According to the classical Menger–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