#### by Steven Hurder and Ana Rechtman

#### 1. Birth of a method

In August 1993, Krystyna Kuperberg wrote out a
three-page sketch of a
construction of what she thought would be a __\( C^3 \)__-counterexample to the
Seifert conjecture. The idea for this construction had occurred to her
earlier in the month, while attending the Georgia Topology Conference
in Athens, Georgia. She faxed these notes to her son
Greg,
with the admonition
that he wait for a more final draft.
In the following days, she wrote out the precise construction,
which expanded to an
eight-page note and realized that it was a
__\( C^{\infty} \)__-counterexample.
Greg then sent an email to several leading topologists announcing the
exciting news, that his mother had found a smooth counterexample to the
Seifert conjecture!

The announcement of Krystyna’s work caused a sensation.
William Thurston
lectured on her construction at the MSRI, Berkeley in September, and observed
that the construction could be realized as a *real-analytic* aperiodic
flow. Krystyna mailed around 60 copies of the manuscript to mathematicians
around the world. Then in October, she gave a plenary lecture on her work
at the conference in honor of
Morris Hirsch
in Berkeley, California.
Elsewhere, a seminar was held in Tokyo in November, as part of the
International Symposium/Workshop: Geometric Study of Foliations. The
participants in this informal seminar included, among many others,
Shigenori Matsumoto,
Étienne Ghys,
Paul Schweitzer,
and the first author.

The manuscript by
Kuperberg was published almost immediately in the *Annals
of Mathematics*
[4].
Moreover, soon afterwards, Ghys
presented Kuperberg’s work in a Séminaire Bourbaki
[e16]
in June 1994. Matsumoto wrote a report in Japanese
[e17],
also in 1994, which provided more results about the dynamical properties
of these “Kuperberg flows”, with detailed proofs of their properties.
The joint paper by Greg and Krystyna
[5]
contained
many new ideas and variations on the basic construction. The brief note
[7]
gives an overview of Krystyna’s work on aperiodic
flows. Finally, Krystyna Kuperberg
[6]
gave a report
to the International Congress of Mathematicians in 1998.

Krystyna commented on her work, “Once you see the construction, you see it.” In addition, her original sketches of the construction, involving a thickened annulus with “rabbit ears”, were replaced in the published version by the now very familiar images drawn by her husband, Włodzimierz (Włodek), which conveyed a strong intuitive feel for the construction.

Krystyna Kuperberg was a student of Karol Borsuk in Warsaw, and many of her works correspondingly use a strong geometric approach to analyzing problems in dynamical systems, and this is more than true for her construction of the Seifert counterexamples.

The goals of this essay are to discuss the antecedents to the Kuperberg construction and give some inkling of the ideas that led Krystyna to her construction. In addition, we speculate on extensions of the construction and its future, especially in the context of other aspects of the theory of smooth dynamical systems. First, in Section 2 we recall the radical idea that made the Kuperberg construction possible, as it was a break from the past approaches to constructing counterexamples in a crucial manner. Section 3 gives a concise overview of known results about the Kuperberg flow. We then discuss in Section 4 questions about alternate flow dynamics for nongeneric constructions of Kuperberg flows. Section 5 discusses related results for flows that preserve some additional geometric structure.

#### 2. Shape of plugs

In his 1950 work
[e2],
Seifert
introduced an invariant
for deformations of nonsingular
flows with a closed orbit on a 3-manifold, which he used to show that
every sufficiently small deformation of the Hopf flow on the 3-sphere
__\( \mathbb{S}^3 \)__ must have a closed orbit. He also remarked in Section 5
of this work:

It is unknown if every continuous (nonsingular) vector field on the 3-dimensional sphere contains a closed integral curve.

This remark became the basis for what is known as the “Seifert conjecture”:

Every nonsingular vector field on the 3-sphere\( \mathbb{S}^3 \)has a periodic orbit.

There must exist at least one minimal set for a flow on a compact manifold — a closed invariant subset which is minimal for this property. A periodic
orbit is a minimal set, but if there are no periodic orbits for a flow,
then a minimal set for the flow will have a possibly far more complicated
topology. So one can ask if there are restrictions on the *shape*
of a minimal set for an aperiodic flow.
Karol Borsuk
defined the shape of a topological space as an inverse
limit continuum defined by a sequence of ANR approximations of the space
[e4],
[e6],
[e13],
[e22].
For an aperiodic flow on a 3-manifold, it seems that the minimal set must
have a complicated shape, though just how complicated is an open question.

Kuperberg’s strategy for the construction of an aperiodic flow was not based on the choice of an invariant minimal set with prescribed shape, but was instead to create a dynamical framework which avoided the implications of the Brouwer fixed-point theorem. We explain this remark in the following discussions.

