The Hilbert cube contains a minimal subshift of full mean dimension
aa r X i v : . [ m a t h . D S ] M a r THE HILBERT CUBE CONTAINS A MINIMAL SUBSHIFTOF FULL MEAN DIMENSION
LEI JIN AND YIXIAO QIAO
Abstract.
We construct a minimal dynamical system of mean dimension equal to 1,which can be embedded in the shift action on the Hilbert cube [0 , Z . Our result clarifiesa seemingly plausible impression and finally enables us to have a full understanding of(a pair of) the exact ranges of all possible values of mean dimension, within which therewill always be a minimal dynamical system that can be (resp. cannot be) embedded inthe shift action on the Hilbert cube. The key ingredient of our idea is to produce a densesubset of the alphabet [0 ,
1] “more gently”. Main result
Mean dimension is a topological invariant of dynamical systems, which originates withMisha Gromov [Gro99] around 1999. It is closely connected with, and has been deeplyapplied to the embedding problem , in particular, deciding if a minimal dynamical systemcan be embedded in the shift action on the Hilbert cube [0 , Z . As follows is a brief reviewof the latest progress in this direction. With the necessary historical background we shallstate our main theorem very quickly. All the precise notions and notations related to thestatements can be found in Section 2, while a constructive proof of the main theorem islocated in Section 3.We denote by ([0 , Z , σ ) the shift action on the Hilbert cube [0 , Z (note that itsmean dimension is equal to 1). Let ( X, T ) be a minimal dynamical system, whose meandimension is denoted by mdim(
X, T ) which takes values within [0 , + ∞ ]. The followingstatements are classically known: • If 1 < mdim( X, T ) ≤ + ∞ then ( X, T ) cannot be embedded in ([0 , Z , σ ). • If 0 ≤ mdim( X, T ) < / X, T ) can be embedded in ([0 , Z , σ ). • If 1 / ≤ mdim( X, T ) <
1, then it is possible for (
X, T ) to be embedded in([0 , Z , σ ), whereas it is also possible that ( X, T ) cannot be embedded in ([0 , Z , σ ). • Suppose mdim(
X, T ) = 1. It may happen that (
X, T ) cannot be embedded in([0 , Z , σ ).Here let us make some remarks on the above assertions. The first implication followsdirectly from definition. The second significant result is due to Gutman and Tsukamoto Mathematics Subject Classification.
Key words and phrases.
Mean dimension; Embedding; Minimal dynamical system. [GT20] (for a corresponding result of Z k -actions we refer to [GQT19]). In relation to thethird and fourth cases, embeddable and non-embeddable examples were constructed byLindenstrauss–Weiss [LW00] and Lindenstrauss–Tsukamoto [LT14], respectively. To beprecise, Lindenstrauss and Tsukamoto [LT14] showed that for any given r ≥ / r , which cannot be embeddedin ([0 , Z , σ ); Lindenstrauss and Weiss [LW00] proved that for any given 0 ≤ r < r , which can be embeddedin ([0 , Z , σ ).A careful reader may observe that in order to have a complete picture of the possibilityof embedding minimal dynamical systems in the Hilbert cube, there is only one issuethat remains open, namely, the embeddable situation for the critical value 1 (with meandimension) is not clear to us. Formally, we study the problem as follows: • Assume that a minimal dynamical system can be embedded in ([0 , Z , σ ). Doesthis imply that it has mean dimension strictly less than 1?In other words, this problem asks if there exists a minimal dynamical system of meandimension equal to 1, which can be embedded in ([0 , Z , σ ). The aim of this paper is tosolve this problem.In fact, we observe the following three aspects:(1) With a view towards mean dimension theory: The approach of any previouslymentioned results [LW00, LT14] reveals that if we try to construct a minimaldynamical system of mean dimension equal to 1, then a construction within theframework of [0 , Z is not adequate for our purpose, and instead, we need consider,with the same method, the shift action on ([0 , ) Z , which has mean dimensionequal to 2, rather than consider the shift action on [0 , Z .(2) With a view towards entropy theory: We note moreover that a highly similarcircumstance took place in topological entropy in connection with the shift overfinite alphabets; strictly speaking, such an “analogue” asserts that if a minimaldynamical system can be embedded in the shift action on { k ∈ Z : 1 ≤ k ≤ N } Z (where N is a positive integer), then it does not have full topological entropy (i.e.its topological entropy must be strictly smaller than log N ).(3) With a view towards dimension theory: A celebrated theorem in dimension theorystates that if a compact metrizable space of (topological) dimension 1 can be(topologically) embedded in [0 , ,
1] (i.e. [0 , not able to possess full mean dimension. However, this impres-sion turns out to be false . Somewhat surprisingly, we do construct a minimal dynamical MINIMAL SUBSHIFT OF FULL MEAN DIMENSION 3 system successfully, which satisfies both embeddable and full mean dimensional condi-tions. This is our main result.
