Ruxi Shi
email: ruxi.shi@upmc.fr

Masaki Tsukamoto
email: tsukamoto@math.kyoto-u.ac.jp

APPLICATION OF WAIST INEQUALITY TO ENTROPY AND MEAN DIMENSION

Keywords: 2020 Mathematics Subject Classification. 37B99, 54F45 Dynamical system, waist inequality, topological conditional entropy, mean dimension, metric mean dimension, conditional metric mean dimension

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps π : (X, T ) → (Y, S) between dynamical systems and assume that the mean dimension of the domain (X, T ) is larger than the mean dimension of the target (Y, S). We exhibit several situations for which the maps π necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss about minimal dynamical systems nonembeddable in [0, 1] Z .

1. Introduction 1.1. Topological conditional entropy. Waist inequality is a deep geometric-topological inequality which incorporates the ideas of isoperimetric inequality with the Borsuk-Ulam theorem in a highly nontrivial way. It states that, for n ≥ m, every continuous map f : S n → R m has some fiber of "large volume" in an appropriate sense. We will review it in Section 2 below. Its origin goes back to the work of Almgren [START_REF] Almgren | The theory of varifolds -a variational calculus in the large for the k-dimensional area integrated[END_REF] on geometric measure theory. Gromov mentioned a version of the waist inequality in his famous "filling" paper [Gro83, Appendix I (F)]. Later he developed its full theory in [START_REF] Gromov | Isoperimetry of waists and concentration of maps[END_REF].

Variants of the waist inequality have appeared in several interesting subjects: convex geometry [Ver06, [START_REF] Giannopoulos | Asymptotic formulas for the diameter of sections of symmetric convex bodies[END_REF][START_REF] Klartag | Convex geometry and waist inequalities[END_REF], generalizations of Morse theory [START_REF] Gromov | Singularities, expanders and topology of maps. part 1: homology versus volume in the spaces of cycles[END_REF], combinatorics [START_REF] Gromov | Singularities, expanders and topology of maps. part 2: from combinatorics to topology via algebraic isoperimetry[END_REF], knots [START_REF] Pardon | On the distortion of knots on embedded surfaces[END_REF][START_REF] Gromov | Generalizations of the Kolmogorov-Barzdin embedding estimates[END_REF] and complex analysis [START_REF] Klartag | Eldan's stochastic localization and tubular neighborhoods of complex-analytic sets[END_REF]. The purpose of this paper is to further add a new item to this list; we explain applications of the waist inequality to topological dynamics.

Our starting point is the theory of topological conditional entropy. We first quickly prepare basic definitions. Details about topological conditional entropy can be found in the book of Downarowicz [START_REF] Downarowicz | Entropy in dynamical systems[END_REF]Section 6].

A pair (X, T ) is called a dynamical system if X is a compact metrizable space and T : X → X is a homeomorphism. Let d be a metric on X. For a natural number N we define a new metric d N on X by d N (x, y) = max 0≤n<N d (T n x, T n y) .

For a positive number ε and a subset E ⊂ X we denote by # (E, d N , ε) the minimum number of n such that there exists an open covering {U 1 , . . . , U n } of E satisfying Diam(U i , d N ) < ε for all 1 ≤ i ≤ n. Here Diam(U i , d N ) is the diameter of U i with respect to the metric d N . When E = ∅, we set # (E, d N , ε) = 0.

We define the topological entropy of (X, T ) by

h top (X, T ) = lim ε→0 lim N →∞ log # (X, d N , ε) N .
This is a topological invariant of dynamical systems. Namely its value is independent of the choice of the metric d. For a closed subset E ⊂ X we also define

h top (E, T ) = lim ε→0 lim sup N →∞ log # (E, d N , ε) N .
Let (X, T ) and (Y, S) be dynamical systems. Let π : X → Y be an equivariant continuous map. Here "equivariant" means π • T = S • π. We often denote such a map by π : (X, T ) → (Y, S) for clarifying the maps T and S. Let d be a metric on X. We define the topological conditional entropy of π by

h top (π, T ) = lim ε→0 lim N →∞ sup y∈Y log # (π -1 (y), d N , ε) N .
This definition is due to [Dow11, Lemma 6.8.2]. It coincides with the supremum of fiberwise entropy [Dow11, Theorem 6.8.3]:

h top (π, T ) = sup y∈Y h top π -1 (y), T .

In the book of Downarowicz [START_REF] Downarowicz | Entropy in dynamical systems[END_REF] the topological conditional entropy h top (π, T ) is denoted by h(T |S).

Bowen [START_REF] Bowen | Entropy for group endomorphisms and homogeneous spaces[END_REF]Theorem 17] proved that for any equivariant continuous map π : (X, T ) → (Y, S) we have

(1•1) h top (X, T ) ≤ h top (Y, S) + h top (π, T ).
This inequality has a clear intuitive meaning; if (X, T ) is "larger than" (Y, S) then some fiber of π must be "large". We are interested in the case that both h top (X, T ) and h top (Y, S) are infinite. In this case Bowen's inequality (1•1) provides no information about the fibers of π. Typical examples of such infinite entropy systems are given by shifts on infinite dimensional cubes. Let a be a natural number, and let [0, 1] a be the a-dimensional cube. We consider the bi-infinite product of [0, 1] a :

([0, 1] a ) Z = • • • × [0, 1] a × [0, 1] a × [0, 1] a × • • • . We define the shift σ : ([0, 1] a ) Z → ([0, 1] a ) Z by σ ((x n ) n∈Z ) = (x n+1 ) n∈Z , (each x n ∈ [0, 1] a ).
The pair ([0, 1] a ) Z , σ is a dynamical system which has infinite topological entropy. This system is called the full-shift on the alphabet [0, 1] a . (Later we will often consider two full-shifts on different alphabets simultaneously. But we use the same letter σ for shift maps of both systems. Hopefully this will not cause any confusion.)

The following theorem is a starting point of our investigation.

Theorem 1.1. Let a > b be two natural numbers. Let ([0, 1] a ) Z , σ and [0, 1] b Z , σ
be the full-shifts on the alphabets [0, 1] a and [0, 1] b respectively. Then for any equivariant continuous map π :

([0, 1] a ) Z → [0, 1] b Z we have h top (π, σ) = +∞.
This theorem has the same spirit as that of Bowen's inequality (1•1): Since we assumed a > b, the shift ([0, 1] a ) Z is "larger than" the shift [0, 1] b Z , and hence some fiber of π : ([0, 1] a ) Z → [0, 1] b Z must be "large". However this result does not follow from Bowen's inequality (1•1) because both the shifts ([0, 1] a ) Z and [0, 1] b Z have infinite topological entropy.

Theorem 1.1 might look an innocent statement. However we can add it a small twist so that it becomes more interesting: Theorem 1.2. Let a > b be two natural numbers. There exists a minimal closed subset X of ([0, 1] a ) Z , σ such that for any equivariant continuous map π : (X, σ)

→ [0, 1] b Z , σ we have h top (π, σ) = +∞.
Here "minimal" means that σ(X) = X and that every orbit of (X, σ) is dense in X. Letting a = 2 and b = 1 in the theorem, we get a minimal closed subset X of the shift ([0, 1] 2 )

Z such that for any equivariant continuous map π : X → [0, 1] Z we have

h top (π, σ) = +∞.
In particular, (X, σ) does not embed in the shift [0, 1] Z . This provides an answer to Auslander's question in [START_REF] Auslander | Minimal flows and their extensions[END_REF]p. 193] from a new angle; Auslander asked whether there exists a minimal dynamical system which does not embed in the shift [0, 1] Z . This problem was first solved in the celebrated paper of Lindenstrauss-Weiss [LW00] by using mean dimension theory. We briefly review their work below. Mean dimension is a topological invariant of dynamical systems introduced by Gromov [START_REF] Gromov | Topological invariants of dynamical systems and spaces of holomorphic maps: I[END_REF]. Let (X, T ) be a dynamical system with a metric d on X. For ε > 0 and a natural number N we define Widim ε (X, d N ) as the minimum number of n such that there exist an n-dimensional finite simplicial complex P and an ε-embedding f : (X, d N ) → P . Here f is called an ε-embedding if we have Diam(f -1 (p), d N ) < ε for all p ∈ P . We define the mean dimension of (X, T ) by mdim(X, T ) := lim

ε→0 lim N →∞ Widim ε (X, d N ) N .
This is a topological invariant, i.e. independent of the choice of d. For a closed subset E ⊂ X we also define mdim(E, T ) := lim ε→0 lim sup

N →∞ Widim ε (E, d N ) N .
If a dynamical system (X, T ) embeds in another (Y, S) then we have mdim(X, T ) ≤ mdim(Y, S). The full-shift ([0, 1] a ) Z , σ has mean dimension a. Lindenstrauss-Weiss [LW00, Proposition 3.5] constructed a minimal closed subset X ⊂ ([0, 1] 2 )

Z such that (X, σ) has mean dimension strictly larger than one. Then such X does not embed in [0, 1] Z because [0, 1] Z has mean dimension one. So this provides a negative answer to Auslander's question. Notice that the statement "X does not embed in [0, 1] Z " means that every equivariant continuous map π : X → [0, 1] Z has some fiber of cardinality strictly larger than one. Theorem 1.2 above provides a substantially stronger conclusion that some fiber of π must have infinite entropy. Indeed the construction of the minimal closed subset X in Theorem 1.2 is the same as that of Lindenstrauss-Weiss [LW00, Proposition 3.5]. But we can deduce a stronger conclusion with the aid of the waist inequality.

