On asymptotic power dimension

Topology and its Applications 201 (2016) 124–130

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Topology and its Applications www.elsevier.com/locate/topol

On asymptotic power dimension Jacek Kucab a , Mykhailo Zarichnyi b,a,∗ a

Faculty of Mathematics and Natural Sciences, University of Rzeszów, Al. Rejtana 16A, 35-959 Rzeszów, Poland b Department of Mechanics and Mathematics, Lviv National University, Universytetska Str. 1, 79000 Lviv, Ukraine

a r t i c l e

i n f o

Article history: Received 31 December 2014 Accepted 8 June 2015 Available online 23 December 2015 MSC: 54F45 54B10 54B20

a b s t r a c t We consider the asymptotic power dimension, i.e., the asymptotic dimension with control power function. It is proved that this dimension is invariant under the coarse bi-Hölder transformations. We also establish estimates for the power asymptotic dimension of the symmetric and hypersymmetric products. © 2015 Elsevier B.V. All rights reserved.

Keywords: Assouad–Nagata dimension Asymptotic dimension Asymptotic power dimension Hölder map Symmetric power

1. Introduction

The asymptotic Assouad–Nagata dimension of a metric space was introduced in [3]. By the definition, the asymptotic Assouad–Nagata dimension of a metric space (X, ρ) does not exceed n (asdimAN X ≤ n) if there exist c > 0 and r0 > 0 such that for all R > r0 one can find R-disjoint and cR-bounded families U0 , U1 , . . . , Un n of subsets of X such that i=0 Ui covers X (see the details below). The asymptotic Assouad–Nagata dimension is investigated by many authors (see, e.g., [1,2,8]). In this paper we consider an analogous concept to this dimension. Namely, in our case the control function is the power function x → xα , where α > 0. One of the motivations of this notion, in addition to its naturalness, consists in its relations to the important class of Hölder maps. The obtained dimension is in

* Corresponding author. E-mail addresses: [email protected] (J. Kucab), [email protected] (M. Zarichnyi). http://dx.doi.org/10.1016/j.topol.2015.12.031 0166-8641/© 2015 Elsevier B.V. All rights reserved.

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between the asymptotic dimension and the asymptotic Assouad–Nagata dimension. An example is provided of a space with these three dimensions different. We also establish estimates for the power asymptotic dimension of the symmetric and hypersymmetric products. 2. Dimensions with power functions controlling diameter A family A in a metric space (X, d) is said to be D-disjoint, where D > 0, if d(A, B) = inf{d(a, b) | a ∈ A, b ∈ B} ≥ D, for every distinct A, B ∈ A. A family A is called B-bounded, for B > 0, if mesh(A) = sup{diam(A) | A ∈ A} ≤ B. By Br (x) we denote the ball of radius r centered at x. Let (X, ρ) be a metric space and let n be a nonnegative integer. We use the notation (λ, B)-dimX ≤ n in the meaning that there exist λ-disjoint and B-bounded families n U0 , U1 , . . . , Un of subsets of X such that i=0 Ui is a cover of X. If for every λ > 0 there exists B(λ) > 0 such that (λ, B)-dimX ≤ n, we say that λ → B(λ) is an n-dimensional control function for X. Definition 2.1. We say that the power dimension of X does not exceed n (asdimP X ≤ n) if there exist α > 0 and r0 > 0 such that (r, rα )-dimX ≤ n for all r > r0 . As usual we say that asdimP X = n if asdimP X ≤ n and it is not true that asdimP X ≤ n − 1. Theorem 2.2. For any metric space X the following inequality holds: asdimP X ≤ asdimAN X. Proof. Let asdimAN ≤ n and c and r0 fulfill the definition. Let α = 2 and λ0 = max(r0 , c). Then for R > λ0 we have (R, cR)-dimX ≤ n and so (R, Rα )-dimX ≤ n as R2 > cR. That means that asdimP X ≤ n, so asdimP X ≤ asdimAN X. 2 It is well-known that, for any metric d on a set X, the function d = ln(1 + d) is also a metric on X. The following result is, in some sense, a special case of [8]. Theorem 2.3. Let (X, d) be a metric space. Then asdimP (X, d) = asdimAN (X, d ). Proof. Suppose that asdimP (X, d) ≤ n, then there exist R0 > 0 and α > 0 such that (R, Rα )-dim(X, d) ≤ n. Without loss of generality, one may assume that α ≥ 1. Let r0 = ln(1 + R0 ). Using the inequality 1 + Rα ≤ (1 + R)α , one easily concludes that (X, d ) is (r, αr)-dim(X, d ) ≤ n. Therefore, asdimP (X, d) ≥ asdimAN (X, d ). The reverse inequality can be proved by similar arguments. 2 Remark 2.4. Note that the obtained metric space (X, d ) is hyperbolic in the sense of Gromov [4]. Remark 2.5. Theorem 2.3 does not mean that one can immediately obtain the properties of the asymptotic power dimension out of those of the asymptotic Assouad–Nagata dimension. The reason is that, in general, there is no way from the latter to the former. Given A ⊂ X, we denote by Br (A) the r-neighborhood of A, i.e., Br (A) = ∪{Br (x) | x ∈ A}.