Theorem 1.1 (Main theorem) . There exists a minimal dynamical system of mean di-mension equal to , which can be embedded in the shift action on the Hilbert cube [0 , Z . In contrast to the seemingly reasonable observation as stated above, Theorem 1.1 en-ables a different (and unexpected) behaviour to become clarified. Furthermore, our maintheorem, together with the previous remarkable results, eventually allows the embeddabil-ity of minimal dynamical systems in the Hilbert cube to be fully understood with a viewtowards mean dimension. More precisely, we have now gotten a full understanding of(a pair of) the exact ranges (I) and (II) of all possible values of mean dimension, suchthat for any value assigned within (I) (resp. within (II)) there always exists a minimaldynamical system of mean dimension equal to this appointed value, which can be (resp.cannot be) embedded in the shift action on the Hilbert cube.The strategy we shall adopt generally follows the technical framework of the “block-type” induction. Nevertheless, we have to make each step in our construction sufficientlydelicate. The key ingredient of our idea is to produce a dense subset of the alphabet [0 , Remark 1.2.
The same statement (as in Theorem 1.1) also applies to the alphabet[0 , D ; namely, the following assertion is true. • Let D be a positive integer (possibly + ∞ ). There exists a minimal dynamicalsystem of mean dimension equal to D , which can be embedded in the shift actionon ([0 , D ) Z .Meanwhile, the alphabet [0 , D of topological dimension D may be replaced by some othercompact metrizable spaces provided they have some nice structure, e.g. polyhedrons P oftopological dimension dim( P ). Besides, we notice that it is straightforward to generalizeour main theorem to actions of infinite (countable discrete) amenable groups (with almostno additional effort while using material on tilings of amenable groups). Acknowledgements.
The initial proof we provided in a previous version of this papercontained a gap. Professor Masaki Tsukamoto pointed out this to us, and meanwhile, healso kindly explained to us how to fix it. We would like to thank him for his warm help.L. Jin was supported by Basal Funding AFB 170001 and Fondecyt Grant No. 3190127.Y. Qiao was supported by NNSF of China No. 11901206.
LEI JIN AND YIXIAO QIAO Terminologies
This section is devoted to a brief review of all the fundamental definitions appearing inthe previous section. By a (topological) dynamical system we shall understand a pair(
X, T ), where X is a compact metrizable space and T : X → X is a homeomorphism. Animportant class of dynamical systems are minimal systems. A dynamical system ( X, T ) issaid to be minimal if for every point x ∈ X the set { T n x : n ∈ Z } is dense in X . Amongother typical examples of dynamical systems, probably the most canonical ones are the shift action on the Hilbert cubes ([0 , D ) Z (where D is a positive integer or + ∞ ), whichwe denote by (([0 , D ) Z , σ ), defined as follows: σ : ([0 , D ) Z → ([0 , D ) Z , ( x n ) n ∈ Z ( x n +1 ) n ∈ Z . For simplicity we assume D = 1. We say that a dynamical system ( X, T ) can be em-bedded in ([0 , Z , σ ) if there is an equivariant topological embedding f : X → [0 , Z ,namely, a homeomorphism of X into [0 , Z satisfying f ◦ T = σ ◦ f . Such a mapping f is usually called an embedding of the dynamical system ( X, T ) in ([0 , Z , σ ). As we haveseen in Section 1, this paper mainly focuses on the embeddability of minimal dynamicalsystems in ([0 , Z , σ ).We denote by dim( P ) the topological dimension (i.e. the Lebesgue covering dimension)of a compact metrizable space P (which is always assumed to be nonempty). Let X and P be two compact metrizable spaces and ρ a compatible metric on X . For ǫ > f : X → P is called an ǫ -embedding with respect to ρ if f ( x ) = f ( x ′ ) implies ρ ( x, x ′ ) < ǫ , for all x, x ′ ∈ X . Let Widim ǫ ( X, ρ ) be the minimum topological dimensiondim( P ) of a compact metrizable space P which admits an ǫ -embedding f : X → P withrespect to ρ . Remark 2.1.