1.2. Conditional metric mean dimension. As we already saw at the end of the last subsection, our study has a connection to mean dimension theory. It turns out that the most fruitful framework is provided by a conditional version of metric mean dimension. Metric mean dimension is a notion introduced by Lindenstrauss-Weiss [LW00, §4] for better understanding the relation between mean dimension and entropy. Its conditional version was introduced by Liang [Lia21, Definition 4.1].

Let (X, T ) be a dynamical system with a metric d on X. We define the upper and lower metric mean dimensions of (X, T, d) by

mdim M (X, T, d) = lim sup ε→0 lim N →∞ log # (X, d N , ε) N log(1/ε) , mdim M (X, T, d) = lim inf ε→0 lim N →∞ log # (X, d N , ε) N log(1/ε) .
These are metric-dependent quantities. When the upper and lower limits coincide, we denote the common value by mdim M (X, T, d). If mdim M (X, T, d) > 0 then h top (X, T ) = +∞. So, in a sense, metric mean dimension evaluates how infinite topological entropy is. Lindenstrauss-Weiss [LW00, Theorem 4.2] proved mdim(X, T ) ≤ mdim M (X, T, d).

See also the paper of Lindenstrauss [START_REF] Lindenstrauss | Mean dimension, small entropy factors and an embedding theorem[END_REF] for a deeper study of metric mean dimension. Let (X, T ) and (Y, S) be dynamical systems, and let π : X → Y be an equivariant continuous map . Let d be a metric on X. We define the upper and lower conditional metric mean dimensions of π by mdim M (π, T, d) = lim sup

ε→0 lim N →∞ sup y∈Y log # (π -1 (y), d N , ε) N log(1/ε) , mdim M (π, T, d) = lim inf ε→0 lim N →∞ sup y∈Y log # (π -1 (y), d N , ε) N log(1/ε) .
These are also metric-dependent quantities. When the upper and lower limits coincide, we denote the common value by mdim M (π, T, d). If mdim M (π, T, d) > 0 then h top (π, T ) = +∞. So conditional metric mean dimension evaluates how infinite topological conditional entropy is.

Let a be a natural number. For x = (x 1 , . . . , x a ) ∈ R a we set

x ∞ := max 1≤i≤a |x i |.
Let ([0, 1] a ) Z be the full-shift on the alphabet [0, 1] a . We define a metric D on it by

(1•2) D (x n ) n∈Z , (y n ) n∈Z = n∈Z 2 -|n| x n -y n ∞ , (x n , y n ∈ [0, 1] a ). Theorem 1.3. Let a > b be two natural numbers. Let ([0, 1] a ) Z , σ and [0, 1] b Z , σ
be the full-shifts on the alphabets [0, 1] a and [0, 1] b respectively. Then for any equivariant continuous map π :

([0, 1] a ) Z → [0, 1] b Z we have mdim M (π, σ, D) ≥ a -b.
Here D is the metric defined by (1•2).

Theorem 1.1 in §1.1 immediately follows from this theorem because mdim

M (π, σ, D) > 0 implies h top (π, σ) = +∞. Example 1.4. The estimate in Theorem 1.3 is sharp: For a > b, let f : [0, 1] a → [0, 1] b be the natural projection to the first b coordinates. We define π : ([0, 1] a ) Z → [0, 1] b Z by π ((x n ) n∈Z ) = (f (x n )) n∈Z . Then mdim M (π, σ, D) = a -b.
We can also prove a statement similar to Theorem 1.3 for maps from ([0, 1] a ) Z to arbitrary dynamical systems: Theorem 1.5. Let a be a natural number and let (Y, S) be a dynamical system. For any equivariant continuous map π : ([0, 1] a ) Z , σ → (Y, S) we have

(1•3) mdim M (π, σ, D) ≥ a -2mdim(Y, S).
Here mdim(Y, S) is the mean dimension of (Y, S).

The factor 2 in the right-hand side of (1•3) looks a bit unsatisfactory. So we propose:

Problem 1.6. In the setting of Theorem 1.5, can one conclude the following stronger inequality?

mdim M (π, σ, D) ≥ a -mdim(Y, S).
See also Remark 4.2 (2).

Here the target system (Y, S) is arbitrary, but the domain is only ([0, 1] a ) Z . It seems difficult for our current technology to try a similar question for maps π : (X, T ) → (Y, S) with arbitrary dynamical systems (X, T ). Nevertheless we would like to propose the following problem because the statement looks beautiful (if it is true).

Problem 1.7. Let (X, T ) and (Y, S) be dynamical systems with mdim(X, T ) > mdim(Y, S).

Let π : (X, T ) → (Y, S) be an equivariant continuous map. Can one always conclude that the topological conditional entropy of π is infinite? When mdim(Y, S) = 0, this can be proved by using [START_REF] Tsukamoto | On an analogue of the Hurewicz theorem for mean dimension[END_REF]Theorem 1.5]. But when mdim(Y, S) > 0, the problem is widely open.

Although it is currently difficult to study maps from general dynamical systems, our method works well for some subsystems of ([0, 1] a ) Z . We can prove the following theorem.

Theorem 1.8. Let a be a natural number. Let s be a real number with 0 ≤ s < a. There exists a minimal closed subset X of ([0, 1] a ) Z , σ satisfying the following two conditions.

(1) mdim(X, σ) = mdim M (X, σ, D) = s.

(2) For any natural number b and any equivariant continuous map π :

(X, σ) → [0, 1] b Z , σ we have mdim M (π, σ, D) ≥ s -b.
Here D is a metric defined in (1•2).

When s > b, the inequality mdim M (π, σ, D) ≥ s -b > 0 implies h top (π, σ) = +∞. Therefore Theorem 1.2 in §1.1 is an immediate corollary of Theorem 1.8. 1.3. On an analogue of the Hurewicz theorem. The Hurewicz theorem [HW41, p.91, Theorem VI 7] in classical dimension theory asserts that for any continuous map f : X → Y between compact metrizable spaces we have

(1•4) dim X ≤ dim Y + sup y∈Y dim f -1 (y).
Here dim(•) denotes topological dimension 1 . Let d be a metric on X. We define the upper and lower Minkowski dimensions of (X, d) by

dim M (X, d) = lim sup ε→0 log # (X, d, ε) log(1/ε) , dim M (X, d) = lim inf ε→0 log # (X, d, ε) log(1/ε) .
When the upper and lower limits coincide, we denote the common value by dim M (X, d).

The topological dimension is always bounded by the lower Minkowski dimension.

dim X ≤ dim M (X, d). So dim f -1 (y) ≤ dim M (f -1 (y), d)
, and hence we also have

(1•5) dim X ≤ dim Y + sup y∈Y dim M f -1 (y), d
for every continuous map f : X → Y . Theorems 1.3 and 1.5 in §1.2 are dynamical analogues of the inequality (1•5). Conditional metric mean dimension is an analogue of the term sup y∈Y dim M (f -1 (y), d). Probably it is also natural to ask whether or not an analogue of the original Hurewicz theorem (1•4) holds for mean dimension. Indeed this question was already proposed by the secondnamed author [START_REF] Tsukamoto | Moduli space of Brody curves, energy and mean dimension[END_REF] more than ten years ago and studied in detail by a recent paper [Tsu]. In this subsection we explain the relation between [Tsu] and the current paper.

The paper [Tsu, Theorem 3.9, Remark 3.10] proved the next theorem.