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Lemma 2.6. Let (X, ρX ) be a metric space and let Y ⊂ X be such that X ⊂ BK (Y ), where K > 0. Then asdimP X = asdimP Y . Proof. Clearly, asdimP Y ≤ asdimP X. Let now asdimP Y ≤ n, for a nonnegative integer n. Then there exist α > 0 and r0 > 0 such that for all R > r0 one can find R-disjoint and Rα -bounded families U0 , U1 , . . . , Un of subsets of Y such that n ˜ = α + 1 and let r˜0 > r0 be large enough that (R + 2K)α + 2K ≤ Rα+1 for all i=0 Ui covers Y . Let α R > r˜0 . Let R > r˜0 . Then R + 2K > r0 . There exist families U0 , U1 , . . . , Un that are (R + 2K)-disjoint, n n (R + 2K)α -bounded and i=0 Ui covers Y . Let Vi = {BK (U ) | U ∈ Ui }, i = 0, 1, . . . , n. Then i=0 Vi covers X and for every i ∈ {0, 1, . . . , n}, the family Vi is R-disjoint and mesh(Vi ) ≤ (R + 2K)α + 2K ≤ Rα+1 = Rα˜ . Therefore asdimP X ≤ n.

2

Lemma 2.7. Let (X, ρ) be a metric space, D > 0 and let ρ˜(x, y) = max(ρ(x, y), D) for every different x, y ∈ X (and obviously ρ˜(x, x) = 0 for every x ∈ X). Then asdimP (X, ρ) = asdimP (X, ρ˜). Proof. Let asdimP (X, ρ) ≤ n for some nonnegative integer n and let α and r0 be as in the definition. Let r˜0 = max(r0 , D1/α ). Let R > r˜0 . Since R > r0 , there exist families U0 , U1 , . . . , Un of subsets of X such that n α ρ)disjoint. i=0 Ui covers X and every Ui is R-(ρ)disjoint and R -(ρ)bounded. Obviously, they are also R-(˜ 1/α Let x, y ∈ U ∈ Ui for any i ∈ {0, 1, . . . , n}. As R > D , we have Rα > D. This combined with ρ(x, y) < Rα means that ρ˜(x, y) < Rα . Therefore Ui is Rα -(˜ ρ)bounded. Hence asdimP (X, ρ˜) ≤ n. Let now asdimP (X, ρ˜) ≤ n for some nonnegative integer n and let α and r0 fulfill the definition. Let n r˜0 = r0 + D. Let R > r˜0 . As R > r0 , there exist families U0 , U1 , . . . , Un of subsets of X such that i=0 Ui covers X and every of them is R-(˜ ρ)disjoint and Rα -(˜ ρ)bounded. Obviously they are also Rα -(ρ)bounded. Let x ∈ U and y ∈ V for any different U, V ∈ Ui and any i ∈ {0, 1, . . . , n}. Then, since ρ˜(x, y) > R > D we have ρ˜(x, y) = ρ(x, y) and therefore ρ(x, y) > R, i.e., Ui is R-(ρ)disjoint. Hence asdimP (X, ρ) ≤ n. 2 Example 2.8. There exists a metric space X with asdim X = 0, asdimP X = 1 and asdimAN X = 2. We let X = ∪∞ i=1 Xi , where Xi = {(im + ci , in) | m, n ∈ N, m ≤ 2i , n ≤ i2 } i−1 and ci = (i2 + i − 2)/2 + k=1 k2k , i ∈ N. The numbers ci are chosen so that the distance between Xi and Xi−1 is i. The metric on X is inherited from R2 . Let Yi = ∪ik=1 Xk . 1) asdim X = 0. Indeed, given i ∈ N, one can cover X by the family {Yi } ∪ {{x} | x ∈ X \ Yi }, which is obviously i-disjoint and uniformly bounded. 