We may verify that the topological dimension of X may be recovered bydim( X ) = lim ǫ → Widim ǫ ( X, ρ ).Let (
X, T ) be a dynamical system with a compatible metric ρ on X . For every positiveinteger n we define on X a compatible metric ρ n as follows: ρ n ( x, x ′ ) = max ≤ i X, T ) is defined bymdim( X, T ) = lim ǫ → lim n → + ∞ Widim ǫ ( X, ρ n ) n . It is well known that the limits in the above definition always exist, and the valuemdim( X, T ) is independent of the choices of a compatible metric ρ on X and a Følnersequence { F n } + ∞ n =1 (instead of { k ∈ Z : 0 ≤ k < n } + ∞ n =1 ) of Z . MINIMAL SUBSHIFT OF FULL MEAN DIMENSION 5 We note that the mean dimension of (([0 , D ) Z , σ ) is equal to D , where D is a positiveinteger or + ∞ . Clearly, if a dynamical system can be embedded in (([0 , D ) Z , σ ), thenits mean dimension must be less than or equal to D .3. Proof of Theorem 1.1 Construction of ( X, σ ) . We are going to construct a minimal dynamical system( X, σ ) such that X is a closed and shift-invariant subset of the Hilbert cube [0 , Z andthat the mean dimension of ( X, σ ) is equal to 1.First of all, we fix a compatible metric d on [0 , Z as follows: d ( x, x ′ ) = X n ∈ Z | x n − x ′ n | | n | , (cid:0) x = ( x n ) n ∈ Z , x ′ = ( x ′ n ) n ∈ Z ∈ [0 , Z (cid:1) . For x = ( x n ) n ∈ Z ∈ [0 , Z and two integers t ≤ t ′ we set x | t ′ t = ( x n ) t ≤ n ≤ t ′ . For a closedinterval I we denote its length by | I | .The construction will be fulfilled by induction. To start with, let us explain the intuitivemeaning of our notations shortly. For a nonnegative integer k we build in the k -th stepa closed shift-invariant subset X k of [0 , Z , which is generated from a “block” B k of theform Q b k j =1 I ( k ) j , where each I ( k ) j is a closed subinterval of [0 , 1] and where the positiveinteger b k indicates the “length” of the block B k . Roughly speaking, employing a familyof closed intervals in our construction is actually to increase dimension. But this is not enough for its mean dimension to be dominated from below. Therefore we are goingthrough the following approach: When we deal with the next block B k +1 in the ( k + 1)-thstep we have to “copy” the block B k sufficiently many times in order to occupy a ratherlarge proportion of positions of the block B k +1 . The positive integer r k is to describe thenumber of copies of the block B k in the block B k +1 , whereas for an integer 0 ≤ s < k + 1,the number η ( s, k + 1) > / s ( s +1) / in the block B k +1 , which will strictly decrease as k increases,while the number η ( s ) > η ( s, k + 1)is always under control as k + 1 ranges over all those integers greater than s , which willfinally converge to 1 as s goes to + ∞ . Step −∞ . We take a two-parameter sequence { η ( s, k ) : 0 ≤ s < k ; s, k ∈ Z } ofreal numbers and a one-parameter sequence { η ( s ) } + ∞ s =0 of real numbers satisfying all thefollowing conditions (the existence of such two sequences is