Theorem 1.9. Let a be a natural number. There exists a zero-dimensional 2 dynamical system (Z, R) for which the following statement holds true: For any δ > 0 there exist a dynamical system (Y, S) and an equivariant continuous map

π : ([0, 1] a ) Z × Z, σ × R → (Y, S) satisfying mdim(Y, S) < δ, sup y∈Y mdim π -1 (y), σ × R = 0.
Here mdim (π -1 (y), σ × R) is the mean dimension of the fiber π -1 (y) ⊂ ([0, 1] a ) Z × Z.

1 Let d be a metric on X. We define the topological dimension by dim X = lim ε→0 Widim ε (X, d).

The value of dim X is independent of the choice of d 2 "Zero-dimensional" means that the topological dimension of Z is zero.

In this statement, the system ([0, 1] a ) Z × Z, σ × R has mean dimension a while mdim(Y, S) < δ may be arbitrarily small. Nonetheless the map π has no mean dimension in the fiber direction. This implies that a direct analogue of the Hurewicz theorem (1•4) does not hold for mean dimension. In other words, mean dimension of the fibers mdim (π -1 (y), σ × R) does not properly evaluate the complexity of the map π.

On the other hand, we can prove the following statement for conditional metric mean dimension:

Proposition 1.10. Let a be a natural number. Let (Z, R) be a dynamical system with a metric ρ on Z. We define a metric d on the product

([0, 1] a ) Z × Z by d ((x, z), (x , z )) = D(x, x ) + ρ(z, z ),
where D is the metric on ([0, 1] a ) Z defined by (1•2). Then for any dynamical system (Y, S) and any equivariant continuous map π :

([0, 1] a ) Z × Z, σ × R → (Y, S) we have mdim M (π, σ × R, d) ≥ a -2mdim(Y, S).
Namely, we can detect a reasonable amount of complexity of the map π by using conditional metric mean dimension. This gives the authors more confidence that the conditional metric mean dimension is a useful tool for studying maps between dynamical systems.

Remark 1.11. In the setting of Proposition 1.10, we can also prove the following statement: Define an equivariant continuous map Π :

([0, 1] a ) Z × Z, σ × R → (Y ×Z, S ×R) by Π(x, z) = (π(x, z), z) .

Then we have mdim

M (Π, σ × R, d) ≥ a -2mdim(Y, S).
This statement is a bit stronger (and arguably more natural) than Proposition 1.10. But the proof is more or less the same. So we omit the details of the proof.

1.4. Organization of the paper. We review the waist inequality in the next section. We prepare a simple fact on conditional metric mean dimension in §3. We prove the main technical theorem in §4. As its corollaries, we prove Theorems 1.3, 1.5 and Proposition 1.10 also in §4. We prove Theorem 1.8 in §5.

Waist inequality

In this section we review the waist inequality. Readers can find much more information in an excellent survey of Guth [START_REF] Guth | The waist inequality in Gromov's work[END_REF]. Let S n = {x ∈ R n+1 | |x| = 1} be the ndimensional sphere. Here |x| is the Euclidean norm. We naturally consider S 0 ⊂ S 1 ⊂ S 2 ⊂ • • • ⊂ S n . For A ⊂ S n and r > 0 we denote by A + r the r-neighborhood of A in the spherical metric.

The following inequality is the original waist inequality proved by Gromov [START_REF] Gromov | Isoperimetry of waists and concentration of maps[END_REF].

Theorem 2.1 [START_REF] Gromov | Isoperimetry of waists and concentration of maps[END_REF]. Let n ≥ m be two natural numbers. For any continuous map f : S n → R m there exists t ∈ R m such that for every r > 0 we have

(2•1) vol f -1 (t) + r ≥ vol S n-m + r .
Here vol(•) denotes the n-dimensional volume on the sphere S n .

For maps f : S n → R, this statement follows from the isoperimetric inequality on the sphere. (See [AG15, Appendix] for the details.) For equidimensional maps f : S n → R n , if we additionally assume that every fiber of f has cardinality at most two, then the above inequality (2•1) for r = π 2 implies that f -1 (t) consists of a pair of antipodal points. This is the statement of the Borsuk-Ulam theorem for such maps f . Therefore the waist inequality has close connections to both the isoperimetric inequality and the Borsuk-Ulam theorem. Its proof is a wonderful mixture of geometry (measure theory and convex geometry) and algebraic topology. Gromov's original paper [START_REF] Gromov | Isoperimetry of waists and concentration of maps[END_REF] is probably not very easily readable. Memarian's paper [START_REF] Memarian | On Gromov's waist of the sphere theorem[END_REF] is helpful for understanding the details of the proof of the waist inequality.

As we already mentioned in the introduction, the ideas of the waist inequality have connections to several subjects. Here we review a connection to convex geometry developed by Klartag [Kla17]. This will be crucial for us.

For subsets A and B of R n we define A + B = {x + y | x ∈ A, y ∈ B}. For r > 0 we set rA = {rx | x ∈ A}. A convex body of R n is a compact convex subset of R n with a non-empty interior. A convex body K is said to be centrally-symmetric if -K = K, where -K := {-x | x ∈ K}. A function ϕ : R n → [0, +∞) is said to be log-concave if for any x, y ∈ R n and 0 < λ < 1

ϕ (λx + (1 -λ)y) ≥ ϕ(x) λ ϕ(y) 1-λ .
Klartag [START_REF] Klartag | Convex geometry and waist inequalities[END_REF] proved several interesting theorems connecting the waist inequality to convex geometry. In particular he proved, among many other things, the following version of the waist inequality [Kla17, Theorem 5.7].

Theorem 2.2 (Klartag, 2017). Let K ⊂ R n be a centrally-symmetric convex body. Let 1 ≤ m ≤ n. Let µ be a probability measure supported in K which has a log-concave density function with respect to the Lebesgue measure. Then for any continuous map f :

K → R m there exists t ∈ R m satisfying µ f -1 (t) + rK ≥ r 2 + r m for all 0 < r < 1.
For A ⊂ R n and r > 0 we define A + ∞ r as the r-neighborhood of A with respect to the ∞ -norm:

A + ∞ r = {x ∈ R n | ∃a ∈ A : x -a ∞ ≤ r}.
Corollary 2.3. Let 1 ≤ m ≤ n. Let µ be the Lebesgue measure restricted to the unit cube [0, 1] n . Then for any continuous map f :

[0, 1] n → R m there exists t ∈ R m satisfying µ f -1 (t) + ∞ r ≥ r 2 m for all 0 < r < 1 2 . Proof. Let K = -1 2 , 1 2
n . This is a centrally-symmetric convex body. Let ν be the Lebesgue measure restricted to K. The density of ν is the characteristic function of K, which is obviously log-concave. We define g :

K → R m by g(x) = f x - 1 2 , 1 2 , . . . , 1 2 .
By Klartag's waist inequality, there exists t ∈ R n satisfying ν g -1 (t) + rK ≥ r 2 + r m for all 0 < r < 1.

We have

g -1 (t) + ∞ r = g -1 (t) + 2rK. Hence, for 0 < r < 1 2 ν g -1 (t) + ∞ r = ν g -1 (t) + 2rK ≥ 2r 2 + 2r m ≥ r 2 m .
We have

f -1 (t) + ∞ r = g -1 (t) + ∞ r + 1 2 , 1 2 , . . . , 1 2 . Thus, for 0 < r < 1 2 µ f -1 (t) + ∞ r = ν g -1 (t) + ∞ r ≥ r 2 m .
For r > 0 and a compact subset A ⊂ R n we define # (A, • ∞ , r) as the minimum number of N such that there exists an open cover

{U 1 , U 2 , . . . , U N } of A satisfying Diam (U i , • ∞ ) < r for all 1 ≤ i ≤ r. Here Diam (U i , • ∞ ) = sup x,y∈U i x -y ∞ . When A is empty, we define # (A, • ∞ , r) = 0. Corollary 2.4. Let 1 ≤ m ≤ n. For any continuous map f : [0, 1] n → R m there exists t ∈ R m satisfying # f -1 (t), • ∞ , r ≥ 1 8 n 1 r n-m
for all 0 < r < 1 2 .

