Mean dimension of product spaces: a fundamental formula
aa r X i v : . [ m a t h . D S ] M a r MEAN DIMENSION OF PRODUCT SPACES:A FUNDAMENTAL FORMULA
LEI JIN AND YIXIAO QIAO
Abstract.
Mean dimension is a topological invariant of dynamical systems, whichoriginates with Mikhail Gromov in 1999 and which was studied with deep applica-tions around 2000 by Elon Lindenstrauss and Benjamin Weiss within the frameworkof amenable group actions. Let a countable discrete amenable group G act continu-ously on compact metrizable spaces X and Y . Consider the product action of G onthe product space X × Y . The product inequality for mean dimension is well known:mdim( X × Y, G ) ≤ mdim( X, G ) + mdim(
Y, G ), while it was unknown for a long time ifthe product inequality could be an equality. In 2019, Masaki Tsukamoto constructed thefirst example of two different continuous actions of G on compact metrizable spaces X and Y , respectively, such that the product inequality becomes strict. However,there is still one longstanding problem which remains open in this direction, askingif there exists a continuous action of G on some compact metrizable space X such thatmdim( X × X, G ) < · mdim( X, G ). We solve this problem. Somewhat surprisingly, weprove, in contrast to (topological) dimension theory, a rather satisfactory theorem: If aninfinite (countable discrete) amenable group G acts continuously on a compact metriz-able space X , then we have mdim( X n , G ) = n · mdim( X, G ), for any positive integer n . Our product formula for mean dimension, together with the example and inequal-ity (stated previously), eventually allows mean dimension of product actions to be fullyunderstood. Main result
Mean dimension is a topological invariant of dynamical systems, which originates withMikhail Gromov in 1999 and which was investigated with deep applications around 2000by Elon Lindenstrauss and Benjamin Weiss within the framework of amenable group ac-tions. The purpose of this paper is to establish a fundamental formula for mean dimensionof product actions . We shall state our main theorem very quickly in this section (Section1). The definition of mean dimension and all the necessary terminologies can be found inSection 2. The proof of the main result is located in Section 3.Let us start with convention. Throughout this paper the symbol N will denote the setof positive integers. All acting groups are always assumed to be countable and discrete . If Mathematics Subject Classification.
Key words and phrases.
Mean dimension; Product space. an amenable group G acts continuously on a compact metrizable space X then we denoteits mean dimension by mdim( X, G ), which takes values in [0 , + ∞ ].Let an amenable group G act continuously on compact metrizable spaces X i , respec-tively, where i ranges over some subset I of N . Consider the product action of G on theproduct space Q i ∈ I X i . The product inequality for mean dimension (due to Lindenstraussand Weiss [LW00]) is well known:mdim( Y i ∈ I X i , G ) ≤ X i ∈ I mdim( X i , G ) . Nevertheless, it was unknown for a long time if the product inequality could always be anequality. In 2019, Masaki Tsukamoto [Tsu19] successfully constructed the first example oftwo different continuous actions of G on compact metrizable spaces X and Y , respectively,such that the product inequality becomes strict:mdim( X × Y, G ) < mdim( X, G ) + mdim(
Y, G ) . A serious reader may observe that in order to have a full understanding of mean dimensionof product actions, there is still one longstanding issue that remains open, asking if itis possible for the two continuous actions (
X, G ) and (
Y, G ) (mentioned in the aboveexample) to be essentially the same . Formally, we study the problem in this direction asfollows: • For an (arbitrarily fixed) amenable group G , does there exist a continuous actionof G on some compact metrizable space X such thatmdim( X × X, G ) < · mdim( X, G )?We solve this problem completely.First of all, let us make two observations on this issue here. On the one hand, we notethat in dimension theory there is an example (to be precise, we refer to Lemma 2.3) of acompact metrizable space K of (topological) dimension dim( K ) so that the product space K × K satisfies dim( K × K ) < · dim( K ) . On the other hand, we have made mention of Tsukamoto’s example which is highly similarto such a classically known analogue that takes place in dimension theory.Apparently, both of these two notable phenomenons lead naturally to a seeminglyplausible impression, i.e. it would be true that we could finally find a compact metrizablespace X (with a continuous action of G on X ) satisfying that mdim( X × X, G ) is strictlyless than 2 · mdim( X, G ), as the (former) example in dimension theory would stimulate usto strengthen the (latter) construction of Tsukamoto’s example in mean dimension theorywith the help of some sufficiently refined method (which seems to be hopeful and whichmight be technically difficult).
