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

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Given a class   of topological spaces,   is universal for   if each member of   embeds in  . Menger stated and proved the case   of the following theorem. The theorem in full generality was proven by Nöbeling.

Theorem:[1] The  -dimensional cube   is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than  .

Nöbeling went further and proved:

Theorem: The subspace of   consisting of set of points, at most   of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than  .

The last theorem was generalized by Lipscomb to the class of metric spaces of weight  ,  : There exist a one-dimensional metric space   such that the subspace of   consisting of set of points, at most   of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than   and whose weight is less than  .[2]

Universal spaces in topological dynamics

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Consider the category of topological dynamical systems   consisting of a compact metric space   and a homeomorphism  . The topological dynamical system   is called minimal if it has no proper non-empty closed  -invariant subsets. It is called infinite if  . A topological dynamical system   is called a factor of   if there exists a continuous surjective mapping   which is equivariant, i.e.   for all  .

Similarly to the definition above, given a class   of topological dynamical systems,   is universal for   if each member of   embeds in   through an equivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem[3]: Let  . The compact metric topological dynamical system   where   and   is the shift homeomorphism  

is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than   and which possess an infinite minimal factor.

In the same article Lindenstrauss asked what is the largest constant   such that a compact metric topological dynamical system whose mean dimension is strictly less than   and which possesses an infinite minimal factor embeds into  . The results above implies  . The question was answered by Lindenstrauss and Tsukamoto[4] who showed that   and Gutman and Tsukamoto[5] who showed that  . Thus the answer is  .

See also

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References

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  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.
  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. Bibcode:2020InMat.221..113G. doi:10.1007/s00222-019-00942-w. ISSN 1432-1297. S2CID 119139371.