Proof. Let µ be the Lebesgue measure restricted to [0, 1] n . By Corollary 2.3 there exists

t ∈ R m satisfying µ f -1 (t) + ∞ r ≥ r 2 m for all 0 < r < 1 2 . Let 0 < r < 1 2 . Set N = # (f -1 (t), • ∞ , r).
There exists an open cover

f -1 (t) ⊂ U 1 ∪ U 2 ∪ • • • ∪ U N satisfying Diam (U i , • ∞ ) < r for all 1 ≤ i ≤ N . Pick a point x i ∈ U i for each U i . We have U i ⊂ x i + [-r, r] n := {y ∈ R n | y -x i ∞ ≤ r} . Hence f -1 (t) ⊂ N i=1 (x i + [-r, r] n ) .
We have

f -1 (t) + ∞ r = N i=1 (x i + [-2r, 2r] n ) . Then r 2 m ≤ µ f -1 (t) + ∞ r ≤ N i=1 µ (x i + [-2r, 2r] n ) ≤ N (4r) n . Therefore N ≥ (4r) -n • r 2 m ≥ 1 8 n • 1 r n-m
. For a convenience of readers, here we recall basic notions used in the definition of mean dimension. Let ε be a positive number, and let (X, d) be a compact metric space. A continuous map f : X → Y from X to a topological space Y is called an ε-embedding if we have Diamf -1 (y) < ε for all y ∈ Y . We define Widim ε (X, d) as the minimum number n such that there exist an n-dimensional finite simplicial complex P and an ε-embedding f : X → P .

We will need the following lemma later. This was proved in [LW00, Lemma 3.2].

Lemma 2.5. For 0 < ε < 1 we have

Widim ε ([0, 1] n , • ∞ ) = n.
Here • ∞ is the metric defined by the ∞ -norm.

This lemma is a basis of the fact that the full-shift ([0, 1] a ) Z has mean dimension a.

A preliminary on conditional metric mean dimension

In this section we prepare a simple fact about conditional metric mean dimension. Let (X, T ) be a dynamical system with a metric d. Recall that, for each N ≥ 1, we defined a new metric d N on X by d N (x 1 , x 2 ) = max 0≤n<N d (T n x 1 , T n x 2 ). Let (Y, S) be another dynamical system with a metric d . Let N be a natural number and let r be a positive real number. For y ∈ Y we set

B r (y, d N ) = {x ∈ Y | d N (x, y) ≤ r} .
Let π : (X, T ) → (Y, S) be an equivariant continuous map. Recall that we defined the upper and lower conditional metric mean dimensions of π by

mdim M (π, T, d) = lim sup ε→0 lim N →∞ sup y∈Y log # (π -1 (y), d N , ε) N log(1/ε) , mdim M (π, T, d) = lim inf ε→0 lim N →∞ sup y∈Y log # (π -1 (y), d N , ε) N log(1/ε) .
In this definition, the limit with respect to N exists because the quantity

sup y∈Y log # π -1 (y), d N , ε
is subadditive in N , as we will see below:

Lemma 3.1. In the above setting, we have the following two statements.

(1) The quantity sup y∈Y log # (π -1 (y), d N , ε) is subadditive in N .

(2) Let ε and δ be positive numbers. Then the quantity

a N := sup y∈Y log # π -1 (B δ (y, d N )) , d N , ε is subadditive in N .
Proof. We only prove (2). The proof of (1) is similar. Let N 1 and N 2 be natural numbers.

We have

B δ y, d N 1 +N 2 = B δ y, d N 1 ∩ S -N 1 B δ S N 1 y, d N 2 . Then π -1 B δ y, d N 1 +N 2 = π -1 B δ y, d N 1 ∩ π -1 S -N 1 B δ S N 1 y, d N 2 = π -1 B δ y, d N 1 ∩ T -N 1 π -1 B δ S N 1 y, d N 2 . Hence # π -1 B δ y, d N 1 +N 2 , d N 1 +N 2 , ε ≤ # π -1 B δ y, d N 1 , d N 1 , ε • # π -1 B δ S N 1 y, d N 2 , d N 2 , ε ≤ e a N 1 • e a N 2 . Therefore we have a N 1 +N 2 ≤ a N 1 + a N 2 .
Lemma 3.2. The upper and lower conditional metric mean dimensions of π : (X, T ) → (Y, S) are given by

mdim M (π, T, d) = lim sup ε→0 lim δ→0 lim N →∞ sup y∈Y log # (π -1 (B δ (y, d N )) , d N , ε) N log(1/ε) , mdim M (π, T, d) = lim inf ε→0 lim δ→0 lim N →∞ sup y∈Y log # (π -1 (B δ (y, d N )) , d N , ε) N log(1/ε) .
Proof. We only prove the formula for the lower conditional metric mean dimension. The upper case is similar. We set

mdim M (π, T, d) = lim inf ε→0 lim δ→0 lim N →∞ sup y∈Y log # (π -1 (B δ (y, d N )) , d N , ε) N log(1/ε) .
It is obvious that mdim M (π, T, d) ≤ mdim M (π, T, d) . We will prove the reverse inequality. Suppose mdim M (π, T, d) < a for some positive number a. We show mdim M (π, T, d) ≤ a. There exists a decreasing sequence

ε 1 > ε 2 > ε 3 > • • • → 0 such that lim N →∞ sup y∈Y log # (π -1 (y), d N , ε k ) N log(1/ε k ) < a.
There exists N k for each k satisfying

sup y∈Y # π -1 (y), d N k , ε k < 1 ε k aN k .
From the compactness of Y , we can find δ k > 0 such that

sup y∈Y # π -1 B δ k (y, d N k ) , d N k , ε k < 1 ε k aN k .
From the subadditivity (2) in Lemma 3.1 lim

N →∞ sup y∈Y log # (π -1 (B δ k (y, d N )) , d N , ε k ) N < a log(1/ε k ). Hence lim δ→0 lim N →∞ sup y∈Y log # (π -1 (B δ (y, d N )) , d N , ε k ) N < a log(1/ε k ).
Thus we have mdim M (π, T, d) ≤ a. This has shown mdim M (π, T, d) ≤ mdim M (π, T, d).

Main technical theorem

Here we formulate the main technical result of this paper. All the theorems in §1 are more or less its corollaries. We prove Theorems 1.3, 1.5 and Proposition 1.10 in this section. Theorem 1.8 will be proved in §5.

Theorem 4.1 (Main Technical theorem). Let s be a nonnegative real number. Let (X, T ) be a dynamical system with a metric d on X. Suppose that there exist sequences of natural numbers {N n } ∞ n=1 , {M n } ∞ n=1 and continuous maps ψ n : [0, 1] Mn → X (n ≥ 1) satisfying the following three conditions.

• N n → +∞ as n → ∞. • The ratio M n /N n converges to s as n → ∞. • For any x, y ∈ [0, 1] Mn we have x -y ∞ ≤ d Nn (ψ n (x), ψ n (y)) .
Then we have the following three conclusions.

(1) mdim(X, T ) ≥ s.

(2) For any natural number b and any equivariant continuous map π : (X, T ) → [0, 1] b Z , σ (where σ is the shift map) we have

mdim M (π, T, d) ≥ s -b.
(3) For any dynamical system (Y, S) and any equivariant continuous map π : (X, T ) → (Y, S) we have

mdim M (π, T, d) ≥ s -2mdim(Y, S).
Proof.

(1) For 0 < ε < 1 we have

Widim ε (X, d Nn ) ≥ Widim ε [0, 1] Mn , • ∞ = M n by Lemma 2.5. Then mdim (X, T ) = lim ε→0 lim n→∞ Widim ε (X, d Nn ) N n ≥ lim n→∞ M n N n = s.
(2) For two integers < m and a point y

= (y n ) n∈Z in [0, 1] b Z we denote y| m = (y , y +1 , . . . , y m-1 , y m ) ∈ [0, 1] b m-+1 .
We take a metric d on [0, 1] b Z . Let 0 < ε < 1/2 and δ > 0 be arbitrary. We take a natural number m such that for y, z

∈ [0, 1] b Z y| m -m = z| m -m =⇒ d (y, z) < δ.
This implies that for any N > 0 and y, z

∈ [0, 1] b Z y| N +m-1 -m = z| N +m-1 -m =⇒ d N (y, z) < δ.
We consider a map

π n : [0, 1] Mn → [0, 1] b Nn+2m , x → π (ψ n (x)) | Nn+m-1 -m
.

We apply the waist inequality to this map. By Corollary 2.4 we can find a point t ∈

[0, 1] b Nn+2m satisfying (4•1) # π -1 n (t), • ∞ , ε ≥ 1 8 Mn • 1 ε Mn-b(Nn+2m)
.

(When M n ≥ b(N n + 2m), this is a direct consequence of Corollary 2.4. When M n < b(N n + 2m), the right-hand side is less than one. So it is obviously true.) Here, of course, the choice of t depends on n. But we suppress its dependence on n in our notation for simplicity.

Take

t ∈ [0, 1] b Z with t | Nn+m-1 -m = t. We claim that (4•2) ψ n π -1 n (t) ⊂ π -1 B δ t , d Nn . Indeed, let x ∈ π -1 n (t). Then π (ψ n (x)) | Nn+m-1 -m = π n (x) = t = t | Nn+m-1 -m
.