2) asdimP X = 1. Let D > 0, D ∈ [i − 1, i], where i ∈ N. Then Yi−1 lies in a strip of the form [0, a] × [0, (i − 1)3 ], where a ≥ i2i . Clearly, the strip and also Yi−1 can be covered by two i-disjoint families consisting of disjoint squares of size [0, (i − 1)3 ] × [0, (i − 1)3 ]. Let us denote these families by U0 and U1 . Put U˜0 = U0 ∪ {{x} | x ∈ X \ Yi−1 }. Then U˜0 and U1 are i-disjoint (i − 1)3 -bounded families whose union is a cover of X. This demonstrates that asdimP X ≤ 1. Simple geometric arguments demonstrate that, given α > 0, for natural i and s large enough, the “rectangle” Xi−1 cannot be covered by any is-disjoint (is)α -bounded family. This proves the reverse inequality. 3) asdimAN X = 2. Since X ⊂ R2 , we conclude that asdimAN X ≤ asdimAN R2 = 2. Suppose that asdimAN X < 2. Then there exist c > 0 and r0 > 0 such that, for every r ≥ r0 , one can find cr-bounded r-disjoint families Ur0 , Ur1 of subsets of X that together cover X. We will use the following fact from the

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dimension theory that is a special case of Ostrand’s characterization of the covering dimension (see [6]): for ε small enough, the unit square cannot be covered by two disjoint families of mesh < ε. There exist r ≥ r0 and natural i ≥ 2 such that √ √ 2i 2 < r < (εi3 − 2i 2)/c. √ ¯ √ (U ) | U ∈ U s }, s = 0, 1, are disjoint and of mesh < cr + 2i 2 < εi3 (here Then the families Vrs = {B r i 2 ¯t (C) stands for the closed t-neighborhood of C in R2 ). The union of the families Vr0 and Vr1 covers a B rectangle of size i2i × i3 in R2 and therefore any square of side i3 in it. This contradicts to our choice of ε. Thus, asdimAN X = 2. 3. Hölder maps and asymptotic power dimension One of the motivations of considering the power dimension is that this dimension is tightly connected with an important class of maps, namely, the Hölder maps. Definition 3.1. Let (X, ρ) and (Y, d) be metric spaces. A map f : X → Y is called large scale Hölder if there exist L > 0 and C > 1 such that d(f (x), f (y)) ≤ ρ(x, y)L + C for any x, y ∈ X. Given a metric space (Y, d), we say that a map f : X → Y is of finite distance from a map g : X → Y if sup{d(f (x), g(x)) | x ∈ X} < ∞. Definition 3.2. Metric spaces X and Y are called large scale Hölder isomorphic if there are large scale Hölder maps f : X → Y and g : Y → X such that the maps gf : X → X and f g : Y → Y are of finite distance to the identity maps 1X and 1Y respectively. Clearly, every Hölder map is large scale Hölder. Given D > 0, we say that a subset Y of a metric space (X, ) is D-discrete, if (x, y) ≥ D, for every distinct x, y ∈ Y . Proposition 3.3. Let f : X → Y be a large scale Hölder map of metric spaces (X, ) and (X, d). There exists D > 0 such that the restriction of f onto any D-discrete subspace of X is a Hölder map. Proof. Let L and C be as in Definition 3.1. Let D > 0 be such that DL (D − 1) ≥ C. Then for any x, y ∈ X such that (x, y) ≥ D we obtain (x, y)L+1 = (x, y)L + (x, y)L ((x, y) − 1) ≥ (x, y)L + DL (D − 1) ≥ (x, y)L + C ≥ d(f (x), f (y)).

