Celebratio Mathematica

K. Kuperberg and C. Reed: “A rest point free dynamical system on \( \mathbf{R}^3 \) with uniformly bounded trajectories,” Fund. Math. 114 : 3 (1981), pp. 229–234. MR 644408 Zbl 0508.58036 article

In this paper, we show that if \( \epsilon > 0 \), then there exists a \( C^{\infty} \) transformation \( G \) from \( R^3 \) into \( R^3 \) such that the unique solution \( \Phi \) to the differential equation \( y^{\prime} = G(y) \) is a dynamical system (a continuous transformation from \( R\times R^3 \) into \( R^3 \) such that \begin{align*} & \Phi(0,p) = p,\\ & \Phi(t_1,\Phi(t_2,p)) = \Phi(t_1+t_2,p), \text{ and}\\ & \tfrac{\partial}{\partial t}\Phi(0,p) = G(p) \end{align*} with the following two properties:

for each point \( p \) in \( R^3 \) and each number \( t \), \( \Phi(t,p) \) is in the \( \epsilon \)-neighborhood for \( p \); and
for each integer \( n\neq 0 \), \( \Phi(n,p) \neq p \).

Notice that the Scottish Book problem number 110 of Ulam follows a corollary where \( f(p) = \Phi(1,p) \) and the manifold is \( R^3 \).

K. M. Kuperberg and C. S. Reed: “A dynamical system on \( \mathbf{R}^3 \) with uniformly bounded trajectories and no compact trajectories,” Proc. Am. Math. Soc. 106 : 4 (August 1989), pp. 1095–1097. MR 965244 Zbl 0676.58047 article

K. M. Kuperberg, W. Kuperberg, P. Minc, and C. S. Reed: “Examples related to Ulam’s fixed point problem,” Topol. Methods Nonlinear Anal. 1 : 1 (1993), pp. 173–181. MR 1215264 Zbl 0787.54041 article

Włodzimierz Kuperberg	Related
Piotr Minc	Related
Coke Stevenson Reed	Related

K. Kuperberg: “A smooth counterexample to the Seifert conjecture,” Ann. Math. (2) 140 : 3 (November 1994), pp. 723–732. MR 1307902 Zbl 0856.57024 article

G. Kuperberg and K. Kuperberg: “Generalized counterexamples to the Seifert conjecture,” Ann. Math. (2) 144 : 2 (September 1996), pp. 239–268. This is a corrected version of the article originally published in Ann. Math. 143:3 (1996). MR 1418899 Zbl 0856.57026 ArXiv math/9802040 article

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere \( S^3 \) has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on \( S^3 \) with no singular points has a periodic trajectory.

K. Kuperberg: “Counterexamples to the Seifert conjecture,” pp. 831–840 in Proceedings of the International Congress of Mathematicians, volume 2: Invited lectures (Berlin, 18–27 August 1998), published as Doc. Math. Extra Volume. Issue edited by G. Fischer and U. Rehmann. Deutsche Mathematiker-Vereinigung (Berlin), 1998. Dedicated to the author’s son Greg. MR 1648130 Zbl 0924.58086 incollection

Ulf Rehmann	Related
Gregory John Kuperberg	Related
Gerd Fischer	Related

K. Kuperberg: “Aperiodic dynamical systems,” Notices Am. Math. Soc. 46 : 9 (October 1999), pp. 1035–1040. MR 1710669 Zbl 0936.37039 article

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Krystyna Kuperberg

Kuperberg dreams

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