By the choice of m we have d Nn (π(ψ n (x)), t ) < δ. Hence π(ψ n (x)) ∈ B δ t , d Nn and

ψ n (x) ∈ π -1 B δ t , d Nn . This has proved (4•2). By x -y ∞ ≤ d Nn (ψ n (x), ψ n (y)) (x, y ∈ [0, 1] Mn ) # π -1 n (t), • ∞ , ε ≤ # ψ n π -1 n (t) , d Nn , ε ≤ # π -1 B δ t , d Nn , d Nn , ε by (4•2). By (4•1) sup y∈([0,1] b ) Z # π -1 B δ y, d Nn , d Nn , ε ≥ 1 8 Mn • 1 ε Mn-b(Nn+2m)
.

Taking the logarithm, we have

sup y∈([0,1] b ) Z log # π -1 B δ y, d Nn , d Nn , ε ≥ -M n log 8 + (M n -bN n -2bm) log 1 ε .
We divide this by

N n log(1/ε) and let n → ∞. By lim n→∞ M n /N n = s lim N →∞ sup y∈([0,1] b ) Z log # (π -1 (B δ (y, d N )) , d N , ε) N log(1/ε) ≥ s -b - s log 8 log(1/ε) .
This holds for arbitrary 0 < ε < 1/2 and δ > 0. So we can let δ → 0 and then ε → 0. By Lemma 3.2 mdim M (π, T, d) ≥ s -b.

(3) Let d be a metric on Y . Let 0 < ε < 1/2 and δ > 0 be arbitrary. We can find N (δ) > 0 such that for any natural number N ≥ N (δ) there exist a finite simplicial complex K N and a δ-embedding ϕ

N : (Y, d N ) → K N satisfying (4•3) dim K N N < mdim(Y, S) + δ.
For each natural number n with N n ≥ N (δ) we consider a map

π n := ϕ Nn • π • ψ n : [0, 1] Mn → K Nn .
We use a famous result of topological dimension theory [HW41, Theorem V.2]: The space K N topologically embeds in the Euclidean space R 2 dim K N +1 . (Since K N is a finite simplicial complex, this is a rather elementary fact. See e.g. [GQT19, Lemma 2.6].) Therefore we can consider that

K Nn ⊂ R 2 dim K Nn +1 .
So π n is a continuous map from [0, 1] Mn to R 2 dim K Nn +1 . We apply the waist inequality to this map. The rest of the argument is quite similar to the case (2). By Corollary 2.4 we can find t ∈ K Nn (depending on n) such that

(4•4) # π -1 n (t), • ∞ , ε ≥ 1 8 Mn • 1 ε Mn-2 dim K Nn -1 . Pick t ∈ Y (depending on n) with ϕ Nn (t ) = t. We claim that (4•5) ψ n π -1 n (t) ⊂ π -1 B δ t , d Nn .
Indeed, take x ∈ π -1 n (t). Then ϕ Nn (π (ψ n (x))) = π n (x) = t = ϕ Nn (t ). Since ϕ Nn is a δ-embedding with respect to d Nn , we have d Nn (π (ψ n (x)) , t ) < δ.

So π (ψ n (x)) ∈ B δ t , d Nn and hence ψ n (x) ∈ π -1 B δ t , d Nn . This has shown (4•5).

Since x -y ∞ ≤ d Nn (ψ n (x), ψ n (y)) (x, y ∈ [0, 1] Mn ),

# π -1 n (t), • ∞ , ε ≤ # ψ n π -1 n (t) , d Nn , ε ≤ # π -1 B δ t , d Nn , d Nn , ε by (4•5).
By (4•4) we have

sup y∈Y # π -1 B δ (y, d Nn ) , d Nn , ε ≥ 1 8 Mn • 1 ε Mn-2 dim K Nn -1 .
Taking the logarithm and dividing it by N n log(1/ε), we have

sup y∈Y log # π -1 B δ (y, d Nn ) , d Nn , ε N n log(1/ε) ≥ M n N n - 2 dim K n N n - 1 N n - M n log 8 N n log(1/ε) ≥ M n N n -2mdim(Y, S) -2δ - 1 N n - M n log 8 N n log(1/ε) .
Here we have used (4•3) in the second inequality. We let n → ∞. Noting M n /N n → s, we get lim

N →∞ sup y∈Y log # (π -1 (B δ (y, d N )) , d N , ε) N log(1/ε) ≥ s -2mdim(Y, S) -2δ - s log 8 log(1/ε) .
We let δ → 0 and then ε → 0. We conclude mdim M (π, T, d) ≥ s-2mdim(Y, S) by Lemma 3.2.

Remark 4.2. There are two points in the above proof which might be able to be improved in a future:

(1) In the above proof, we used the waist inequality in (4•1) and (4•4). We notice that they are slightly weaker than the original waist inequalities. In both the inequalities (4•1) and (4•4) the parameter ε is fixed. On the other hand, it can vary in the original waist inequalities. It is an interesting problem to discover how to apply the full-power of the waist inequality to the study of mean dimension. A related question is the following: For an equivariant continuous map π : (X, T ) → (Y, S) between dynamical systems, we define the fiber-wise lower conditional metric mean dimension by mdim M (π, T, d) fiber = sup

y∈Y lim inf ε→0 lim sup N →∞ log # (π -1 (y), d N , ε) N log(1/ε) .
Can one replace mdim M (π, T, d) with mdim M (π, T, d) fiber in the statements (2) and (3) in Theorem 4.1? (2) The multiplicative factor 2 of 2mdim(Y, S) in the statement of Theorem 4.1 (3) comes form the embedding K Nn ⊂ R 2 dim K Nn +1 used in the above proof. (See also Problem 1.6 in §1.2.) If we would like to reduce the factor 2 to a smaller value (hopefully, one), then we need to develop a method to bypass this embedding. It seems possible to find such a method if the local topology of the simplicial complex K Nn is not very complicated. For example, if K Nn is a smooth manifold then there exists a smooth map f : K Nn → R dim K Nn for which every fiber has cardinality at most 4 dim K Nn [Gro10, p. 447]. Then we apply the waist inequality to f • π n and get the desired result. Another idea is to develop a waist inequality directly applicable to the map π n : [0, 1] Mn → K Nn . But currently we do not know how to do it.

Theorems 1.3, 1.5 and Proposition 1.10 in §1 are immediate corollaries of the above main technical theorem. They are all included in the following statement.

Corollary 4.3 (= Theorems 1.3, 1.5 and Proposition 1.10). Let a be a natural number, and let ([0, 1] a ) Z , σ be the full-shift on the alphabet [0, 1] a . We define a metric D on it by

D ((x n ) n∈Z , (y n ) n∈Z ) = n∈Z 2 -|n| x n -y n ∞ .
(1) Let b be a natural number. For any equivariant continuous map π : (

[0, 1] a ) Z , σ → [0, 1] b Z , σ we have mdim M (π, σ, D) ≥ a -b.
(2) Let (Y, S) be a dynamical system. For any equivariant continuous map π : (

[0, 1] a ) Z , σ → (Y, S) we have mdim M (π, σ, D) ≥ a -2mdim(Y, S).
Notice that this statement is logically contained in the next claim (3) (letting Z to be one-point there). But we separately state it for making the exposition easier to understand.

(3) Let (Z, R) be a dynamical system with a metric ρ. We define a metric d on the product ([0, 1] a ) Z ×Z by d ((x, z), (x , z )) = D(x, x )+ρ(z, z ). Then for any dynamical system (Y, S) and any equivariant continuous map π :

([0, 1] a ) Z × Z, σ × R → (Y, S) we have mdim M (π, σ × R, d) ≥ a -2mdim(Y, S).
Proof. For each natural number n we define a map

ψ n : ([0, 1] a ) n → ([0, 1] a ) Z by ψ n (x 0 , x 1 , . . . , x n-1 ) k =    x k (0 ≤ k ≤ n -1) 0 (otherwise)
.

This satisfies that for any x, y ∈ (

[0, 1] a ) n x -y ∞ ≤ D n (ψ n (x), ψ n (y)) .
Then the statements (1) and (2) immediately follow from Theorem 4.1 with the parameters N n := n and M n := an. For (3), we fix a point p ∈ Z and consider a map

ψ n : ([0, 1] a ) n → ([0, 1] a ) Z × Z, x → (ψ n (x), p) .
This satisfies that for any x, y ∈ ([0, 1] a ) n

x -y ∞ ≤ d n (ψ n (x), ψ n (y)) .

