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The dynamics of generic Kuperberg flows. (English) Zbl 1380.37001

Astérisque 377. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-831-2/pbk). viii, 250 p. (2016).
The goal of this monograph is to study the dynamical properties of aperiodic Kuperberg flows on 3-manifolds. The authors introduce some new concepts and general hypotheses, use new techniques and prove that the only minimal set for such flows is an invariant zippered lamination. Topological and dynamical properties of the minimal set, including the presence of non-zero invariants of entropic type and chaotic behavior are investigated. It is established that the minimal set does not have a stable form, but it satisfies the Mittag-Leffler conditions for homological groups.
Section 1 includes the statement of this problems and reviews previous results.
In Sections 2, 3 and 4 the authors introduce and describe some new notions and generic hypotheses such as: modified Wilson Plug, the Kuperberg Plug, transition points and the radius function.
Various properties of the level function and how it connects the dynamical behavior of the Wilson flow with the much more complicated dynamical behavior of the Kuperberg flow is investigated in Sections 5 and 6. This relationship is one of the main research directions in this work.
In Section 7 the analysis of the trapped orbits is given. The basic result here is a general formulation of Kuperberg’s statement [G. Kuperberg and K. Kuperberg, Ann. Math. (2) 143, No. 3, 547–576 (1996; Zbl 0856.57026)].
The authors prove that the flow in the Kuperberg Plug is aperiodic in Section 8. Some results on the non-wandering set of the flow are obtained.
In Section 9 the authors define a pseudogroup \({G}_K\) acting on a rectangle which captures the dynamics of the flow. The study of the action of \({G}_K\) leads to a deeper understanding of the geometry and topology of the minimal set. It is shown that the global dynamics of the Kuperberg flow is determined by the action of the generators \({G}_K\) for the Kuperberg pseudogroup. Group analysis methods allow to give a new interpretation of the results and methods developed in the previous sections. The properties of the tree model for the dynamics of \({G}_K\) are discussed further in Sections 14 and 22 .
In Section 10 the study of the Kuperberg flow from a topological point of view is presented. A topological interpretation for many results of previous sections is given. The final task of this approach is given in Section 19, where the dynamics of the flow in terms of the structures of the zippered lamination is described.
One of the main new ideas of this work is introduced, described and illustrated in Sections 11, 12 and 13: the notions of finite, infinite and double propellers. The authors analyze the Kuperberg flow dynamics and the dynamics of the Kuperberg pseudogroup by using the concept of propellers.
In Section 14 the authors introduce an algebraic normal form for the elements of the pseudogroup and estimate the number of normal forms as a function of the word length. They show that this function has sub-exponential growth.
The aim of Section 15 is to study the properties of the bubbles that arise when the interior of a propeller intersects an insertion region, resulting in an internal notch, as discussed in Section 12. It is shown that the bubbles obtained from internal notches admit a uniform bound.
In Section 16 the authors explore the subtle dynamical properties of sets of wandering points. The proof of these results requires the introduction of a new class of (double) propellers.
In Section 17 the authors consider an additional regularity hypothesis on the intersection maps and use it to estimate the behavior of the Kuperberg flow near some special orbits.
The goal of Section 18 is to investigate the geometry of the collection of curves, formed by the intersection of the class of all finite orbits of wandering points and rectangles. For a generic Kuperberg flow the estimates on the geometry of these curves are obtained.
In Section 19 the authors introduce the notion of zippered laminations, which is a type of stratified laminations with a possible pathological behavior for the strata. After some preliminary notions the main theorem of this section is proved: if the flow is a generic Kuperberg flow then the set of wandering points is a zippered lamination.
The authors give a geometric proof of Katok’s theorem: the entropy for \(C^2\)-flows on compact 3-manifolds imply that the topological entropy of the Kuperberg flow is zero [A. Katok, Publ. Math., Inst. Hautes Étud. Sci. 51, 137–173 (1980; Zbl 0445.58015)] in Section 20. Here the key idea is to relate the flow entropy to another type of entropy invariant, which is derived from the action of the pseudogroup \({G}_K\) on the rectangle. Entropy of the flows introduces the entropy associated to a finite symmetric set of pseudogroup generators [E. Ghys et al., Acta Math. 160, No. 1–2, 105–142 (1988; Zbl 0666.57021)].
In Section 21 the zippered lamination structure is used to define invariants for the lamination itself. These invariants are entropy-like. Growth-type invariants for the leaves and entropy-like invariants for revealing the subtleties of the class generic Kuperberg flows are studied in Section 22.
In Section 23 the topological properties of the minimal set for the generic Kuperberg flow are considered. It is established that the minimal set does not have a stable form.
An added value of this work is the presence of a large number of diagrams that illustrate the theoretical material. They help understanding flows from different points of view and give a complete picture of the complexity of the flows dynamics.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37Bxx Topological dynamics
37Axx Ergodic theory
37Cxx Smooth dynamical systems: general theory
54H20 Topological dynamics (MSC2010)
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