Density of the Level Sets of the Metric Mean Dimension for Homeomorphisms

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AbstractLetNbe ann-dimensional compact riemannian manifold, with$$n\ge 2$$n≥2. In this paper, we prove that for any$$\alpha \in [0,n]$$α∈[0,n], the set consisting of homeomorphisms onNwith lower and upper metric mean dimensions equal to$$\alpha $$αis dense in$$\text {Hom}(N)$$Hom(N). More generally, given$$\alpha ,\beta \in [0,n]$$α,β∈[0,n], with$$\alpha \le \beta $$α≤β, we show the set consisting of homeomorphisms onNwith lower metric mean dimension equal to$$\alpha $$αand upper metric mean dimension equal to$$\beta $$βis dense in$$\text {Hom}(N)$$Hom(N). Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal tonis residual in$$\text {Hom}(N)$$Hom(N).