We use Theorem 4.1 with these maps ψ n and get the statement (3).

Construction of a minimal closed subset

We prove Theorem 1.8 in this section. The construction of a minimal closed subset itself is the same as that of [LW00, Proposition 3.5]. It uses an idea of block-system reviewed below.

5.1. Review of block-system. Let a be a natural number. As before, we define a metric D on the full-shift ([0, 1] a ) Z by

D ((x n ) n∈Z , (y n ) n∈Z ) = n∈Z 2 -|n| x n -y n ∞ .
Let x be a point in ([0, 1] a ) Z . For integers < m we set x| m = (x , x +1 , . . . , x m-1 , x m ) ∈ ([0, 1] a ) m-+1 .

Let N be a natural number, and let K be a closed subset of ([0, 1] a ) N . We define a block-system X(K) in ([0, 1] a ) Z by X(K) = x ∈ ([0, 1] a ) Z ∃ ∈ Z : ∀n ∈ Z : x|

+(n+1)N -1 +nN ∈ K .
This is a shift-invariant closed subset of ([0, 1] a ) Z . Lemma 5.1.

mdim M (X(K), σ, D) ≤ dim M (K, • ∞ ) N .
Here σ is the shift-map, and dim M (K, • ∞ ) is the upper Minkowski dimension of K with respect to the ∞ -distance.

Proof. For 0 ≤ ≤ N -1 we set

X = x ∈ ([0, 1] a ) Z ∀n ∈ Z : x| +(n+1)N -1 +nN ∈ K .
We have

X(K) = X 0 ∪ X 1 ∪ X 2 ∪ • • • ∪ X N -1 .
Let ε be a positive number. Take m = m(ε) > 0 satisfying |n|>m 2 -|n| < ε/2. Let L be an arbitrary natural number. We set Hence, for any x, y ∈ X

D L (x, y) < 3 f (x) -f (y) ∞ + ε 2 . Then # (X , D L , ε) ≤ # K t-s+1 , • ∞ , ε 9 ≤ # K, • ∞ , ε 9 t-s+1
.

We have

t -s + 1 ≤ L -+ m N - --m N + 1 + 1 ≤ L + 2m N + 2. Hence # (X , D L , ε) ≤ # K, • ∞ , ε 9 L+2m N +2 . Therefore # (X(K), D L , ε) ≤ N • # K, • ∞ , ε 9
L+2m N

+2

.

Taking the logarithm and dividing it by L log(1/ε), we get log # (X(K), D L , ε) L log(1/ε)

≤ log N L log(1/ε) + 1 N + 2m N L + 2 L log # K, • ∞ , ε 9
log(1/ε) .

We let L → +∞ and then let ε → 0. We conclude that mdim M (X(K), σ, D) ≤ dim M (K, • ∞ ) N .

5.2. Proof of Theorem 1.8. Now we prove Theorem 1.8. We write the statement again.

Theorem 5.2 (= Theorem 1.8). Let a be a natural number and s a real number with 0 ≤ s < a. There exists a minimal closed subset X of ([0, 1] a ) Z , σ satisfying the following two conditions.

(1) mdim (X, σ) = mdim M (X, σ, D) = s.

(2) For any natural number b and any equivariant continuous map π : (X, σ) → [0, 1] b Z , σ we have mdim M (π, σ, D) ≥ s -b.

Proof. As we already noted, the construction of X is the same as that of [LW00, Proposition 3.5]. The difference is that we analyze the resulting minimal closed subset by using the waist inequality. We take and fix a sequence of rational numbers 0 < r n < 1 (n ≥ 1) satisfying

a ∞ n=1
(1 -r n ) = s.

Set N 1 = 1 and K 1 = [0, 1] a . Starting from these, we inductively construct an increasing sequence of natural numbers N n and closed subsets K n ⊂ ([0, 1] a ) Nn (n ≥ 1) satisfying the following conditions.

(i) There are natural numbers p n > q n (n ≥ 1) larger than one satisfying N n+1 = p n N n and r n = q n /p n . (ii) K n+1 ⊂ (K n ) pn . Moreover, there is a point (w

(n) 1 , w (n) 2 , . . . , w (n) qn ) in (K n ) qn such that • K n+1 = (K n ) pn-qn × {(w (n) 1 , w (n)
2 , . . . , w

qn )}. • For any x ∈ K n × K n there exists 1 ≤ k < q n satisfying x -(w

(n) k , w (n) k+1 ) ∞ < 1 n .
A moment thought shows that such an inductive construction works well. We consider the block-system X n = X(K n ) associated with K n . We set

X = ∞ n=1 X n ⊂ ([0, 1] a ) Z .

  , + (n + 1)N ) = [ + sN, + (t + 1)N ) ⊃ [-m, L + m).

R.S was partially supported by M.T. was supported by JSPS KAKENHI JP21K03227.

By the above condition (ii) the set X is minimal. We will show that it satisfies the required conditions (1) and (2).

We define a sequence of natural numbers M n (n ≥ 1) by

We have

Nn and hence

From the construction K n is naturally homeomorphic to the cube

So we have mdim M (X, σ, D) ≤ s.

It also follows from the construction that there exists a continuous map ψ n : ([0, 1] a ) Mn → X satisfying

Then we can apply to the maps ψ n Theorem 4.1 with the parameters aM n and N n . Noting lim n→∞ aMn Nn = s, we get the following conclusions: • mdim(X, σ) ≥ s.

• For any natural number b and any equivariant continuous map π : (X, σ) →

Since mean dimension is bounded from above by metric mean dimension [LW00, Theorem 4.2], we have s ≤ mdim(X, σ) ≤ mdim M (X, σ, D) ≤ mdim M (X, σ, D) ≤ s.

Thus mdim(X, σ) = mdim M (X, σ, D) = s. So (X, σ, D) satisfies all the required conditions.


Bingbing Liang
email: bbliang@suda.edu.cn

Ruxi Shi
email: ruxi.shi@upmc.fr

MEAN DIMENSION OF NATURAL EXTENSION OF ALGEBRAIC SYSTEMS

Keywords: Mathematics Subject Classification. Primary 37B02, 20K30 dynamical system, mean dimension, Pontryagin dual, natural extension

Mean dimension may decrease after taking natural extension. In this paper we show that mean dimension stays the same after taking natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of algebraic cellular automaton coincides with its natural extension, which strengthens a result of Burguet and Shi [1] with a different proof.

Introduction

By a dynamical system, we mean a pair (X, ϕ) for a compact metrizable space X and ϕ : X → X a continuous map. Denote by ‹ X the natural extension of (X, ϕ) with the extension map ϕ (see Definition 2.5). It is well known that the topological entropy of (X, ϕ) coincides with the topological entropy of ( ‹ X, ϕ). Mean (topological) dimension is a dynamical invariant which can distinguish the dynamical systems of infinite entropy. Gromov introduced this invariant when measuring the size of some holomorphic function spaces [START_REF] Gromov | Topological invariants of dynamical systems and spaces of holomorphic maps[END_REF]. Mean dimension takes a crucial role in the dynamical embedding problems [START_REF] Lindenstrauss | Mean topological dimension[END_REF][START_REF] Gutman | Mean dimension and Jaworski-type theorems[END_REF][START_REF] Gutman | Mean dimension of Z k -actions[END_REF] and has close relations with other areas like information theory [START_REF] Lindenstrauss | From rate distortion theory to metric mean dimension: variational principle[END_REF] and operator algebras [START_REF] Li | Mean dimension, mean rank, and von Neumann-Lück rank[END_REF][START_REF] Elliott | The C*-algebra of a minimal homeomorphism of zero mean dimension[END_REF]. As is pointed out by Burguet and the second author, the mean dimension of the natural extension ( ‹ X, ϕ) is no greater than the mean dimension of (X, ϕ) and the inequality can be strict [START_REF] Burguet | Mean dimension of continuous cellular automata[END_REF]Section 3.2]. In fact, it roughly follows from the facts that the natural extension is defined as the inverse limit (see Definition 2.5) and that the inverse limit of compact metrizable spaces can decrease the (covering) dimension [START_REF] Ingram | Inverse limits. From continua to chaos[END_REF]Example 183]. This motivates the following question: Question 1.1. For which dynamical system (X, ϕ) do we have the equality mdim(X, ϕ) = mdim( ‹ X, ϕ)?

In other words, we concern which dynamical properties would guarantee the non-decreasing of mean dimension of its natural extension. In this paper, we show that the above equality holds when the dynamical system admits an algebraic structure. That is, Theorem 1.2. For a compact metrizable abelian group X and a continuous endomorphism ϕ : X → X, we have mdim(X, ϕ) = mdim( ‹ X, ϕ).

