Celebratio Mathematica

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 $C^{\mathrm{\infty }}$-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.

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 ${\mathrm{\Phi }}_t$ 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.

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 $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$. As the min­im­al set $\mathrm{\Sigma }$ 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 $C^{\mathrm{\infty }}$-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 $\mathrm{\Sigma }$. Thus, the to­po­lo­gic­al shape of the min­im­al set $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ is taken in [e33]. Con­versely, the in­clu­sion $\mathrm{\Sigma }\subset \mathfrak{M}$ 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 $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }=\mathfrak{M}$, 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 $\mathfrak{M}$.

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 $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }=\mathfrak{M}$? Ghys gave a con­struc­tion of a flow on a plug and sketched the proof that $\mathrm{\Sigma }=\mathfrak{M}$ for this flow at the end of Sec­tion 7 in [e16]. The Ku­per­bergs gave an ex­pli­cit real ana­lyt­ic flow on a plug for which $\mathrm{\Sigma }=\mathfrak{M}$ 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:

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 $\mathrm{\Sigma }=\mathfrak{M}$ 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 $\mathrm{\Sigma }=\mathfrak{M}$ 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 $\mathrm{\Sigma }=\mathfrak{M}$ has proven dif­fi­cult.

If $\mathrm{\Sigma }\subset \mathfrak{M}$ is a prop­er in­clu­sion, then the to­po­lo­gic­al types of $\mathrm{\Sigma }$ 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:

The­or­em 19 in [5] gives the con­struc­tion of aperi­od­ic PL plugs for which $\mathrm{\Sigma }$ is 1-di­men­sion­al. In ad­di­tion, there is a dis­cus­sion of the sym­bol­ic dy­nam­ics for these spe­cial flows they con­struct. Con­sid­er­ing the to­po­lo­gic­al type of $\mathrm{\Sigma }$, 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 $\mathrm{\Sigma }$ 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]:

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 $\mathrm{HD}(\mathfrak{M})$ of $\mathfrak{M}$ as the least up­per bound of the ex­po­nents $0\le \alpha \le 1$ such that the $\alpha$-slow en­tropy of the flow is pos­it­ive [e20]. Thus, for a gen­er­ic flow we have $\mathrm{HD}(\mathfrak{M})\ge 1/2$. There are many ques­tions one can ask about these in­vari­ants, for ex­ample:

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].

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 $C^{\mathrm{\infty }}$-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 $C^{\mathrm{\infty }}$-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.

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 $C^{\mathrm{\infty }}$ 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:

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.