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Definition

Given a class C {\displaystyle \textstyle {\mathcal {C}}} of topological spaces, U ∈ C {\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}} is universal for C {\displaystyle \textstyle {\mathcal {C}}} if each member of C {\displaystyle \textstyle {\mathcal {C}}} embeds in U {\displaystyle \textstyle \mathbb {U} } . Menger stated and proved the case d = 1 {\displaystyle \textstyle d=1} of the following theorem. The theorem in full generality was proven by Nöbeling.

Theorem:¹ The ( 2 d + 1 ) {\displaystyle \textstyle (2d+1)} -dimensional cube [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} .

Nöbeling went further and proved:

Theorem: The subspace of [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} consisting of set of points, at most d {\displaystyle \textstyle d} of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} .

The last theorem was generalized by Lipscomb to the class of metric spaces of weight α {\displaystyle \textstyle \alpha } , α > ℵ 0 {\displaystyle \textstyle \alpha >\aleph _{0}} : There exist a one-dimensional metric space J α {\displaystyle \textstyle J_{\alpha }} such that the subspace of J α 2 d + 1 {\displaystyle \textstyle J_{\alpha }^{2d+1}} consisting of set of points, at most d {\displaystyle \textstyle d} of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} and whose weight is less than α {\displaystyle \textstyle \alpha } .²

Universal spaces in topological dynamics

Consider the category of topological dynamical systems ( X , T ) {\displaystyle \textstyle (X,T)} consisting of a compact metric space X {\displaystyle \textstyle X} and a homeomorphism T : X → X {\displaystyle \textstyle T:X\rightarrow X} . The topological dynamical system ( X , T ) {\displaystyle \textstyle (X,T)} is called minimal if it has no proper non-empty closed T {\displaystyle \textstyle T} -invariant subsets. It is called infinite if | X | = ∞ {\displaystyle \textstyle |X|=\infty } . A topological dynamical system ( Y , S ) {\displaystyle \textstyle (Y,S)} is called a factor of ( X , T ) {\displaystyle \textstyle (X,T)} if there exists a continuous surjective mapping φ : X → Y {\displaystyle \textstyle \varphi :X\rightarrow Y} which is equivariant, i.e. φ ( T x ) = S φ ( x ) {\displaystyle \textstyle \varphi (Tx)=S\varphi (x)} for all x ∈ X {\displaystyle \textstyle x\in X} .

Similarly to the definition above, given a class C {\displaystyle \textstyle {\mathcal {C}}} of topological dynamical systems, U ∈ C {\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}} is universal for C {\displaystyle \textstyle {\mathcal {C}}} if each member of C {\displaystyle \textstyle {\mathcal {C}}} embeds in U {\displaystyle \textstyle \mathbb {U} } through an equivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem³: Let d ∈ N {\displaystyle \textstyle d\in \mathbb {N} } . The compact metric topological dynamical system ( X , T ) {\displaystyle \textstyle (X,T)} where X = ( [ 0 , 1 ] d ) Z {\displaystyle \textstyle X=([0,1]^{d})^{\mathbb {Z} }} and T : X → X {\displaystyle \textstyle T:X\rightarrow X} is the shift homeomorphism ( … , x − 2 , x − 1 , x 0 , x 1 , x 2 , … ) → ( … , x − 1 , x 0 , x 1 , x 2 , x 3 , … ) {\displaystyle \textstyle (\ldots ,x_{-2},x_{-1},\mathbf {x_{0}} ,x_{1},x_{2},\ldots )\rightarrow (\ldots ,x_{-1},x_{0},\mathbf {x_{1}} ,x_{2},x_{3},\ldots )}

is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than d 36 {\displaystyle \textstyle {\frac {d}{36}}} and which possess an infinite minimal factor.

In the same article Lindenstrauss asked what is the largest constant c {\displaystyle \textstyle c} such that a compact metric topological dynamical system whose mean dimension is strictly less than c d {\displaystyle \textstyle cd} and which possesses an infinite minimal factor embeds into ( [ 0 , 1 ] d ) Z {\displaystyle \textstyle ([0,1]^{d})^{\mathbb {Z} }} . The results above implies c ≥ 1 36 {\displaystyle \textstyle c\geq {\frac {1}{36}}} . The question was answered by Lindenstrauss and Tsukamoto⁴ who showed that c ≤ 1 2 {\displaystyle \textstyle c\leq {\frac {1}{2}}} and Gutman and Tsukamoto⁵ who showed that c ≥ 1 2 {\displaystyle \textstyle c\geq {\frac {1}{2}}} . Thus the answer is c = 1 2 {\displaystyle \textstyle c={\frac {1}{2}}} .

See also

• Universal property
• Urysohn universal space
• Mean dimension

References

1. Hurewicz, Witold; Wallman, Henry (2015) [1941]. "V Covering and Imbedding Theorems §3 Imbedding of a compact n-dimensional space in I2n+1: Theorem V.2". Dimension Theory. Princeton Mathematical Series. Vol. 4. Princeton University Press. pp. 56–. ISBN 978-1400875665. 978-1400875665 ↩

2. Lipscomb, Stephen Leon (2009). "The quest for universal spaces in dimension theory" (PDF). Notices Amer. Math. Soc. 56 (11): 1418–24. https://www.ams.org/notices/200911/rtx091101418p.pdf ↩

3. Lindenstrauss, Elon (1999). "Mean dimension, small entropy factors and an embedding theorem. Theorem 5.1". Inst. Hautes Études Sci. Publ. Math. 89 (1): 227–262. doi:10.1007/BF02698858. S2CID 2413058. http://www.numdam.org/item/PMIHES_1999__89__227_0/ ↩

4. Lindenstrauss, Elon; Tsukamoto, Masaki (March 2014). "Mean dimension and an embedding problem: An example". Israel Journal of Mathematics. 199 (2): 573–584. doi:10.1007/s11856-013-0040-9. ISSN 0021-2172. S2CID 2099527. https://doi.org/10.1007%2Fs11856-013-0040-9 ↩

5. Gutman, Yonatan; Tsukamoto, Masaki (2020-07-01). "Embedding minimal dynamical systems into Hilbert cubes". Inventiones Mathematicae. 221 (1): 113–166. arXiv:1511.01802. Bibcode:2020InMat.221..113G. doi:10.1007/s00222-019-00942-w. ISSN 1432-1297. S2CID 119139371. https://doi.org/10.1007/s00222-019-00942-w ↩