The first general approach to the Seifert conjecture was made in the 1966
paper by
Wesley Wilson
[e3],
where he showed that every closed
3-manifold __\( M \)__ has a flow with only finitely many closed orbits. Wilson
introduced the use of a special kind of “plug” to modify a given flow,
and this idea profoundly influenced the thinking about the constructions
of flows with special properties, and the Seifert conjecture in particular.

A *Wilson plug* for a flow on the 3-manifold __\( M \)__ is a neighborhood
__\( P \subset M \)__ diffeomorphic to an *annulus*
cross an interval,
__\( P \cong \mathbb{A}^2 \times [-1,1] \)__, where orbits enter one end of the
plug, __\( \partial^- P = \mathbb{A}^2 \times \{-1\} \)__, and exit the other end,
__\( \partial^+P = \mathbb{A}^2 \times \{+1\} \)__. The key idea introduced by
Wilson was that the plug flow should be symmetric: an orbit which traverses
the plug should enter and exit at symmetric points on the faces. No such
restriction is placed on an orbit which enters the plug and never exits,
of course, and one assumes there are such orbits. Wilson introduced the
technique of using a plug reflected on itself to achieve this symmetry
property. As a result, the plugs he uses have two periodic orbits, as
illustrated in Figure 1.

A plug can be inserted into any nonsingular flow on a manifold, not just once but any finite number of times as needed to modify a given nonsingular flow. Wilson showed that any flow can be modified using finitely many plugs to obtain a flow with only isolated periodic orbits. It is an observation in [e9] that by appropriately arranging the insertion of Wilson plugs, one can obtain a flow with exactly two periodic orbits.

Subsequently, in the early 1970’s, Paul Schweitzer had the inspired idea to construct a modified Wilson-type plug, but instead of the top and bottom boundaries of the plug being annuli, they were diffeomorphic to a 2-torus with an open disk removed, as illustrated in Figure 2. The paradigm shift of a Schweitzer plug is that the faces of the plug are immersed in the transverse space, but still transverse to the flow. Using a plug with this shape, the second key point was to double the construction using a mirror symmetry as with the Wilson plug, so that the flow in the plug has two Denjoy-type minimal sets, instead of two circles.

A Denjoy minimal set, as introduced by
Denjoy
in
[e1],
has the
shape of a wedge of two circles. The Schweitzer plug has the celebrated
“double doughnut” shape as seen in Figure 2.
This plug can then be inserted into flows on 3-manifolds to create new flows
which have all minimal sets of Denjoy type. In particular, using this plug
one can construct flows with no periodic orbits. Thus, Schweitzer showed
that every 3-manifold carries a __\( C^1 \)__-flow with no periodic orbits! This
celebrated result appeared in the *Annals of Mathematics*
[e5]
and in his talk
[e7]
to the
International Congress of Mathematicians in 1974.

Mike Handel
showed in his 1980 paper in the *Annals of Mathematics*
[e10]
a result which severely limits the shape of a minimal
set for a smooth aperiodic flow — it cannot be embedded as the minimal
set for a flow on a surface. Thus, a flow with a Denjoy-type minimal
set carried by a smooth surface cannot be __\( C^2 \)__. This was an attempt to prove that
plugs were not the method to find smooth aperiodic flows on __\( \mathbb{S}^3 \)__.

Jenny Harrison
showed in
[e14]
that the Schweitzer
plug could be modified to obtain an aperiodic plug with a
__\( C^{2+\delta} \)__-regular flow by embedding the Denjoy minimal sets in
a 3-dimensional fashion. However, this result required delicate
arguments, and obtaining any further improvement in the differentiability
of the flow using this method seemed completely out of reach.
The Handel and Harrison works suggested that any counterexample to the
Seifert conjecture was going to require a radically new approach.

Meanwhile, in 1979, the Kuperbergs were attending the Scottish Book Conference in Denton, Texas, organized by Dan Mauldin [e31]. At this conference, Stanislav Ulam told Krystyna and Włodek about an unsolved problem, posed by Ulam:

[e31], Problem 110:

Let\( M \)be a manifold. Does there exist a numerical constant\( K \)such that every continuous mapping\( f : M \to M \)which satisfies the condition\( |f^n(x) -x| < K \)for\( n=1,2,\dots \)must possess a fixed point\( f^n(x_0) = x_0 \)?

The mathematician/inventor Coke Reed was a colleague of Ulam at Los Alamos and later a colleague of Krystyna at Auburn University. Kuperberg and Reed discussed this problem and subsequently solved it. They gave counterexamples to the assertion in Problem 110, as published in the 1981 work [1]. The methods they developed are analogous to the methods introduced by Wilson in [e3]. Kuperberg and Reed improved their result in the 1989 paper [2] via an application of the Schweitzer plug. An analysis of the solutions to the Ulam problem 110 were recalled and analyzed in the paper [3] by K. Kuperberg, W. Kuperberg, P. Minc and C. Reed. There is a connection between the solutions of the Ulam problem and the construction of counterexamples to the Seifert conjecture in that the plug technique can be applied to construct counterexamples to both, as remarked in the joint work of Greg and Krystyna [5], Theorem 8.