EAN DIMENSION OF PRODUCT SPACES: A FUNDAMENTAL FORMULA 3
However, this assertion turns out to be false . Somewhat surprisingly, we prove a rathersatisfactory theorem:
Theorem 1.1 (Main theorem) . If an infinite amenable group G acts continuously on acompact metrizable space X , then we have mdim( X n , G ) = n · mdim( X, G ) for all positive integers n . Remark 1.2.
Theorem 1.1 also applies to n ∈ N ∪ { } ∪ { + ∞} provided 0 · (+ ∞ ) =(+ ∞ ) · n = 0 if we set X to bethe one-point set. Moreover, with a slightly more effort assuming Theorem 1.1 for any n ∈ N we are able to show that the statement is true for n = + ∞ . In fact, there aretwo cases. If mdim( X, G ) = 0, then mdim( X ∞ , G ) = 0 follows directly from the productinequality for mean dimension. Now we suppose mdim( X, G ) >
0. Since it is clear thatmdim( X ∞ , G ) ≥ mdim( X n , G ) = n · mdim( X, G ) for every n ∈ N (by definition and bythe statement of Theorem 1.1 for all n ∈ N ), we have mdim( X ∞ , G ) = + ∞ . Remark 1.3. If G is a finite group (which is automatically amenable) then Theorem 1.1may be false. Notice that in this case we have by definition mdim( X, G ) = dim( X ) / | G | .As follows is an entire picture of the situation: If X satisfies dim( X ) = + ∞ , thenmdim( X, G ) = + ∞ . So does any of its self-product. Thus, the statement remains truefor every n ∈ N ∪ { , + ∞} in this case. Now let us suppose that X is finite dimensional.It follows from Lemma 2.3 that for each n ∈ N mdim( X n , G ) = n · mdim( X, G ) , if dim( X × X ) = 2 dim( X ) n · mdim( X, G ) − ( n − / | G | , otherwise . Thus, in this case the statement fails if and only if X does not satisfy dim( X × X ) =2 dim( X ) and meanwhile n does not belong to { , , + ∞} . In short, the exact range towhich the statement of Theorem 1.1 does not apply is where G is a finite group, X satisfiesdim( X × X ) < X ), and n ∈ N \ { } .In contrast to dimension theory, Theorem 1.1 enables an unexpected behaviour in meandimension theory to become clarified. Furthermore, our main theorem, together withLindenstrauss–Weiss’ inequality and Tsukamoto’s example (stated previously), eventuallyallows mean dimension of product actions to be fully understood.Our result is new even for Z -actions. A novel point of the theorem is that the statementapplies to the context of amenable group actions, whereas the proof goes through theframework of its sofic nature. The key ingredient of our idea is to produce different sofic LEI JIN AND YIXIAO QIAO approximation sequences for the acting group, with respect to which, we consider the soficmean dimension of a group action.
Acknowledgements.
L. Jin was supported by Basal Funding AFB 170001 and FondecytGrant No. 3190127. Y. Qiao was supported by NNSF of China No. 11901206.2.
A brief review of mean dimension
Both mean dimension and sofic groups originate with Misha Gromov around 1999. Asystematic study of mean dimension in the context of amenable group actions was givenaround 2000 by Lindenstrauss and Weiss [LW00]. In 2013, Hanfeng Li [Li13] introducedthe notion of sofic mean dimension which is a successful extension of the definition ofmean dimension to the setting of sofic group actions, and further, Li built its connectionwith classical mean dimension. This section is devoted to all the precise notions andnotations in relation to our result, and to collecting fundamental material on them.2.1.
