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Register a new userThe embedding problem in topological dynamics and Takens theorem
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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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We study the problem of embedding arbitrary $mathbb{Z}^k$-actions into the shift action on the infinite dimensional cube $left([0,1]^Dright)^{mathbb{Z}^k}$. We prove that if a $mathbb{Z}^k$-action satisfies the marker property (in particular if it is a minimal system without periodic points) and if its mean dimension is smaller than $D/2$ then we can embed it in the shift on $left([0,1]^Dright)^{mathbb{Z}^k}$. The value $D/2$ here is optimal. The proof goes through signal analysis. We develop the theory of encoding $mathbb{Z}^k$-actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of the Euclidean space into a band-limited function which is constructed from another tiling.
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