behavior of periodic Reeb orbits “representing” ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.

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Received 27 May 2019

Accepted 13 October 2020

Published 21 July 2021