It is natural to consider the space of continuous functions
on the real line C(R,R) as a dynamical system w.r.t. translation in
time. Not surprisingly this point of view is ubiquitous in
mathematics, notable examples are given by various spaces of almost
periodic functions studied by Besicovitch, Bochner, Weyl, von Neumann
and others. A celebrated result of Bebutov and Kakutani states that
C(R,R) is a universal embedding space for all topological flows whose
fixed point set embeds in the unit interval. In 1973 Eberlein showed
that the compact space of $1$-Lipschitz functions from the real line
to the unit interval, is a topological model for all free measurable
flows. This served as a fundamental step in his proof - together with
Denker - of the Jewett-Krieger theorem for flows (1974), demonstrating
the usefulness of the Lipschitz representation approach.
We generalize Eberlein’s theorem to multidimensional flows, in
particular giving a new proof for the one-dimensional case. This
necessitates the development of the multidimensional version of a
Lipschitz representation theorem by Gutman, Jin and Tsukamoto (2019).
The theory of topological local sections also plays a role in the
proof. Based on a joint work with Qiang Huo.
Meeting ID: 852 4277 3200
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