obvious):(1) For any two integers 0 ≤ s < k : 0 < η ( s, k ) < 1; 0 < η ( s ) < s the one-parameter sequence { η ( s, k ) } + ∞ k = s +1 is strictly decreasing, namely η ( s, k + 1) < η ( s, k ) , ∀ ≤ s < k ( s, k ∈ Z ) . LEI JIN AND YIXIAO QIAO (3) For any nonnegative integer s the one-parameter sequence { η ( s, k ) } + ∞ k = s +1 is boundedby η ( s ) from below, i.e. η ( s, k ) > η ( s ) , ∀ ≤ s < k ( s, k ∈ Z ) . (4) η ( s ) → s → + ∞ .We fix them throughout this section. We would like to remind the reader that we shalluse these conditions implicitly in the sequel. Step 0. We take b = 1, B = I (0)1 = [0 , 1] and X = [0 , Z . Step 1. We divide I (0)1 = [0 , 1] equally into 2 closed subintervals of length 1 / I (1)1 , = [0 , / , I (1)1 , = [1 / , . We take a positive integer r with r r + 2 ≥ η (0 , . We put b = r + 2 and set B = ( B ) r × I (1)1 , × I (1)1 , ⊂ [0 , b . We rename the b closed subintervals of [0 , 1] appearing in the above product with indices,and rewrite B as follows: B = Q b j =1 I (1) j . More precisely, we notice here (in relation tothe indices) that B = I (1)1 and that the sets I (1) b − and I (1) b form a cover of B . Let X bethe set of all those x = ( x n ) n ∈ Z ∈ [0 , Z satisfying the following condition: • There exists some integer 0 ≤ l ≤ b − x | l + b ( m +1) l + b m +1 ∈ B for all m ∈ Z .Obviously, X is a nonempty closed and shift-invariant subset of X = [0 , Z .To proceed, we assume for a positive integer k that b k − , B k − and X k − have alreadybeen generated in Step ( k − b k , B k and X k . Step k . We write B k − = Q b k − j =1 I ( k − j , where every I ( k − j is a closed subinterval of[0 , ≤ j ≤ b k − we divide I ( k − j equally into 2 k closed subintervals of lengthequal to | I ( k − j | / k , which are denoted by I ( k ) j,i with 1 ≤ i ≤ k . We take a positive integer r k − sufficiently large such that for every integer 0 ≤ s ≤ k − (cid:16) { ≤ j ≤ b k − : | I ( k − j | ≥ / s ( s +1) / } (cid:17) · r k − b k − · ( r k − + 2 kb k − ) ≥ η ( s, k ) , where the symbol b k = b k − · ( r k − + 2 kb k − )and set B k = ( B k − ) r k − × Y i ,...,i bk − ∈{ ,..., k } b k − Y j =1 I ( k ) j,i j ⊂ ( B k − ) r k − +2 kbk − . MINIMAL SUBSHIFT OF FULL MEAN DIMENSION 7 We note that B k is a closed subset of [0 , b k . We rewrite B k as B k = Q b k j =1 I ( k ) j , where I ( k )1 , . . . , I ( k ) b k are closed subintervals of [0 , r k − +2 kbk − − [ m = r k − b k − ( m +1) Y j = b k − m +1 I ( k ) j = B k − = b k − Y j =1 I ( k ) j . We let X k be the set consisting of all the points x = ( x n ) n ∈ Z ∈ [0 , Z satisfying thefollowing condition: • There is some integer 0 ≤ l ≤ b k − x | l + b k ( m +1) l + b k m +1 ∈ B k for all m ∈ Z .It follows that X k is a (nonempty) closed and shift-invariant subset of X k − ⊂ [0 , Z . Step + ∞ . The induction has now been completed, as we have already generated b k , B k and X k for all nonnegative integers k . To end the construction, we finally take theintersection as