Universal space

In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics.

Definition

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

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

Nöbeling went further and proved:

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

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

Universal spaces in topological dynamics

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

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

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

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

In the same article Lindenstrauss asked what is the largest constant ${\displaystyle \textstyle c}$ such that a compact metric topological dynamical system whose mean dimension is strictly less than ${\displaystyle \textstyle cd}$ and which possesses an infinite minimal factor embeds into ${\displaystyle \textstyle ([0,1]^{d})^{\mathbb {Z} }}$. The results above implies ${\displaystyle \textstyle c\geq {\frac {1}{36}}}$. The question was answered by Lindenstrauss and Tsukamoto[4] who showed that ${\displaystyle \textstyle c\leq {\frac {1}{2}}}$ and Gutman and Tsukamoto[5] who showed that ${\displaystyle \textstyle c\geq {\frac {1}{2}}}$. Thus the answer is ${\displaystyle \textstyle c={\frac {1}{2}}}$.

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 I_2n+1: Theorem V.2". Dimension Theory. Princeton Mathematical Series. 4. Princeton University Press. pp. 56–. ISBN 978-1400875665.
2. Lipscomb, Stephen Leon (2009). "The quest for universal spaces in dimension theory" (PDF). Notices Amer. Math. Soc. 56 (11): 1418–24.
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.
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.
5. Gutman, Yonatan; Tsukamoto, Masaki (2020-07-01). "Embedding minimal dynamical systems into Hilbert cubes". Inventiones Mathematicae. 221 (1): 113–166. arXiv:1511.01802. doi:10.1007/s00222-019-00942-w. ISSN 1432-1297. S2CID 119139371.