Sofic groups.
We denote by | F | the cardinality of a set F . For every d ∈ N wewrite [ d ] for the set { k ∈ N : 1 ≤ k ≤ d } and Sym( d ) for the group of permutations of [ d ].A group G is sofic if there is a sequenceΣ = { σ i : G → Sym( d i ) } i ∈ N together with a sequence { d i } i ∈ N ⊂ N such that the following three conditions are satisfied: • lim i →∞ d i |{ k ∈ [ d i ] : σ i ( st )( k ) = σ i ( s ) σ i ( t )( k ) }| = 1 for all s, t ∈ G ; • lim i →∞ d i |{ k ∈ [ d i ] : σ i ( s )( k ) = σ i ( t )( k ) }| = 1 for all distinct s, t ∈ G ; • lim i →∞ d i = + ∞ . Such a sequence Σ is called a sofic approximation sequence for G . Remark 2.1.
Note that the third condition will be fulfilled automatically if G is aninfinite group. Remark 2.2.
The sofic groups are a fairly extensive class, which contain in particular allamenable groups and all residually finite groups. However, it has not yet been confirmedif there exists a non-sofic group.2.2.
Product actions.
Let G be a group. By the terminology “ G acts continuouslyon a compact metrizable space X ” we understand a continuous mappingΦ : G × X → X, ( g, x ) gx satisfying Φ( e, x ) = x, Φ( gh, x ) = Φ( g, Φ( h, x )) , ∀ x ∈ X, ∀ g, h ∈ G, EAN DIMENSION OF PRODUCT SPACES: A FUNDAMENTAL FORMULA 5 where e is the identity element of the group G .Let a group G act continuously on compact metrizable spaces X n , respectively, where n ranges over some R ∈ { [ r ] : r ∈ N } ∪ { N } . The product action of G on the productspace Q n ∈ R X n is defined as follows: g ( x n ) n ∈ R = ( gx n ) n ∈ R , ∀ g ∈ G, ∀ ( x n ) n ∈ R ∈ Y n ∈ R X n . Dimension.
We denote by dim( K ) the topological dimension (i.e. the Lebesguecovering dimension) of a compact metrizable space K . If the space K is empty, then weset dim( K ) = −∞ . For a finite dimensional (nonempty) compact metrizable space K ,since it was classically known that2 dim( K ) − ≤ dim( K × K ) ≤ K )and since dim( K ) must be a nonnegative integer, we have • either dim( K × K ) = 2 dim( K ), • or dim( K × K ) = 2 dim( K ) − Lemma 2.3.
Let K be a finite dimensional compact metrizable space. Then for every n ∈ N dim( K n ) = n dim( K ) , if K satisfies dim( K × K ) = 2 dim( K ) ,n dim( K ) − n + 1 , otherwise . Let X and P be two compact metrizable spaces. Let ρ be a compatible metric on X .For ǫ > f : X → P is called an ǫ -embedding with respectto ρ if f ( x ) = f ( x ′ ) implies ρ ( x, x ′ ) < ǫ , for all x, x ′ ∈ X . Let Widim ǫ ( X, ρ ) be theminimum topological dimension dim( P ) of a compact metrizable space P which admitsan ǫ -embedding f : X → P with respect to ρ . Remark 2.4.
