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1. Contact three-manifolds with exactly two simple Reeb orbits

   https://doi.org/10.2140/gt.2023.27.3801  

   Cristofaro-Gardiner, Daniel; Hryniewicz, Umberto; Hutchings, Michael; Liu, Hui (January 2023, Geometry & Topology)

   It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid. 

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2. Computing Reeb dynamics on four-dimensional convex polytopes

   https://doi.org/10.3934/jcd.2021016  

   Chaidez, Julian; Hutchings, Michael (January 2021, Journal of Computational Dynamics)

   null (Ed.)

   We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$ 1 $$\end{document}$. 

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3. Surface groups, infinite generating sets, and stable commutator length

   https://doi.org/10.1017/prm.2019.22  

   Margalit, Dan; Putman, Andrew (April 2019, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Abstract We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves. 

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4. Measure-Theoretic Reeb Graphs and Reeb Spaces

   https://doi.org/10.4230/LIPIcs.SoCG.2024.80  

   Wang, Qingsong; Ma, Guanqun; Sridharamurthy, Raghavendra; Wang, Bei (June 2024, 40th International Symposium on Computational Geometry (SoCG 2024), Leibniz International Proceedings in Informatics (LIPIcs))

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively. 

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5. An avoidance principle and Margulis functions for expanding translates of unipotent orbits

   https://doi.org/10.3934/jmd.2024011  

   Sanchez, Anthony; Seong, Juno (January 2024, Journal of Modern Dynamics)

   We prove an avoidance principle for expanding translates of unipotent orbits for some quotients of semisimple Lie groups. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proofs of our results rely on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups. 

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