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 [math]\displaystyle{ \textstyle \mathcal{C} }[/math] of topological spaces, [math]\displaystyle{ \textstyle \mathbb{U}\in\mathcal{C} }[/math] is universal for [math]\displaystyle{ \textstyle \mathcal{C} }[/math] if each member of [math]\displaystyle{ \textstyle \mathcal{C} }[/math] embeds in [math]\displaystyle{ \textstyle \mathbb{U} }[/math]. Menger stated and proved the case [math]\displaystyle{ \textstyle d=1 }[/math] of the following theorem. The theorem in full generality was proven by Nöbeling.
Theorem:[1] The [math]\displaystyle{ \textstyle (2d+1) }[/math]-dimensional cube [math]\displaystyle{ \textstyle [0,1]^{2d+1} }[/math] is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than [math]\displaystyle{ \textstyle d }[/math].
Nöbeling went further and proved:
Theorem: The subspace of [math]\displaystyle{ \textstyle [0,1]^{2d+1} }[/math] consisting of set of points, at most [math]\displaystyle{ \textstyle d }[/math] of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than [math]\displaystyle{ \textstyle d }[/math].
The last theorem was generalized by Lipscomb to the class of metric spaces of weight [math]\displaystyle{ \textstyle \alpha }[/math], [math]\displaystyle{ \textstyle \alpha\gt \aleph_{0} }[/math]: There exist a one-dimensional metric space [math]\displaystyle{ \textstyle J_{\alpha} }[/math] such that the subspace of [math]\displaystyle{ \textstyle J_{\alpha}^{2d+1} }[/math] consisting of set of points, at most [math]\displaystyle{ \textstyle d }[/math] of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than [math]\displaystyle{ \textstyle d }[/math] and whose weight is less than [math]\displaystyle{ \textstyle \alpha }[/math].[2]
Universal spaces in topological dynamics
Consider the category of topological dynamical systems [math]\displaystyle{ \textstyle (X,T) }[/math] consisting of a compact metric space [math]\displaystyle{ \textstyle X }[/math] and a homeomorphism [math]\displaystyle{ \textstyle T:X\rightarrow X }[/math]. The topological dynamical system [math]\displaystyle{ \textstyle (X,T) }[/math] is called minimal if it has no proper non-empty closed [math]\displaystyle{ \textstyle T }[/math]-invariant subsets. It is called infinite if [math]\displaystyle{ \textstyle |X|=\infty }[/math]. A topological dynamical system [math]\displaystyle{ \textstyle (Y,S) }[/math] is called a factor of [math]\displaystyle{ \textstyle (X,T) }[/math] if there exists a continuous surjective mapping [math]\displaystyle{ \textstyle \varphi:X\rightarrow Y }[/math] which is equivariant, i.e. [math]\displaystyle{ \textstyle \varphi(Tx)=S\varphi(x) }[/math] for all [math]\displaystyle{ \textstyle x\in X }[/math].
Similarly to the definition above, given a class [math]\displaystyle{ \textstyle \mathcal{C} }[/math] of topological dynamical systems, [math]\displaystyle{ \textstyle \mathbb{U}\in\mathcal{C} }[/math] is universal for [math]\displaystyle{ \textstyle \mathcal{C} }[/math] if each member of [math]\displaystyle{ \textstyle \mathcal{C} }[/math] embeds in [math]\displaystyle{ \textstyle \mathbb{U} }[/math] through an equivariant continuous mapping. Lindenstrauss proved the following theorem:
Theorem[3]: Let [math]\displaystyle{ \textstyle d\in\mathbb{{N}} }[/math]. The compact metric topological dynamical system [math]\displaystyle{ \textstyle (X,T) }[/math] where [math]\displaystyle{ \textstyle X=([0,1]^{d})^{\mathbb{{Z}}} }[/math] and [math]\displaystyle{ \textstyle T:X\rightarrow X }[/math] is the shift homeomorphism [math]\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) }[/math]
is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than [math]\displaystyle{ \textstyle \frac{d}{36} }[/math] and which possess an infinite minimal factor.
In the same article Lindenstrauss asked what is the largest constant [math]\displaystyle{ \textstyle c }[/math] such that a compact metric topological dynamical system whose mean dimension is strictly less than [math]\displaystyle{ \textstyle cd }[/math] and which possesses an infinite minimal factor embeds into [math]\displaystyle{ \textstyle ([0,1]^{d})^{\mathbb{{Z}}} }[/math]. The results above implies [math]\displaystyle{ \textstyle c \geq \frac{1}{36} }[/math]. The question was answered by Lindenstrauss and Tsukamoto[4] who showed that [math]\displaystyle{ \textstyle c \leq \frac{1}{2} }[/math] and Gutman and Tsukamoto[5] who showed that [math]\displaystyle{ \textstyle c \geq \frac{1}{2} }[/math]. Thus the answer is [math]\displaystyle{ \textstyle c=\frac{1}{2} }[/math].
See also
References
- ↑ Hurewicz, Witold; Wallman, Henry (2015). "V Covering and Imbedding Theorems §3 Imbedding of a compact n-dimensional space in I2n+1: Theorem V.2". Dimension Theory. Princeton Mathematical Series. 4. Princeton University Press. pp. 56–. ISBN 978-1400875665. https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA56.
- ↑ Lipscomb, Stephen Leon (2009). "The quest for universal spaces in dimension theory". Notices Amer. Math. Soc. 56 (11): 1418–24. https://www.ams.org/notices/200911/rtx091101418p.pdf.
- ↑ 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. http://www.numdam.org/item/PMIHES_1999__89__227_0/.
- ↑ Lindenstrauss, Elon; Tsukamoto, Masaki (March 2014). "Mean dimension and an embedding problem: An example" (in en). Israel Journal of Mathematics 199 (2): 573–584. doi:10.1007/s11856-013-0040-9. ISSN 0021-2172.
- ↑ Gutman, Yonatan; Tsukamoto, Masaki (2020-07-01). "Embedding minimal dynamical systems into Hilbert cubes" (in en). Inventiones Mathematicae 221 (1): 113–166. doi:10.1007/s00222-019-00942-w. ISSN 1432-1297. Bibcode: 2020InMat.221..113G. https://doi.org/10.1007/s00222-019-00942-w.
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