We may verify that the topological dimension of X may be recovered bydim( X ) = lim ǫ → Widim ǫ ( X, ρ ).Let K be a compact metrizable space with a compatible metric ρ . For every n ∈ N wedefine on the product space K n two compatible metrics ρ and ρ ∞ as follows: ρ (cid:0) ( x i ) i ∈ [ n ] , ( y i ) i ∈ [ n ] (cid:1) = s n X i ∈ [ n ] ( ρ ( x i , y i )) ,ρ ∞ (cid:0) ( x i ) i ∈ [ n ] , ( y i ) i ∈ [ n ] (cid:1) = max i ∈ [ n ] ρ ( x i , y i ) . We do not include n ∈ N in the notations ρ and ρ ∞ because it does not cause anyambiguity. LEI JIN AND YIXIAO QIAO
Mean dimension.
A group G is amenable if there exists a sequence { F n } n ∈ N ofnonempty finite subsets of G such that for any g ∈ G lim n →∞ | F n △ gF n || F n | = 0 . Such a sequence { F n } n ∈ N is called a Følner sequence of the group G .Let an amenable group G act continuously on a compact metrizable space X . Take aFølner sequence { F n } n ∈ N of G and a compatible metric ρ on X . For a nonempty finitesubset F of G we set ρ F ( x, x ′ ) = ρ ∞ (( gx ) g ∈ F , ( gx ′ ) g ∈ F ) , ∀ x, x ′ ∈ X. It is clear that ρ F is also a compatible metric on X . The mean dimension of ( X, G ) isdefined by mdim(
X, G ) = lim ǫ → lim n →∞ Widim ǫ ( X, ρ F n ) | F n | . It is well known that the limits in the above definition always exist. The value mdim(
X, G )is independent of the choices of a Følner sequence { F n } n ∈ N of G and a compatible metric ρ on X .2.5. Sofic mean dimension.
Suppose that Σ = { σ i : G → Sym( d i ) } i ∈ N is a sofic ap-proximation sequence for a sofic group G which acts continuously on a compact metrizablespace X equipped with a compatible metric ρ . For a finite subset F of G , δ > σ : G → Sym( d ) (where d ∈ N ) we defineMap( ρ, F, δ, σ ) = { φ : [ d ] → X : ρ ( φ ◦ σ ( s ) , sφ ) ≤ δ, ∀ s ∈ F } . We consider the set Map( ρ, F, δ, σ ) as a compact subspace of the product space X d . The sofic mean dimension of ( X, G ) with respect to Σ is defined bymdim Σ ( X, G ) = sup ǫ> inf F ⊂ G finite, δ> lim sup i →∞ Widim ǫ (Map( ρ, F, δ, σ i ) , ρ ∞ ) d i . The definition of mdim Σ ( X, G ) does not depend on the compatible metrics ρ on X . Nev-ertheless, it is not clear yet if there is an example of a sofic approximation sequence Σ ′ different from Σ, which leads to a different value mdim Σ ′ ( X, G ). We shall make use ofthe following theorem [Li13, Section 3].
Lemma 2.5.
If an infinite amenable group G acts continuously on a compact metriz-able space X and if Σ is a sofic approximation sequence for G , then mdim Σ ( X, G ) =mdim(
X, G ) . EAN DIMENSION OF PRODUCT SPACES: A FUNDAMENTAL FORMULA 7 Proof of the main theorem