2

Theorem 3.4. Let (X, ρ) and (Y, d) be large scale Hölder equivalent metric spaces. Then asdimP X = asdimP Y . Proof. Suppose that asdimP X ≤ n for some nonnegative integer n and let α and r0 be taken from Definition 2.1. Suppose that maps f : X → Y and g : Y → X are as in Definition 3.2, i.e., there exist K, R.λ, μ, l, m > 0 such that (x, gf (x)) < K, d(y, f g(y)) < R, d(f (x), f (x )) ≤ (x, x )λ + l, (g(y), g(y  )) ≤ d(y, y  )μ + m, for all x, x ∈ X and y, y  ∈ Y . Let Uri , i = 0, 1, . . . , n, be r-disjoint families of mesh ≤ rα such that ∪ni=0 Uri is a cover of X. Let Vri = {g −1 (U ) | U ∈ Uri }, i = 0, 1, . . . , n. Clearly, ∪ni=0 Vri is a cover of Y . We are going to estimate the mesh of Vri , i = 0, 1, . . . , n. Without loss of generality, we may suppose that r0 ≥ max{l + 2R, 4, 2m}. Let U ∈ Uri and x, y ∈ g −1 (U ). Then

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d(x, y) ≤ d(f g(x), x) + d(f g(x), f g(y)) + d(f g(y), y) ≤ d(f g(x), f g(y)) + 2r ≤ (g(x), g(y)λ + l + 2R ≤ rαλ + r ≤ 2rmax{αλ,1} ≤ rmax{αλ,1}+1 . Now, suppose that x ∈ g −1 (U ), y ∈ g −1 (V ), where U, V ∈ Uri , U = V . Then r ≤ (g(x), g(y)) ≤ d(x, y)μ + m and, as r ≥ r0 ≥ 2m, we see that d(x, y) ≥ (r/2)1/μ . Since r0 ≥ 4, for r ≥ r0 , (r/2)1/μ ≥ r1/(2μ) and, finally, d(x, y) ≥ r1/(2μ) . 1/(2μ) Let s = r1/(2μ) , s0 = r0 . We have shown that the family Vri is s-discrete and of mesh ≤ s(max{αλ,1}+1)/(2μ) . We conclude that asdimP Y ≤ n. Therefore, asdimP Y ≤ asdimP X. The reverse inequality is established similarly. 2 4. Symmetric and hypersymmetric products Recall that, for any subgroup G of the symmetric group Sn , the G-symmetric product SPGn X is defined as follows. Denote by ∼ the following equivalence relation on X n : (x1 , . . . , xn ) ∼ (y1 , . . . , yn ) if there exists σ ∈ G such that yi = xσ(i) , for every i = 1, . . . , n. Denote by [x1 , . . . , xn ] the equivalence class containing (x1 , . . . , xn ). Then SPGn X = {[x1 , . . . , xn ]|(x1 , . . . , xn ) ∈ X n } = X n /G. The metric d on SPGn X is defined as follows:  1 , . . . , xn ], [y1 , . . . , yn ]) = min max d(xi , yσ(i) ). d([x σ∈G 1≤i≤n