Following the successful solution to the Ulam problem, Kuperberg began to consider what was required to construct a smooth plug without periodic points for the flow. As she remarked (personal communication, November 15, 2021), the problem was always the Brouwer fixed-point theorem: any map of a (transverse) disk to itself must have a fixed point, and so a corresponding suspension flow will have a periodic orbit. Thus, to build a flow in a plug without periodic orbits, the transverse holonomy for the flow must be a translation on an infinite line or region.

The inspired geometric observation is that such a translation is already available in the Wilson plug, as the flow on the Reeb cylinder, as pictured in Figure 3.

The flow of a point below the orbit __\( \mathcal{O}_1 \)__ will climb up to
the periodic orbit __\( \mathcal{O}_1 \)__ and the return map of this flow is
a translation on a vertical interval in the cylinder transverse to the
flow. A similar dynamical behavior happens for points in the cylinder above
the periodic orbit __\( \mathcal{O}_2 \)__, but in reverse time.

With this point of view, one can then ask how to get the flow of the periodic
orbit to “break open” and “start climbing the holonomy staircase”. That
is, break open the periodic orbit using its own holonomy!
The solution is the second inspired idea behind Kuperberg’s construction,
to insert the flow on the Wilson cylinder into itself, so that the orbit
__\( \mathcal{O}_1 \)__ is inserted into one of the flow lines climbing up to
__\( \mathcal{O}_1 \)__. This is precisely what the insertion maps illustrated in
Figure 5
accomplish. To make this work and keep the
plug properties, one must first do a double twist
of the embedding of
the Reeb cylinder, as illustrated in Figure 4.
(See also the illustration on the left side of Figure 8.)
According to Krystyna, the idea for
this construction occurred to
her during a beer party at the Georgia Topology Conference.

The essence of the novel strategy behind the aperiodic property of the
flow __\( \Phi_t \)__ on a Kuperberg plug is perhaps best described by a quote
from the paper by
Matsumoto
[e17]:

We therefore must demolish the two closed orbits in the Wilson plug beforehand. But producing a new plug will take us back to the starting line. The idea of Kuperberg is to

let closed orbits demolish themselves. We set up a trap within enemy lines and watch them settle their dispute while we take no active part.

The reader is exhorted to read the proof that these flows are aperiodic in [4], or in the reports by Ghys [e16] or Matsumoto [e17], as the elegance of Kuperberg’s idea is revealed in the simplicity of this proof.

Note that there are many choices of the vector field that can be made for the flow in a Wilson plug. What is essential for Kuperberg’s construction is that there are the two closed orbits which attract/repel a set of orbits entering or leaving a face of the plug.

A Kuperberg plug can also be constructed for which its
flow __\( \mathcal{K} \)__ is real-analytic. An explicit construction of such a flow
is given in Section 6 of the paper
[5]
by Greg and
Krystyna Kuperberg. There is the added
difficulty that the insertion of the plug in an analytic manifold must
also be analytic, which requires some subtlety. Analytic aperiodic flows
are discussed in the second author’s Ph.D. thesis
[e27].

The construction of smooth aperiodic flows on compact manifolds using the
Kuperberg method produces orbits that never close up, so they wander
around a compact region of a plug that has been inserted. There is much
that can be said about the dynamical properties of these flows, deduced
from the way the flows in the plugs are constructed.
These are the *known knowns* about Kuperberg flows, as discussed in
Section 3.

There are also many *known unknowns* about Kuperberg flows, which
generate interesting open questions about the dynamics of the Kuperberg
flows, as discussed in Section 4. There are also the
*unknown unknowns*, which are speculations of dynamical phenomena
yet to be discovered for this class of smooth flows.

#### 3. Aperiodic flow dynamics

Ghys observed in [e16] that a Kuperberg flow has zero topological entropy. This follows from the remark that for a smooth flow on a compact 3-manifold, Katok’s results in [e11] imply that if a smooth flow has positive entropy, then it must have periodic orbits. Thus, aperiodic flows belong to a class of smooth dynamical systems which might be considered “less than chaotic”, or more precisely “at worst, slowly chaotic”.