Let G be an infinite amenable group which acts continuously on a compact metrizablespace X . We fix a positive integer n in this section. Recall that ( X n , G ) denotes theproduct action of G on the product space X n . We shall provemdim( X n , G ) = n · mdim( X, G ) . Since the group G is amenable, it is sofic. Therefore we may take a sofic approximationsequence for G : Σ = { σ i : G → Sym( d i ) } i ∈ N , where { d i } i ∈ N is a sequence of positive integers with d i → + ∞ as i → + ∞ . We generatea new sofic approximation sequence for G (confirmed below) as follows:Σ ( n ) = { σ ( n ) i : G → Sym( nd i ) } i ∈ N , where for every i ∈ N the map σ ( n ) i : G → Sym( nd i )is defined by: σ ( n ) i ( g ) (( j − n + l ) = ( σ i ( g )( j ) − n + l, ∀ g ∈ G, ∀ j ∈ [ d i ] , ∀ l ∈ [ n ] . Lemma 3.1. Σ ( n ) is a sofic approximation sequence for G .Proof. Clearly, for every i ∈ N and g ∈ G the map σ ( n ) i ( g ) : [ nd i ] → [ nd i ] is a permutationof [ nd i ]. Besides, it is straightforward to verify that for any s, t ∈ G , j ∈ [ d i ] and l ∈ [ n ]we have σ ( n ) i ( st ) (( j − n + l ) = σ ( n ) i ( s ) σ ( n ) i ( t ) (( j − n + l ) ⇐⇒ σ i ( st )( j ) = σ i ( s ) σ i ( t )( j ) ,σ ( n ) i ( s ) (( j − n + l ) = σ ( n ) i ( t ) (( j − n + l ) ⇐⇒ σ i ( s )( j ) = σ i ( t )( j ) . Since Σ is a sofic approximation sequence for G , the assertion follows. (cid:3) Let us consider the sofic mean dimension of (
X, G ) and ( X n , G ) with respect to the soficapproximation sequences Σ ( n ) and Σ, respectively. These two values share the followingrelation. Lemma 3.2 (Key lemma) . mdim Σ ( n ) ( X, G ) = 1 n · mdim Σ ( X n , G ) . LEI JIN AND YIXIAO QIAO
Proof.
We fix a compatible metric ρ on X in the proof. Let ρ ( n ) be the compatible metric ρ ∞ on X n . Let us consider two compact metric spaces as follows: ( X, ρ ) and ( X n , ρ ( n ) ).We take ǫ > δ >
0, a finite subset F of G , and a positive integer i , arbitrarily and fixthem temporarily.We note that both of the following two sets:Map( ρ, F, δ, σ ( n ) i ) = { φ : [ nd i ] → X : ρ ( φ ◦ σ ( n ) i ( s ) , sφ ) ≤ δ, ∀ s ∈ F } Map( ρ ( n ) , F, δ, σ i ) = { φ : [ d i ] → X n : ρ ( n )2 ( φ ◦ σ i ( s ) , sφ ) ≤ δ, ∀ s ∈ F } can be regarded as compact subspaces of the product space X nd i = ( X n ) d i . More explic-itly, the point here is that we identify X nd i with ( X n ) d i . We notice that the constructionof the sofic approximation sequence Σ ( n ) for G and the definition of the product action( X n , G ) ensure that the terms φ ◦ σ ( n ) i ( s ) and φ ◦ σ i ( s ) agree, i.e. φ ◦ σ ( n ) i ( s ) = φ ◦ σ i ( s ) , ∀ φ ∈ X nd i = ( X n ) d i , ∀ s ∈ F. Further, we also remark that ρ ∞ defined on X nd i corresponds to ρ ( n ) ∞ defined on ( X n ) d i ,namely ρ ∞ ( ψ, ψ ′ ) = ρ ( n ) ∞ ( ψ, ψ ′ ) , ∀ ψ, ψ ′ ∈ X nd i = ( X n ) d i , while ρ defined on X nd i and ρ ( n )2 defined on ( X n ) d i satisfy the inequality:1 √ n · ρ ( n )2 ( ψ, ψ ′ ) ≤ ρ ( ψ, ψ ′ ) ≤ ρ ( n )2 ( ψ, ψ ′ ) , ∀ ψ, ψ ′ ∈ X nd i = ( X n ) d i . The above observation implies thatMap( ρ ( n ) , F, δ, σ i ) ⊂ Map( ρ, F, δ, σ ( n ) i ) ⊂ Map( ρ ( n ) , F, √ nδ, σ i ) . It follows that Widim ǫ (cid:0) Map( ρ ( n ) , F, δ, σ i ) , ρ ( n ) ∞ (cid:1) ≤ Widim ǫ (cid:16) Map( ρ, F, δ, σ ( n ) i ) , ρ ∞ (cid:17) ≤ Widim ǫ (cid:0) Map( ρ ( n ) , F, √ nδ, σ i ) , ρ ( n ) ∞ (cid:1) . Since ǫ > δ >
0, a finite subset F ⊂ G and i ∈ N (which we took in the beginning ofthe proof) are arbitrary, we deduce thatmdim Σ ( n ) ( X, G ) = 1 n · mdim Σ ( X n , G ) . Thus, we end the proof. (cid:3)
Remark 3.3.
