Abstract
We prove that every -action of mean dimension less than admitting a factor of Rokhlin dimension not greater than L embeds in , where , and σ is the shift on the Hilbert cube ; in particular, when is an irrational -rotation on the k-torus, embeds in , which is compared to a previous result in Gutman, Lindenstrauss and Tsukamoto (2016 Geom. Funct. Anal. 3 778–817). Moreover, we give a complete and detailed proof of Takens’ embedding theorem with a continuous observable for -actions and deduce the analogous result for -actions. Lastly, we show that the Lindenstrauss–Tsukamoto conjecture for -actions holds generically, discuss an analogous conjecture for -actions in Gutman, Qiao and Tsukamoto (2017 arXiv:1709.00125) and verify it for -actions on finite dimensional spaces.
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