Examples related to Ulam's fixed point problem (Q1310502)

The paper deals with several examples related to the following problem posed by \textit{S. Ulam}: given a continuum (connected compact metric space) \(M\), does there exist a constant \(\varepsilon > 0\) such that any \(\varepsilon\)-map \(f: M\to M\) (i.e. continuous map whose trajectories \(\{f^ n(x)\}^ \infty_{n=0}\), \(x\in M\), are bounded by \(\varepsilon\)), possess a fixed point? An example of a one-dimensional continuum such that for any \(\varepsilon\) there is a fixed point free \(\varepsilon\)-involution is given. Then it is shown that the Cartesian product of the circle and the Hilbert cube for any \(\varepsilon > 0\) admits a fixed point free \(\varepsilon\)-homeomorphism. At last an example of the rest point free dynamical system with trajectories uniformly bounded by arbitrary \(\varepsilon > 0\) defined on a three-dimensional manifold is provided.