follows: X = + ∞ \ k =0 X k . Since { X k } + ∞ k =0 is a decreasing sequence of nonempty closed shift-invariant subsets of [0 , Z , X is a nonempty closed shift-invariant subset of [0 , Z as well. Thus, ( X, σ ) becomesa dynamical system. In what follows we need verify that it does satisfy the requiredconditions.3.2. Minimality of ( X, σ ) . We will show that the dynamical system ( X, σ ) is minimal.In fact, it suffices to show that for any x, y ∈ X and any ǫ > M ∈ Z such that d ( σ M ( x ) , y ) < ǫ . Let us fix x = ( x n ) n ∈ Z , y = ( y n ) n ∈ Z ∈ X and ǫ > k we set M k = max n | I ( k ) j | : b k − · r k − + 1 ≤ j ≤ b k o . This sequence { M k } + ∞ k =1 is deceasing and converges to 0 as k goes to + ∞ . Actually, itfollows from the construction that the value M k is equal to 1 / k .We choose a positive integer L depending only on ǫ > x ′ =( x ′ n ) n ∈ Z and x ′′ = ( x ′′ n ) n ∈ Z coming from [0 , Z satisfy that | x ′ n − x ′′ n | < ǫ/ − L ≤ n ≤ L , then they will satisfy d ( x ′ , x ′′ ) < ǫ .We take an integer N > L + 1 with 1 / N < ǫ/ 2. Let us look at the N -th, ( N + 1)-thand ( N − x, y ∈ X ⊂ X N , there must exist two integers 0 ≤ p, q ≤ b N − x | p + b N ( m +1) p + b N m +1 , y | q + b N ( m +1) q + b N m +1 ∈ B N , ∀ m ∈ Z . LEI JIN AND YIXIAO QIAO There are two integers L ∈ [ − L, L ] and m ∈ {− , } , both of which are uniquelydetermined, satisfying that[ − L, L − ⊂ [ q + b N ( m − 1) + 1 , q + b N m ] , [ L , L ] ⊂ [ q + b N m + 1 , q + b N ( m + 1)] . Here we assume by convention [ − L, − L − 1] = ∅ .Let us make it clearer with a short remark. Strictly speaking, the general case is where L ∈ [ − L + 1 , L ] (and hence where we will have L = q + b N m + 1), with an exception(which turns out to be simpler than the general case) if such an integer L ∈ [ − L + 1 , L ]does not exist, for which we set L = − L . As we will see in a moment, the exceptionalcase is contained in (a part of) the general case. Therefore we assume without loss ofgenerality − L + 1 ≤ L ≤ L .Since y | q + b N ( m +1) q + b N m +1 ∈ B N = r N +2 ( N +1) bN − [ m = r N b N ( m +1) Y j = b N m +1 I ( N +1) j , there is an integer r N ≤ m ≤ r N + 2 ( N +1) b N − y | q + b N ( m +1) q + b N m +1 ∈ b N ( m +1) Y j = b N m +1 I ( N +1) j . Since x ∈ X ⊂ X N +1 and since B N +1 = ( B N ) r N × b N +1 Y j = b N r N +1 I ( N +1) j , there is some integer m depending on m such that x | p + b N ( m +1) p + b N m +1 ∈ b N ( m +1) Y j = b N m +1 I ( N +1) j . Thus, for every integer 1 ≤ i ≤ b N | x p + b N m + i − y q + b N m + i | ≤ M N +1 . This implies that for every integer n ∈ [ q + b N m + 1 , q + b N ( m + 1)], in particular, forevery integer n ∈ [ L , L ] | σ b N ( m − m )+ p − q ( x ) n − y n | ≤ M N +1 = 1 / N +1 < ǫ/ . Since σ b N ( m − m )+ p − q ( x ) | q + b N m q + b N ( m − , y | q + b N m q + b N ( m − ∈ B N and since 2 L + 1 ≤ N − ≤ b N − , MINIMAL SUBSHIFT OF FULL MEAN DIMENSION 9 we deduce that for every integer n ∈ [ − L, L − | σ b N ( m − m )+ p − q ( x ) n − y n | ≤ M N = 1 / N < ǫ/ . Thus, we conclude that for all integers n ∈ [ − L, L ] | σ b N ( m − m )+ p − q ( x ) n − y n | < ǫ/ , which implies d ( σ b N ( m − m )+ p − q ( x ) , y ) < ǫ. Mean dimension of ( X, σ ) . We shall show that mdim( X, σ ) = 1. Since it isobvious that the dynamical system ( X, σ ) has been embedded in ([0 , Z , σ ) already, itsmean dimension does not exceed 1. Thus, it suffices to prove mdim( X, σ ) ≥ 1. This willend the paper.We borrow a practical lemma [Gro99, Lemma 1.1.1][LT14, Example 2.1] as follows. Thepoint which will be crucial to our argument is that the equality presented in this lemmais true for all positive integers n and all sufficiently small δ > strictly less than τ > Lemma 3.1. For any < δ < τ and any positive integer n Widim δ ([0 , τ ] n , d l ∞ ) = n. Here d l ∞ is the compatible metric on [0 , τ ] n defined by d l ∞ (( x i ) ni =1 , ( x ′ i ) ni =1 ) = max ≤ i ≤ n | x i − x ′ i | , ∀ ( x i ) ni =1 , ( x ′ i ) ni =1 ∈ [0 , τ ] n . We fix a point z ∈ X satisfying that z | b k ( m +1) − b k m ∈ B k for any nonnegative integer k and any m ∈ Z . For each nonnegative integer k we define a mapping as follows: F k : B k → X, x = ( x n ) b k − n =0 F k ( x ) = ( F k ( x ) n ) n ∈ Z ,F k ( x ) n = x n , ≤ n ≤ b k − z n , n ∈ Z \ [0 , b k − . It is clear that F k ( x ) (where x ∈ B k ) is indeed in X . Moreover, the mapping F k : B k → X is continuous and distance-increasing with respect to the compatible metric d l ∞ on B k and the compatible metric d b k on X , i.e. it satisfies that d l ∞ ( x, x ′ ) ≤ d b k ( F k ( x ) , F k ( x ′ )) , ∀ x, x ′ ∈ B k . It follows that for every ǫ > k Widim ǫ ( B k , d l ∞ ) ≤ Widim ǫ ( X, d b k ) . Now let us take ǫ > s = s ( ǫ ) dependingonly on ǫ > / ( s +1)( s +2) / ≤ ǫ < / s ( s +1) / . Note that for all integers k > s { ≤ j ≤ b k : | I ( k ) j | ≥ / s ( s +1) / } b k ≥ η ( s, k ) . By Lemma 3.1 this implies that for all integers k > s Widim ǫ ( B k , d l ∞ ) b k ≥ η ( s, k ) ≥ η ( s ) . Thus, for all integers k > s Widim ǫ ( X, d b k ) b k ≥ η ( s ) . Since ǫ > s → + ∞ as ǫ → 0, we conclude withmdim( X, σ ) = lim ǫ → lim k → + ∞ Widim ǫ ( X, d b k ) b k ≥ lim ǫ → η ( s ) = 1 . References [GQT19] Yonatan Gutman, Yixiao Qiao, Masaki Tsukamoto. Application of signal analysis to the em-bedding problem of Z k -actions. Geometric and Functional Analysis 29 (2019), 1440–1502.[Gro99] Misha Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps:I. Math. Phys. Anal. Geom. 2 (1999), 323–415.[GT20] Yonatan Gutman, Masaki Tsukamoto. Embedding minimal dynamical systems into Hilbertcubes. Inventiones Mathematicae 221 (2020), 113–166.[LT14] Elon Lindenstrauss, Masaki Tsukamoto. Mean dimension and an embedding problem: anexample. Israel Journal of Mathematics 199 (2014), 573–584.[LW00] Elon Lindenstrauss, Benjamin Weiss. Mean topological dimension. Israel Journal of Mathe-matics 115 (2000), 1–24. 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 ::