The flow in the Wilson plug in Figure 1 has two periodic
orbits, labeled __\( \mathcal{O}_1 \)__ and __\( \mathcal{O}_2 \)__. Both of these orbits
are broken open by the insertion surgery illustrated in
Figure 5, to yield what are called the *special
orbits* for the flow. Not as obvious without a closer inspection is that
each of the special orbits is trapped inside of the Kuperberg plug formed
by the surgery, and they limit on each other. Thus, their closures form a
minimal set for the flow in the plug. Furthermore, every orbit entering
the plug either escapes from the plug on the opposite face, or else it
limits on the closure of a special orbit. Thus, there exists a unique
minimal set for the flow in the plug. If the aperiodic flow is constructed
using multiple Kuperberg plugs, then there may be multiple minimal sets,
each disjoint from the other. For simplicity of exposition, we assume
there is only one plug, and let __\( \Sigma \)__ denote the unique minimal set.

Another highly nonobvious result due to
Matsumoto
[e17],
Proposition 7.2,
(see also the discussion in
[e16],
Section 8)
is that there is an open set of points in the
entrance of a Kuperberg plug whose forward orbits limit on the minimal set
__\( \Sigma \)__. As the minimal set __\( \Sigma \)__ is contained in the interior of the
plug, this open set consists of nonrecurrent points for the flow. In
particular, this implies that there is no invariant probability measure
in the Lebesgue measure class for the flow in the Kuperberg plug. It
is remarked after the proof of
[5],
Proposition 18,
that the Matsumoto argument works for __\( C^1 \)__-flows and some but not all
piecewise-linear self-insertions
[5],
Section 9.

Remarkably,
Greg Kuperberg
showed in Theorem 1 of
[e18]
that on every closed 3-manifold there is a volume-preserving __\( C^1 \)__-vector
field with no closed orbits, obtained by modifying the Schweitzer plug.
He also proved in Theorem 2 of
[e18]
that for every closed
3-manifold, there is a volume-preserving __\( C^\infty \)__-flow with a finite
number of periodic orbits. Also in the volume-preserving setting,
Viktor L. Ginzburg
and
Bașak Gürel
[e24]
constructed a
version of Schweitzer’s plug that
allows one to produce examples
of __\( C^2 \)__-functions
on symplectic 4-manifolds having a level set whose Hamiltonian vector
field has no periodic orbits. A Hamiltonian vector field always preserves
volume and in this case is __\( C^1 \)__; hence it gives examples of aperiodic
volume-preserving flows.

The final general remark about the geometry of the Kuperberg flows is more
complicated to explain but has profound consequences for their dynamics.
The Reeb cylinder is an annular region bounded by the two periodic
orbits for the Wilson flow, as pictured on the right side of
Figure 1.
The process of doing flow surgery as illustrated in Figure 5
cuts two “notches” out of the cylinder. This is the region labeled
by __\( \mathcal{R}^{\prime} \)__ on the left side of Figure 6,
called the “notched Reeb cylinder” as pictured on the right side of
Figure 1, before the cylinder is deformed to appear as in
Figure 4, which is then twisted as in
Figure 5.

The Kuperberg flow in the Wilson cylinder preserves the region __\( \mathcal{R}^{\prime} \)__,
called the *notched Reeb cylinder*, until it reaches a notch,
then it continues to flow but the orbits are
no longer in the cylinder,
as in Figure 5. This produces a tongue-shaped region as
illustrated on the right side of Figure 6. This tongue-shaped
region continues to flow until it intersects the insertion regions created
by the surgery, and the pattern repeats ad infinitum, attaching tongues to
the tongues.

The complete flow of the notched Reeb cylinder
is a surface with
boundary embedded in the Kuperberg plug. Figure 7
gives three illustrations of this central object for Kuperberg
flows, which
Siebenmann
called a “chou-fleur” in his communication
[e19].
This was illustrated by
Ghys
as a “fractal-like
cluster” in
[e16],
as pictured
in the lower left side of Figure 7. In the monograph
[e33],
the authors called the surface as laid out on the plane, a “propeller” , which
is approximately illustrated by the thickened 2-dimensional, tree-like
structure on the right side of Figure 7, but again not
to scale (all the branches have approximately the same width). We call
this infinite surface __\( \mathfrak{M}_0 \)__.

The basic observation is that the boundary of __\( \mathfrak{M}_0 \)__ is the union
of the special orbits for the flow! Thus, to understand the dynamics of
the flow, one analyzes the structure of the propellers in the Kuperberg
plug, which are defined by a recursion process that is generated by the
insertion surgery. The recursion process which defines the branches
of the propeller depends in a sensitive manner on the precise surgery
process used to construct the plug. This makes the analysis very delicate,
and more will be said about that later.

The Kuperberg flow preserves the embedded infinite surface
__\( \mathfrak{M}_0 \)__ whose boundary contains the special orbits. The
fact that the flow is aperiodic corresponds to the fact that the propellers
are infinite.