The equality established in Lemma 3.2 is generally true for all sofic groupactions (
X, G ) and all positive integers n . The acting group G in this lemma is notrequired to be infinite. EAN DIMENSION OF PRODUCT SPACES: A FUNDAMENTAL FORMULA 9
We are now able to prove Theorem 1.1. The key lemma (Lemma 3.2) indicates thatmdim Σ ( n ) ( X, G ) = 1 n · mdim Σ ( X n , G ) . By Lemma 2.5, sofic mean dimension (with respect to any sofic approximation sequence)will coincide with (classical) mean dimension, as the acting group G is infinite. Thus, weconclude with mdim( X, G ) = 1 n · mdim( X n , G ) . Remark 3.4.
We explain shortly about the difficulty with this problem. Let G be aninfinite amenable group which acts continuously on a compact metrizable space X . Wefix a positive integer n and a Følner sequence { F k } k ∈ N of G . We recall thatmdim( X n , G ) = lim ǫ → lim n →∞ Widim ǫ ( X n , ( ρ ∞ ) F k ) | F k | ,n · mdim( X, G ) = lim ǫ → lim n →∞ n · Widim ǫ ( X, ρ F k ) | F k | , where ρ and ρ ∞ are compatible metrics on X and X n , respectively. To showmdim( X n , G ) ≥ n · mdim( X, G ) , a main issue is how to estimate the term n · Widim ǫ ( X, ρ F k ) from above with terms suchas some variants of Widim ǫ ( X n , ( ρ ∞ ) F k ). We overcome this obstacle. The strategy weadopted is to consider different approximation sequences in the limits. For a systematictreatment we went through an approach of sofic mean dimension. To make it clearer, letus focus on the case of Z -actions. More precisely, let ( X, d ) be a compact metric spaceand T : X → X a homeomorphism. For convenience we change our notations here, whichapply only to the remark. For every positive integer k we write d k for the compatiblemetric on X defined by d k ( x, x ′ ) = max ≤ i Therefore, in order to show2 · mdim( X, T ) ≤ mdim( X × X, T × T ) = lim ǫ → lim N → + ∞ Widim ǫ ( X × X, ( d × d ) N ) N it suffices to prove Widim ǫ ( X, d N ) ≤ Widim ǫ ( X × X, ( d × d ) N )for ǫ > N ∈ N . This will be deduced from the following statement: The continuousmapping X → X × X, x ( x, T N x )is distance-increasing (actually it is distance-preserving) with respect to ( X, d N ) and( X × X, ( d × d ) N ), i.e. d N ( x, x ′ ) = max ≤ i ≤ N − d ( T i x, T i x ′ ) = ( d × d ) N (cid:0) ( x, T N x ) , ( x ′ , T N x ′ ) (cid:1) , ∀ x, x ′ ∈ X. References [Li13] Hanfeng Li. Sofic mean dimension. Advances in Mathematics 244 (2013), 570–604.[LW00] Elon Lindenstrauss, Benjamin Weiss. Mean topological dimension. Israel Journal of Mathe-matics 115 (2000), 1–24.[Tsu19] Masaki Tsukamoto. Mean dimension of full shifts. Israel Journal of Mathematics 230 (2019),183–193. Lei Jin: Center for Mathematical Modeling, University of Chile and UMI 2807 - CNRS Email address : [email protected] Yixiao Qiao (Corresponding author): School of Mathematical Sciences, South ChinaNormal University, Guangzhou, Guangdong 510631, China Email address ::