To show this theorem, taking advantage of the abelian group structure of X, we can translate this question into its dual action of Pontryagin dual space. We first study the surjective endomorphism ϕ on X. By introducing a proper notion of mean rank, we will see that mean dimension of (X, ϕ) coincides with the mean rank of its Pontryagin dual. With the help of a result of Gutman [START_REF] Gutman | Embedding topological dynamical systems with periodic points in cubical shifts[END_REF] regarding the mean dimension of nonwandering subsystems, we can finally deal with the nonsurjective case.

It is worth-mentioning that the approach of interplay between an action on a compact abelian group and its dual action on Pontryagin dual space already appears in [START_REF] Li | Mean dimension, mean rank, and von Neumann-Lück rank[END_REF], where the addition formula for mean dimension of algebraic actions is hard to prove directly. However, when transforming the problem into the algebraic setting, the discreteness of algebraic objects makes the proof accessible.

On the other hand, Burgeut and the second author [1, Theorem 6.4] showed that the mean dimension of a unit cellular automaton coincides with that of its natural extension. In general, they [1, Question 3.6] asked that Question 1.3. Does the mean dimension of a general cellular automaton coincide with the mean dimension of its natural extension?

Based on a computation result concerning cellular automaton [1, Lemma 7.3], it is implicitly proved that the mean dimension of an algebraic permutative cellular automaton (see Definition 4.3) stays the same as the mean dimension of its natural extension.

Our result improves the result of [START_REF] Burguet | Mean dimension of continuous cellular automata[END_REF] with a different proof and give a partial answer to Question 1.3.

Corollary 1.4. For an algebraic cellular automaton

F : X Z → X Z , we have mdim(X Z , F ) = mdim( › X Z , F ).
The paper is organized as follows. In Section 2 we recall the definitions of mean dimension and natural extension. As a preparation, in Section 3 we introduce the notion of mean rank for a general group homomorphism, show some nice properties, and establishes an equality relating mean dimension and mean rank. In Section 4 we finish the proof of Theorems 1.2 and Corollary 1.4. Acknowledgements. The first author is supported by the School of Mathematical Sciences at Soochow University.

background

2.1. Mean dimension. Let X be a compact metrizable space with a compatible metric d. Fix ε > 0. Recall that a continuous map f : (X, d) → Y for another topological space Y is an (ε, d)-embedding if diam(f -1 (y), d) < ε for every y ∈ Y where diam denotes the diameter of set. Denote by Wdim ε (X, d) the minimal dimension of a polyhedron P bearing an (ε, d)-embedding f : (X, d) → P .

The following important lemma is proved first by Gromov [6, Section 1.11 and 1.12] and admits a proof via Brouwer's fixed point theorem [2, Proposition 4.6.5].

Lemma 2.1. For every 0 < ε < a and d ∈ N, we have

Wdim ε ([0, a] d , | • | ∞ ) = d.
Let ϕ : X → X be a continuous map. For a compatible metric d on X and n ∈ N, we obtain a larger compatible metric

d n (x, x ) := max 0≤i≤n-1 d(ϕ i (x), ϕ i (x )).
It is easy to check that the sequence a n := Wdim ε (X, d n ) is subadditive in the sense that a n+m ≤ a n + a m for any n, m ∈ N. Thus we obtain a well-defined notation:

mdim ε (X, d) := lim n→∞ Wdim ε (X, d n ) n . Definition 2.2. The mean (topological) dimension of (X, ϕ) is mdim(X, ϕ) := sup ε>0 mdim ε (X, d).
It is easy to see that the definition is independent of choices of compatible metrics.

Remark 2.3. Lindenstrauss and Weiss introduced another equivalent definition of mean (topological) dimension in terms of the overlapping number of finite open covers [START_REF] Lindenstrauss | Mean topological dimension[END_REF]. This definition works for actions on general compact Hausdorff spaces. See a friendly introduction of mean dimension theory in [START_REF] Coornaert | Topological Dimension and Dynamical Systems[END_REF]. 2.2. Inverse limit. Let {X n } ∞ n=1 be a sequence of compact metrizable spaces with surjective continuous maps ψ n : X n+1 → X n for all n ∈ N. The inverse limit of ψ n 's , denoted as lim ← -

X n , is defined as the subset of n≥1 X n consisting of all elements (x n ) n∈N satisfying ψ n (x n+1 ) = x n for all n ∈ N. Now let ϕ n : X n → X n be a surjective continuous map such that

ψ n • ϕ n+1 = ϕ n • ψ n for every n ∈ N. In other words, ψ n : (X n+1 , ϕ n+1 ) → (X n , ϕ n )'s is a sequence of factor maps of dynamical systems (X n , ϕ n )'s. Define Φ : lim ← - X n → lim ← - X n by sending (x n ) n∈N to (ϕ n (x n )) n∈N .
Definition 2.4. The dynamical system given by (lim ← -X n , Φ) is called the inverse limit of dynamical systems (X n , ϕ n )'s.

For a (not necessarily surjective) dynamical system (X, ϕ), it is easy to check that the continuous map ϕ restricted on ∩ k≥1 ϕ k (X) is surjective. Notice that if ϕ is surjective, then ∩ k≥1 ϕ k (X) = X. Definition 2.5. For a dynamical system (X, ϕ), setting X n = ∩ k≥1 ϕ k (X) and ψ n = ϕ n = ϕ, the induced inverse limit is called the natural extension of (X, ϕ), which we denote by ( ‹ X, ϕ).

The following proposition shows that the mean dimension of the natural extension ( ‹ X, ϕ) is no greater than the mean dimension of (X, ϕ). Proposition 2.6. [15, Proposition 5.8] For an inverse limit of dynamical systems (X n , ϕ n )'s, we have

mdim(lim ← - X n , lim ← - ϕ n ) ≤ lim n→∞ mdim(X n , ϕ n ).
In particular, for a dynamical system (X, ϕ), we have mdim( ‹ X, ϕ) ≤ mdim(X, ϕ).

Mean rank and mean dimension

In this section, similar to [START_REF] Li | Mean dimension, mean rank, and von Neumann-Lück rank[END_REF], we introduce the notion of mean rank, discuss its nice properties, and establish its connection with mean dimension.

Let A be a discrete abelian group and ϕ : A → A a group homomorphism. For a finite subset E of A, consider the subgroup of A generated by {ϕ i (E) : i = 0, • • • , n -1}, which we denote by T n (E, ϕ).

We call T n (E, ϕ) the n-trajectory of E. For a subgroup B of A, we denote by rk(B) the rank of B. Since the sequence {rk(T n (E, ϕ))} n∈N is subadditive, the limit lim n→∞ rk(Tn(E,ϕ)) n

exists, which we denote by mrk(E, ϕ). Definition 3.1. For a discrete abelian group A and ϕ : A → A a group homomorphism, the mean rank of A is defined as

mrk(A, ϕ) := sup E mrk(E, ϕ)
for E ranging over all finite nonempty subsets of A.

Let {A n } ∞ n=1 be a sequence of discrete abelian groups and ϕ n ∈ End(A n ) space of endomorphisms for every n ∈ N. Suppose that {ψ n,m :

A n → A m } is a directed system of discrete abelian groups A n 's, i.e. ψ n,m • ϕ n = ϕ m • ψ n,m for any n, m ∈ N with n ≤ m. Denote by λ n the natural embedding A n → ⊕ n≥1 A n . Then an explicit construction of the colimit of {ψ nm : A n → A m } is lim -→ A n := (⊕ n≥1 A n )/S
where S is the subgroup of ⊕ n≥1 A generated by all elements

λ m • ψ n,m (a n ) -λ n (a n )
with n ≤ m and a n ∈ A n (see [START_REF] Rotman | Advanced Modern Algebra. Third Edition[END_REF]).