The closure __\( \mathfrak{M} = \overline{\mathfrak{M}_0} \)__ is a type of
2-dimensional *lamination with boundary* that contains the minimal
set __\( \Sigma \)__. Thus, the topological shape of the minimal set __\( \Sigma \)__
is dominated by the shape of the lamination __\( \mathfrak{M} \)__, which can
be estimated using the recursion on the reinsertions of the notched Reeb
cylinder for the Kuperberg plug. This approach to the study of the shape
of __\( \Sigma \)__ is taken in
[e33].
Conversely, the inclusion __\( \Sigma
\subset \mathfrak{M} \)__ suggests possible relations between the shape of
__\( \mathfrak{M} \)__ and the dynamical properties of the flow, as discussed in
Section 4.

We describe the technical setting for comparing __\( \Sigma \)__ and __\( \mathfrak{M} \)__.
The drawing on the left side of Figure 8 gives an
expanded view of the insertions illustrated in Figure 5.
A key point is that the image of a boundary orbit, denoted by
__\( \tau(\mathcal{O}_1) \)__ in the illustration, has image under the insertion
as a twisted curve that is tangent to the cylinder at the boundary curve
__\( \mathcal{O}_1 \)__, but intersects transversally __\( \tau(\mathcal{O}_1) \)__. This
results in a picture as on the right side of Figure 8, where
the two thickened curves are the boundaries of the cylinder and its insertion.

The graph on the right side of Figure 8 is called the
*radius function* (actually, what is illustrated is the inverse
of this function) and the quadratic shape of the graph is the
basis for showing that the Kuperberg flow is aperiodic. Understanding
these remarks takes some effort and is explained in the papers
[4],
[e16],
[e17],
[e33],
but reveals the great
beauty of the Kuperberg construction.

The authors introduced the notion of a *generic* Kuperberg flow in the
work
[e33],
which formulates a collection of optimal conditions
on the choices for the construction of the flow. One of these conditions
is that the radius function for the insertion is actually
quadratic,
as pictured on the right side of Figure 8.
We
showed in
[e33]
that for a generic Kuperberg flow, there is
equality __\( \Sigma = \mathfrak{M} \)__, and thus one can study the shape of the
minimal set for the flow by studying the properties of the 2-dimensional
lamination with boundary __\( \mathfrak{M} \)__.

#### 4. Nongeneric flows

Ghys wrote in his 1995 survey [e16], page 302:

Par ailleurs, on peut construire beaucoup de pièges de Kuperberg et il n’est pas clair qu’ils aient la même dynamique.

In this section, we discuss some ways in which the dynamical properties
of an aperiodic smooth Kuperberg plug vary with the choices made
in its construction. The properties considered include the *shape*
and the *Hausdorff dimension* of the minimal set __\( \Sigma \)__ and the
*slow entropy* of the flow.

The construction of a Kuperberg flow begins with the choice of a flow
__\( \mathcal{W} \)__ on a cylinder, with an invariant cylinder bounded by the two
periodic orbits that is called the *Reeb cylinder*, as illustrated
on the left side of Figure 1.
This flow is then extended smoothly to a thickened cylinder, or
an interval times an annulus, as on the right side of Figure 1. There
are further symmetry conditions imposed on the flow __\( \mathcal{W} \)__, as given in
[4],
Section 3,
or for example as specified by (P1)
to (P4) in
[e33],
Section 2.
The symmetry conditions still allow
for a wide variation of choices for the flow __\( \mathcal{W} \)__.

The flow of the Kuperberg plug that results from the insertion process
consists of orbit segments of the gradient-like flow __\( \mathcal{W} \)__ which
are “patched together”. This view of the orbits, as a union of finite
flows, is key to the analysis of the Kuperberg flow in the works of
Ghys
[e16]
and
Matsumoto
[e17].
For a different take on this,
Section 5 of the paper
[5]
describes the result of the
self-insertion patching, as the analog of a
stacking subroutine for
a computer program. Then the fact that no orbit is periodic corresponds
to showing that the routine does not terminate for the given starting
point. So in essence, the solution to the Seifert conjecture is realized
by building a dynamical computer program that is nonterminating.

This patching of flow segments
occurs orbitwise, so if there is
to be any hope of organizing this process in a unified description for
the entire flow on the plug, one needs to impose assumptions on the flow
__\( \mathcal{W} \)__ that its orbits are as uniform as possible.
Such additional conditions on __\( \mathcal{W} \)__ are formulated in the works
[4],
[5],
and even more restrictive conditions are imposed in
[e33].

