According to the classical Mengerâ€“Nöbeling (1932) theorem, a
compact metric space $X$ of (Lebesgue covering) dimension less than $r/2$
admits a topological embedding into $r$-dimensional Euclidean space.
Generalizing this to the dynamical setting we prove that whenever an
(arbitrary) group $G$ acts on a finite-dimensional compact metric
space $X$, there exists an equivariant topological embedding of $X$
into $([0,1]^r)^G$, provided that for every positive integer $N$, the
dimension of the space of points in $X$ with orbit size at most $N$
is strictly less than $Nr/2$. Note that the equivariant topological
embedding is
necessarily of the form $x\mapsto (f(gx))_{g\in G}$ for some
continuous map $f:X\rightarrow [0,1]^r$.
Going further we derive a topological Takens theorem for finitely
generated group action, that is under the assumptions above when the group $G$
is finitely generated one may find a continuous map $f:X\rightarrow
[0,1]^r$ so that
$x\mapsto (f(gx))_{g\in G'}$ is injective for some finite $G'\subset G$
where the cardinality of $G'$ is bounded by a function of $r$, the
dimension of $X$ and the number of generators of $G$.

Based on a joint work with Michael Levin and Tom Meyerovitch.

Meeting ID: 852 4277 3200
Passcode: 103121