The embedding problem in topological dynamics and Takens' theorem

We prove that every ${\mathbb{Z}}^k$-action $(X,{\mathbb{Z}}^k,T)$ of mean dimension less than $D\mathrm{/}2$ admitting a factor $(Y,{\mathbb{Z}}^k,S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1{]}^{(L+1)D}{)}^{{\mathbb{Z}}^k}\times Y,\sigma \times S)$, where $D\in \mathbb{N}$, $L\in \mathbb{N}\cup \{0\}$ and $\sigma$ is the shift on the Hilbert cube $([0,1{]}^{(L+1)D}{)}^{{\mathbb{Z}}^k}$; in particular, when $(Y,{\mathbb{Z}}^k,S)$ is an irrational ${\mathbb{Z}}^k$-rotation on the $k$-torus, $(X,{\mathbb{Z}}^k,T)$ embeds in $(([0,1{]}^{2^{k}D+1}{)}^{{\mathbb{Z}}^k},\sigma )$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens’ embedding theorem with a continuous observable for $\mathbb{Z}$-actions and deduce the analogous result for ${\mathbb{Z}}^k$-actions. Lastly, we show that the Lindenstrauss–Tsukamoto conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous conjecture for ${\mathbb{Z}}^k$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for ${\mathbb{Z}}^k$-actions on finite dimensional spaces.