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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
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:
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 }
.
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:
Theorem: 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 Tsukamoto who showed that
c
≤
1
2
{\displaystyle \textstyle c\leq {\frac {1}{2}}}
and Gutman and Tsukamoto 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