Universal space explained
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
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
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.
style\varphi(Tx)=S\varphi(x)
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
style(\ldots,x-2,x-1
,x2,\ldots) → (\ldots,x-1,x0
,x3,\ldots)
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
Notes and References
- Book: Dimension Theory. Hurewicz. Witold. Princeton Mathematical Series . 4 . Princeton University Press . 1941 . 2015 . 978-1400875665 . 56– . V Covering and Imbedding Theorems §3 Imbedding of a compact n-dimensional space in I2n+1: Theorem V.2 . https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA56 . Wallman. Henry.
- The quest for universal spaces in dimension theory . Lipscomb . Stephen Leon . 2009 . Notices Amer. Math. Soc. . 56 . 11 . 1418–24 .
- Mean dimension, small entropy factors and an embedding theorem. Theorem 5.1 . Lindenstrauss . Elon . 1999 . Inst. Hautes Études Sci. Publ. Math. . 89 . 1 . 227–262 . 10.1007/BF02698858 . 2413058 .
- Lindenstrauss. Elon. Tsukamoto. Masaki. March 2014. Mean dimension and an embedding problem: An example. Israel Journal of Mathematics. en. 199. 2. 573–584. 10.1007/s11856-013-0040-9. free. 2099527. 0021-2172.
- Gutman. Yonatan. Tsukamoto. Masaki. 2020-07-01. Embedding minimal dynamical systems into Hilbert cubes. Inventiones Mathematicae. en. 221. 1. 113–166. 10.1007/s00222-019-00942-w. 1432-1297. 1511.01802. 2020InMat.221..113G. 119139371.
