Celebratio Mathematica — Kuperberg

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Works connected to Coke Stevenson Reed

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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