The second step in the construction of a Kuperberg flow is to make a choice for each of the two insertion maps, as pictured in Figures 5 and 8. In this step, the embedding of the Reeb cylinder makes a transverse contact at the periodic orbit, either the bottom or the top orbit, so as to break open these closed orbits. The result is that the special orbits which result by breaking open the closed orbits intersect the vertex of the parabola pictured on the right side of Figure 8. They return infinitely often to a neighborhood of the vertex of the parabola as a result of the translation holonomy of the flow. Hence, the germinal shape at its vertex of the parabolic map has a strong influence on the dynamics of the special orbit, and so on the dynamics of the entire flow.

A basic question is, when does the minimal set satisfy __\( \Sigma =
\mathfrak{M} \)__? Ghys gave a construction of a flow on a plug and sketched
the proof that __\( \Sigma = \mathfrak{M} \)__ for this flow at the end of
Section 7 in
[e16].
The Kuperbergs gave an explicit real analytic
flow on a plug for which __\( \Sigma = \mathfrak{M} \)__ in their joint paper
[5].
In both cases, the idea of the proof is to show
that the closure of a special orbit contains the notched Reeb cylinder
__\( \mathcal{R}^{\prime} \)__,
and hence contains the flow of __\( \mathfrak{M}_0 \)__ and thus equals
its closure __\( \mathfrak{M} \)__.

The authors began our investigations of the shape of the minimal set in the Kuperberg flows in 2010, and for several years after that every time we met Krystyna we had to admit we couldn’t answer one of the mysteries of the Kuperberg construction:

*always*2-dimensional for a smooth Kuperberg flow?

This is the conclusion for the examples by Ghys in
[e16]
and in the joint paper
[5],
but the situation in general was not known.
In order to answer Krystyna’s question, we introduced the notion of a
“generic Kuperberg flow” in
[e33],
which assumes that the flow
satisfies the conditions of
Hypotheses 12.2 and 17.2 formulated in that work.
Hypothesis 12.2 assumes in particular that the vertical component of
the vector field __\( \mathcal{W} \)__ on the Reeb cylinder has quadratic germ
near the two periodic orbits at which its “vertical component” must
vanish. This condition is used to show that the closure of the special
orbits contain __\( \mathcal{R}^{\prime} \)__. But it is unknown, for example,
if one can still
show that __\( \mathcal{R}^{\prime} \)__ is contained in the closure of a special orbit
if the vector field vanishes to higher order at the periodic orbits?

The equality __\( \Sigma = \mathfrak{M} \)__ can be considered as a type of
*Denjoy theorem* for 2-dimensional laminations. The usual
Denjoy theorem states that the closure of an orbit for a __\( C^2 \)__-flow
without closed orbits on the 2-torus is all of __\( \mathbb{T}^2 \)__. Then the
equality __\( \Sigma = \mathfrak{M} \)__ can be considered as analogous, for it
states that the closure of an orbit is not a 1-dimensional submanifold of
__\( \mathfrak{M} \)__, but fills up its 2-dimensional leaves. The known proofs of
this conclusion cited above
all make assumptions on both the flow
__\( \mathcal{W} \)__ and on the insertion map. Formulating general criteria which
suffice to imply the equality __\( \Sigma = \mathfrak{M} \)__ has proven difficult.

If __\( \Sigma \subset \mathfrak{M} \)__ is a proper inclusion, then the
topological
types of __\( \Sigma \)__ and __\( \mathfrak{M} \)__ will differ; for example
they could have distinct shapes. A __\( C^1 \)__-flow on __\( \mathbb{T}^2 \)__ without
closed orbits provides a good model for this difference. If the minimal set
__\( \mathfrak{D} \)__ for the flow is not all of __\( \mathbb{T}^2 \)__ then __\( \mathfrak{D} \)__
is a 1-dimensional continuum, as used in the construction of the Schweitzer
plug. If __\( p \in \mathbb{T}^2 - \mathfrak{D} \)__ then there is a retract
of the open punctured torus __\( \mathbb{T}^2 - \{p\} \)__ onto __\( \mathfrak{D} \)__,
so that __\( \mathfrak{D} \)__ has the shape of a wedge of two circles. It seems
a very interesting problem to compare the shapes of the closed invariant
sets for a general aperiodic flow:

__\( \Sigma \subset \mathfrak{M} \)__which is a proper inclusion. What is the relation between the shapes of

__\( \Sigma \)__and

__\( \mathfrak{M} \)__?

Theorem 19 in
[5]
gives the construction of aperiodic PL
plugs for which __\( \Sigma \)__ is 1-dimensional. In addition, there is a discussion
of the symbolic dynamics for these special flows they construct.
Considering the topological type of __\( \Sigma \)__, in addition to the
known unknowns about its shape, one senses that its study leads into
the realm of the unknown unknowns, that new dynamical phenomena will be
discovered.

