We prove that every $mathbb{Z}^{k}$-action $(X,mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{mathbb{Z}^{k}}times Y,sigmatimes S)$, where $Dinmathbb{N}$, $Linmathbb{N}cup{0}$ 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 $(([0,1]^{2^kD+1})^{mathbb{Z}^k},sigma)$, 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.
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