Note that, in the case of trivial group G, one obtains the l∞ -metric on the space SPGn X = X n . Let (X, d) be a metric space. The nth hypersymmetric product of X is the set expn X of all nonempty subsets of X of cardinality ≤ n. We endow expn X with the Hausdorff metric dH , dH (C, D) = inf{r > 0 | C ⊂ Or (D), D ⊂ Or (C)}. The following theorem is a counterpart of a result from [7]. In its proof we will use the Kolmogorov trick (see, e.g., [2]). Given a metric space X and k ≥ n + 1 ≥ 1, we say that a function DX : R+ → R+ is an (n, k)-dimensional control function for X if for any r > 0 there is a family U = ∪ki=1 Ui satisfying the following conditions: (1) each Ui is r-disjoint, (2) each Ui is DX (r)-bounded, (3) each element x ∈ X belongs to at least k − n elements of U (equivalently, ∪i∈T Ui is a cover of X for every subset T of {1, . . . , k} consisting of n + 1 elements). The following is proved in [2]. Theorem 4.1. If D(n+1) X is an n-dimensional control function of X and one defines a sequence of functions (i) (i+1) (i) (k) {DX }i≥n+1 inductively by DX (r) = DX (3r)+2r for all i ≥ n+1, then each DX is an (n, k)-dimensional control function of X for all k ≥ n + 1. (n+1)

Corollary 4.2. If an n-dimensional control function DX (r) = rα1 , r ≥ 1, is a power function, then, for all k ≥ n + 1 there exists a power function rαk , r ≥ 3, which is an (n, k)-dimensional control function of X. Proof. Without loss of generality, one may assume that α1 ≥ 1. Use induction. Given αi , we obtain (i+1)

DX

(i)

(r) = DX (3r) + 2r = 3αi rαi + 2r ≤ r2αi +1 = rαi+1 .