We next consider another dynamical aspect of Kuperberg flows, their entropy and the Hausdorff dimension of their minimal sets. While Ghys observed in [e16] that an aperiodic flow must have entropy equal to zero using a well-known deep result of Katok [e11], the dynamical behavior of the flow appears to be chaotic upon closer inspection.

The geometry of the propeller as pictured in Figure 7
suggests that the flow in the minimal set __\( \Sigma \)__ should exhibit
exponential behavior, as it follows the boundary of the propeller which
appears to have exponentially growing area. This intuition is flawed,
though, as the flow cannot simply travel down the core of the propeller,
but must instead travel on the boundary to get to the extremities of the
tree. The time required to get to the extremes of the tree is not linear,
but grows approximately as the square of the distance to be traveled. This
observation is the basis for Theorem 21.10 in
[e33]:

__\( \alpha = 1/2 \)__.

The notion of slow entropy was introduced by Katok and
Thouvenot
[e21].
The exponent __\( \alpha=1/2 \)__ is essentially saying
that the entropy would be positive if the time variable is sped up by
taking the square of time. We also note that for this result, the generic
hypotheses include an extra condition, the assumption that the insertion
maps for the construction of the plug have “slow growth” themselves. This
is a technical property, but its requirement emphasizes that the proof
of Theorem 4.3 is quite technically involved. In particular,
the proof uses an estimate on the growth rates of leaves in the lamination
__\( \mathfrak{M} \)__, that they grow at
the subexponential rate __\( 1/2 \)__ as well.
Define the growth rate of __\( \mathfrak{M} \)__ as the exponential growth rate of the
dense leaf __\( \mathfrak{M}_0 \)__, so for a generic flow this rate is __\( 1/2 \)__. Also,
define the entropy dimension __\( \operatorname{HD}(\mathfrak{M}) \)__ of __\( \mathfrak{M} \)__ as the
least upper bound of the exponents __\( 0 \leq \alpha \leq 1 \)__ such that the
__\( \alpha \)__-slow entropy of the flow is positive
[e20].
Thus, for a generic flow we have __\( \operatorname{HD}(\mathfrak{M}) \geq 1/2 \)__.
There are many questions one can ask about these invariants, for example:

__\( \operatorname{HD}(\mathfrak{M}) \)__can be realized by aperiodic flows on 3-manifolds? Is there an aperiodic flow with

__\( \operatorname{HD}(\mathfrak{M}) = 0 \)__?

For a Kuperberg flow, one can also ask about the Hausdorff dimension of
the lamination __\( \mathfrak{M} \)__.
This problem seems almost intractable, but
Daniel Ingebretson
developed an
approach to calculating this dimension for a particular class of flows in
his thesis work
[e35],
[e34].

__[e35]__The Hausdorff dimension of the minimal set for a generic Kuperberg flow satisfies

__\( 2 < d_h(\mathfrak{M}) < 3 \)__.

The dimension of __\( \mathfrak{M} \)__ must be at least 2, as __\( \mathfrak{M} \)__
is a union of 2-dimensional “leaves” by the generic assumption. But the
fact that the dimension is greater than 2 is a measure of the transverse
complexity of __\( \mathfrak{M} \)__. The actual Hausdorff dimension appears to
depend in a sensitive manner on the choices made. The thesis work also
gives a method to numerically calculate this number.
We can then formulate a question which falls into the category of an unknown
unknown about Kuperberg flows:

We should also mention a question that belongs to the class of unknown-unknown problems.

Another remarkable fact about the Kuperberg flows, is that they lie at the
“boundary of chaos” in the __\( C^{\infty} \)__-topology on flows. The idea behind
this remark is that when making the insertion as in Figure 8,
one can stop the insertion “too soon”. That is, if the two cylinders do
not make contact, then the insertion does not break open the periodic orbits
for the Wilson flow __\( \mathcal{W} \)__. We
showed in
[e37]
that the flow
for such a truncated insertion again has two periodic orbits. However,
the lengths of these orbits tends to infinity
as the construction limits to a Kuperberg insertion.
Thus, we obtain smooth families of variations of the
Kuperberg plug with simple dynamics, and exactly two periodic orbits,
and the limit of the family is an aperiodic flow.

In such a family, the period of the periodic orbits must blow up at the limit. Palis and Pugh in [e8] asked whether this dynamical phenomenon can occur in families of smooth flows on closed manifolds, and called a closed orbit whose length “blows up” to infinity under deformation a “blue sky catastrophe”. The first examples of a family of flows with this property was found by Medvedev in [e12]. The constructions in [e37] show that deformations of Kuperberg flows provide a new class of examples. The work of A. Shilnikov, L. Shilnikov and D. Turaev in [e25] further discusses blue sky catastrophe phenomenon.

