We prove that any dynamical system $(X,T)$ that admits the marker property and has mean dimension strictly less than $d$ admits a continuous, finite-to-one equivariant map into $(([0,1]^d)^\mathbb{Z},\operatorname{shift})$. Moreover, in the above situation a generic continuous equivariant map from $X$ to $([0,1]^d)^\mathbb{Z}$ is finite-to-one. In particular when $\operatorname{mdim}(X,T) < \frac{1}{2}d$, we show that such a a generic continuous equivariant map is an embedding and this strengthens the optimal embedding theorem of Gutman, Qiao, and Tsukamoto (2019), for $\mathbb{Z}$-actions.

Unlike earlier works, our proof relies on classical topological techniques originating in the work of Ostrand (1965), Kolmogorov (1957), and Arnold (1957).

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