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Theorem 4.3. Let (X, d) be a metric space of finite asymptotic power dimension. Then asdimP SPGn (X) ≤ n asdimP X. Proof. Suppose that asdimP (X, d) = m. There exists r0 ≥ 1 and an m-dimensional control power function K (m+1) (r) = rα1 , r ≥ r0 , of X. By Corollary 4.2, there exists an (m, mn + 1)-dimensional control function of X of the form K mn+1 (r) = rαmn+1 , r ≥ k0 , for some αmn+1 > 0 and k0 > 0. Consider any r > k0 , and choose r-disjoint rαmn+1 -bounded families U0 , U1 , . . . , Unm such that any m + 1 families cover X. In particular, for each x1 , . . . , xn ∈ X, there exists i ∈ {0, 1, . . . , mn} such that {x1 , . . . , xn } ⊂ Xi = ∪Ui . n Therefore, SPGn (X) = ∪mn i=0 SPG (Xi ). Let U be a disjoint family of subsets of X. We say that [x1 , . . . , xn ], [y1 , . . . , yn ] ∈ SPGn (X) are U-near, if there exists σ ∈ G with the following property: for every i there exists U ∈ U such that {xi , yσ(i) } ⊂ U . Clearly, the U-nearness is an equivalence relation on SPGn (X). We denote by U the family of equivalence classes of this relation. s , s = 0, 1, . . . , mn, is r-disjoint. Consider [x1 , . . . , xn ], [y1 , . . . , yn ] ∈ We first show that every family U ˆ 1 , . . . , xn ], [y1 , . . . , yn ]) < r. The latter means that there exists σ ∈ G such that s with d([x ∪U max{d(xi , yσ(i) ) | i = 1, . . . , n} < r. Since d(xi , yσ(i) ) < r, for every i, the r-disjointness of Us implies s . that [x1 , . . . , xn ], [y1 , . . . , yn ] belong to the same element of U  Now, we show that every family Us , s = 0, 1, . . . , mn, is rαmn+1 -bounded. Suppose that [x1 , . . . , xn ], s . Then there exists σ ∈ G with the following property: for every [y1 , . . . , yn ] belong to an element of U ˆ i ∈ {1, . . . , n} there exists Ui ∈ Us such that {xi , yσ(i) } ⊂ Ui . By the definition of the metric d, ˆ 1 , . . . , xn ], [y1 , . . . , yn ]) ≤ max{d(xi , yσ(i) ) | i = 1, . . . , n} < rαmn+1 , d([x and we are done. s , where s = 0, 1, . . . , mn, form a cover of SP n (X). Thus, the r-disjoint, rαmn+1 -bounded families U G n Therefore, asdimP SPG (X) ≤ mn. 2 Remark 4.4. Our proof follows the line of that of the main result in [7]. Our case is even simpler, as we have an explicit expression for the metrics on the symmetric powers. Remark 4.5. If G is a trivial group, Theorem 4.3 becomes the product theorem for the asymptotic power dimension. The following result can be proved similarly. Theorem 4.6. Let (X, d) be a metric space of finite asymptotic power dimension. Then asdimP expn X ≤ n asdimP X. Remark 4.7. The counterparts of Theorems 4.3 and 4.6 can be similarly proved for the asymptotic Assouad– Nagata dimension. Note that in [9], the result concerning the Assouad–Nagata dimension was proved for the symmetric power SP n (i.e., the case G = Sn ). The proof uses the characterization of asymptotic dimension in terms of maps into polyhedra. 5. Remarks and open questions One can find equivalent characterizations of the asymptotic power dimension in the spirit of [3, Proposition 1.7], in particular, in terms of maps into simplicial complexes. Question 5.1. Is there a finitely generated group with different dimensions asdim, asdimP , and asdimAN ?

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Dranishnikov and Smith [3] established connections between the asymptotic Assouad–Nagata dimension of a proper metric space and the covering dimension of the sublinear corona. In [5], the authors introduced the subpower coronas of proper metric spaces. Question 5.2. Do the asymptotic power dimension of a proper metric space and the covering dimension of the subpower corona of this space coincide? Acknowledgement The authors are grateful to the referee for his/her considerable improvement of the text of the manuscript. References [1] G. Bell, A. Dranishnikov, Asymptotic dimension, Topol. Appl. 155 (12) (2008) 1265–1296. [2] N. Brodskiy, J. Dydak, M. Levin, A. Mitra, Hurewicz theorem for Assouad–Nagata dimension, J. Lond. Math. Soc. 77 (3) (2008) 741–756. [3] A.N. Dranishnikov, J. Smith, On asymptotic Assouad–Nagata dimension, Topol. Appl. 154 (4) (2007) 934–952. [4] Mikhail Gromov, Hyperbolic groups, in: Essays in Group Theory, in: Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. [5] Jacek Kucab, Mykhailo Zarichnyi, Subpower Higson corona of a metric space, Algebra Discrete Math. 17 (2) (2014) 280–287. [6] Phillip A. Ostrand, Dimension of metric spaces and Hilbert’s problem 13, Bull. Am. Math. Soc. 71 (4) (1965) 619–622. [7] T.M. Radul, O. Shukel’, Functors of finite degree and asymptotic dimension, Mat. Stud. 31 (2009) 204–206. [8] Damian Sawicki, Remarks on coarse triviality of asymptotic Assouad–Nagata dimension, Topol. Appl. 167 (2014) 69–75. [9] Oksana Shukel’, Mykhaylo Zarichnyi, Asymptotic dimension of symmetric powers, Math. Bull. 5 (2008) 304–310.