On the other hand, when making the insertion as in Figure 8,
one can take the insertion “too far”. That is, the two cylinders intersect
not along one arc *but along two arcs*. We
showed in
[e37]
that the flow for these over-extended insertions
has countably many
embedded horseshoes, so an abundance of hyperbolic behavior. Moreover,
as the insertion maps are deformed to one that just makes contact at a
single point, then these horseshoes increase in number. The dynamics of
the generic Kuperberg flow is thus the limit of the dynamical chaos in
a continuous family of horseshoes for neighboring smooth flows.

The immediate conclusion is that the Kuperberg flows are not stable in
the __\( C^{\infty} \)__-topology on flows.

The recitation of remarkable properties of the class of flows created by Kuperberg could continue, as they are zero-entropy dynamical systems, which are simply not boring [e36]! Instead, we want to also discuss a variety of other constructions and results following from Seifert’s conjecture and the questions they engender.

#### 5. Decorated flows

As explained by Ghys [e16], the construction of nonsingular aperiodic flows is particularly complicated in dimension 3. The Seifert conjecture, which is false in full generality, can be decorated with invariant structures: we can ask, if the flow preserves something, then must it have a periodic orbit? On one hand, there are families of nonsingular flows that always have periodic orbits; on the other hand we have the examples discussed previously in this text. Here we discuss this broader subject of flows on closed 3-manifolds, and we review some results on the existence of periodic orbits. We do not intend to make an exhaustive list.

Let us start with the class of Reeb vector fields associated to a contact
form. It has been shown by
Hofer
in
[e15],
that the flow
of a Reeb vector field on __\( \mathbb{S}^3 \)__ must have a periodic orbit,
and this was generalized to every closed 3-manifold by
Taubes
in
[e26].
But we know more; in fact every Reeb vector field
on a closed 3-manifold has at least two periodic orbits as proved by
Cristofaro-Gardiner
and
Hutchings
[e32]
and the examples with
exactly two periodic orbits are completely understood
[e39].
The second author, together with
Colin
and
Dehornoy,
proved that if a
Reeb field is nondegenerate then it has either two or infinitely many
periodic orbits
[e38].

A Reeb vector field always preserves a volume form. A big open question in
the subject is whether a __\( C^\infty \)__ volume-preserving flow on a 3-manifold
has a periodic orbit. It seems impossible to mimic Krystyna’s construction
of a plug in this setting, but we can do a Wilson plug for volume-preserving
flows. The lack of examples of possible minimal sets makes it also impossible
to mimic other plug constructions.

In the case of Reeb flows, Alves and Pirnapasov [e40] proved recently the existence of some knots such that if a Reeb flow has a periodic orbit realizing the knot then it has positive entropy. In particular, by the already-mentioned theorem of Katok [e11], such Reeb flows have lots of periodic orbits. It is an observation that such a result cannot hold for categories of flows for which we can construct plugs. Using a suitable Wilson plug one can prove the following:

__\( K \)__be a given knot. Every 3-manifold that admits a zero-entropy flow without fixed points admits a zero-entropy flow without fixed points having (at least) two periodic orbits whose knot type is

__\( K \)__. Moreover, if the original flow is generated by a Reeb vector field, then the new flow preserves a plane field that fails to be a contact structure along the two periodic orbits of knot type

__\( K \)__.

The existence of periodic orbits was extended to geodesible volume-preserving flows (also known as Reeb vector fields of stable Hamiltonian structures) on manifolds that are not torus bundles over the circle by Hutchings and Taubes [e28], [e29] and Rechtman [e30], and for real-analytic geodesible flows by Rechtman [e30]. The existence of a periodic orbit was also established for real-analytic solutions of the Euler equation by Etnyre and Ghrist [e23]. Geodesible volume-preserving flows are solutions to the Euler equation, but we do not know if the last ones have periodic orbits. We do know that it is impossible to construct plugs whose flow satisfies the Euler equation [e41].

*Steven Hurder received his PhD in mathematics from the
University of Illinois in 1980. He was a Member of the Institute for
Advanced Studies, Princeton from 1980–81, an Instructor at Princeton
University from 1981–83, and a faculty member of the University of
Illinois at Chicago from 1983–2012, Professor Emeritus since
2012. He was an Alfred P. Sloan Foundation fellow from 1985–1987,
and an inaugural Fellow of the American Mathematical Society in
2012. His research interests are concentrated on the interactions of
dynamics, topology and geometry.*

*Ana Rechtman is a Mexican mathematician working at the
Strasbourg University. She did her PhD in Mathematics at the École
Normale Supérieure in de Lyon under the supervision of Étienne Ghys.
She was Boas Assistant Professor at Northwestern University (2010–11).
She was a faculty member at National University of Mexico from
2016–18, from 2011–16 and since 2018 she has been at Strasbourg
University. Her research is on the dynamics and the topology of
flows.*