The compatible ϕ n 's then induce a homomorphism Φ : lim Proposition 3.2. Let {ψ n,m : A n → A m } be a directed system of discrete abelian groups and

-→ A n → lim -→ A n sending (a n ) n to (ϕ n (a n )) n .
ϕ n ∈ End(A n ) such that ψ n,m • ϕ n = ϕ m • ψ n,m for any n, m ∈ N with n ≤ m. Denote by A n the quotient group A n / ∪ m≥n ker(A n ψn,m → A m ) and ϕ n the map A n → A n sending x to ϕ n (x). Then we have mrk(lim -→ A n , Φ) = sup n≥1 mrk(A n , ϕ n ).
In particular, if A is a discrete abelian group and a group homomrophism ϕ :

A → A is injective, then mrk(A, ϕ) = mrk( - → A , - → ϕ ).
Proof. Consider the injective homomorphism ι n : A n → lim -→ A n by sending x to λ n (x). We have that ι n intertwines ϕ n with Φ and ι n (A n ) increases with ∪ n≥1 ι n (A n ) = lim -→ A n . Since mrk is upper continuous, we see that mrk(A n , ϕ n ) increases to mrk(lim -→ A n , Φ). On the other hand, for a single injective homomorphism ϕ : A → A, we have

A n = A and ϕ n = ϕ. Thus mrk( - → A , - → ϕ ) = sup n≥1 mrk(A, ϕ) = mrk(A, ϕ).
Now we consider a compact metrizable abelian group X. The Pontryagin dual of X, denoted as " X, is defined as the collection of continuous group homomorphisms from X to the unit circle T. Under the pointwise multiplication "

X becomes an abelian group and under the compact-open topology " X makes a discrete group. The classical Pontryagin duality says that the Pontryagin dual of "

X is isomorphic to X as topological groups.

Let ϕ : X → X be a group homomorphism. Then ϕ induces a group homomorphsim ϕ : " X → " X sending χ to χ • ϕ. For convenience, we may write ϕ(χ), x = χ, ϕ(x)

for every χ ∈ " X and x ∈ X. The following theorem establishes the connection between mean dimension and mean rank, which already appears in [START_REF] Li | Mean dimension, mean rank, and von Neumann-Lück rank[END_REF]Theorem 4.1]. For readers' convenience, we present a proof here. Theorem 3.3. Let X be a compact metrizable group and ϕ : X → X a continuous endomorphism. Then mdim(X, ϕ) = mrk( " X, ϕ).

Proof. Let us first estimate the upper bound of mean dimension. Fix a compatible metric d on X. We may consider X as the Pontryagin dual of " X. Then for any ε > 0, there exists A ⊆ " X finite such that whenever χ|

A = χ | A for χ, χ ∈ X we have d(χ, χ ) < ε. Fix n ∈ N and pick a maximal independent subset E of ∪ n-1 i=0 ϕ i (A) and hence |E| = rk( ϕ i (A) : i = 0, • • • , n -1 ). Consider the continuous map π E : X → T E sending χ to the restriction map χ| E . Assume that χ| E = χ | E for some χ, χ ∈ X. By our selection of E, for every a ∈ A and i = 0, 1, • • • , n -1, we have ϕ i (χ), a = χ, ϕ i (a) = χ , ϕ i (a) = ϕ i (χ ), a .
It follows that π E is an (ε, d n )-embedding and hence Wdim ε (X, d n ) ≤ dim(T E ) = |E| = rk(T n (A, ϕ)).

Let n → ∞ and then ε → 0, we obtain mdim(X, ϕ) ≤ mrk( " X, ϕ).

In the rest of this section, we will prove Corollay 1.4. Firstly, we recall the definition of algebraic cellular automatons. Definition 4.3. For a compact metric space X and a continuous map f : X I → X with a finite set I ⊆ Z, the cellular automaton on X Z with local rule f is the continuous map F : X Z → X Z sending (x n ) n∈Z to (f ((x n+j ) j∈I )) n . We say F is algebraic if X is a compact metric abelian group and f is a group homomorphism.

Next, we give a proof of Corollay 1.4.

Corollary 4.4 (Corollay 1.4). For an algebraic cellular automaton, we have mdim(X Z , F ) = mdim( › X Z , F ).

Proof. Since f is a continuous homomorphism between compact metric abelian groups, so is F . Applying Theorem 1.2 to the induced dynamical systems (X Z , F ), we see that the mean dimension of the natural extension ( › X Z , F ) stays the same as mdim(X Z , F ).

  In particular, as A n = A, ψ n,m = id, and ϕ n = ϕ for all n, m ∈ N, we denote by -→ A the colimit and -→ ϕ the corresponding map on -→ A . The following proposition is in the same line of [3, Proposition 3.5].

Now we estimate the lower bound of mean dimension. Fix a finite subset A ∈ " X and n ∈ N. Let E ⊆ ∪ n-1 i=0 ϕ i (A) be a maximal linearly independent subset and hence |E| = rk(T n (A, ϕ)). We then desire to embed [0, 1/2] E into X topologically.

Consider the collection {H ≤ " X : H ∩ E = 0} with the partial order of set containment. By Zorn's Lemma, there exists a maximal subgroup

Then the desired metric d can be defined via

This completes the proof.

Remark 3.4. In light of [4, Definition 4.2], we can extend the notion of mean rank to the actions by amenable semigroups and expect the above connection holds.

mean dimension of the natural extension

In this section we give the proof of Theorem 1.2 and Corollary 1.4. For a dynamical system (X, ϕ), recall that a point x ∈ X is called non-wandering if for every open set U containing x and N ≥ 1, there is k ≥ N such that U ∩ ϕ -k U = ∅. Denote by Ω(X) the collection of non-wandering points of (X, ϕ). Lemma 4.1. For a dynamical system (X, ϕ), we have mdim(X, ϕ) = mdim(∩ n≥1 ϕ n (X), ϕ).

Proof. Following the same argument of [START_REF] Gutman | Embedding topological dynamical systems with periodic points in cubical shifts[END_REF]Lemma 7.2] for the noninvertible case, we still have mdim(X, ϕ) = mdim(Ω(X), ϕ). Since X ⊇ ∩ n≥1 ϕ n (X) ⊇ Ω(X), it follows that mdim(X, ϕ) ≥ mdim(∩ n≥1 ϕ n (X), ϕ) ≥ mdim(Ω(X), ϕ) = mdim(X, ϕ). Now we are ready to complete the proof of the main theorem. We repeat our main result as follows.

Theorem 4.2 (Theorem 1.2). For any compact metrizable abelian group X and a continuous homomorphism ϕ : X → X, we have the equality mdim(X, ϕ) = mdim( ‹ X, ϕ).

Proof. First we deal with the case that ϕ is surjective. From Proposition 2.6, it suffices to show mdim(X, ϕ) ≤ mdim( ‹ X, ϕ).

By definition, for each element (χ

Thus χ 1 (x 1 ) = χ n (x n ) for all n ∈ N. Moreover we see that the map π is well defined and injective. Note that

X is an injective group homomorphism that intertwines -→ ϕ with ϕ. Thus we have mrk(

). Since ϕ is surjective, we get that the map ϕ is injective. By Proposition 3.2, we obtain mrk( "

This completes the proof when the map ϕ is surjective. Now we consider the general case that ϕ is not necessarily surjective. Denote by Y the subspace ∩ n≥1 ϕ n (X). Since ϕ| Y is surjective, we have mdim( ‹ Y , ϕ) = mdim(Y, ϕ). By Lemma 4.1 and the definition of ( ‹ X, ϕ), we have mdim(X, ϕ) = mdim(Y, ϕ) = mdim( ‹ Y , ϕ) = mdim( ‹ X, ϕ).


Rebeca Da

Rocha Grangeiro

Lucas Emmanuel

Nascimento Silva

Catherine Esnard

Rebeca Da Rocha Grangeiro

Lucas Emmanuel Nascimento

I broke the glass ceiling, now what? Overview of metaphors to explain gender inequality in organizations

Keywords: Gender, Inequality, Literature Review, Metaphors, Women's Careers

Purpose: This paper aims to identify and systematically summarize the relevant research on metaphors that are used to explain gender inequalities in the organizational context.

Design/methodology/approach: The researchers conducted a systematic literature review using bibliometric techniques and content analysis.

Findings:

The systematic literature review identified a total of 1,269 papers in 688 journals written by 2,441 authors. The first paper was published in 1971 and the publication growth can be observed in the five decades analyzed. Gender in Management: An International Journal, was the journal that published most papers about this thematic. Michelle Ryan was the most influential researcher regarding the number of papers and citations she had. The content analysis performed with the 27 most influential papers showed four research streams (Metaphors; Gender and Leadership; Challenges, Stereotypes and Toxic Environments; and Gender in Academia). Concerning the metaphors explored in those papers, the glass ceiling is the most prominent. Furthermore, 26 kinds of barriers were identified regarding the challenges that women face in reaching positions of power.

Practical implications:

The scrutiny of the metaphors and barriers enables access to what is being a hindrance to female progression in the organizational structure. Thus, this study may instrumentalize organizations and women to improve gender diversity practices in the workplace.

Originality/value: The value of the paper lays in the extensive literature review, using a bibliometric approach and content analysis.

Introduction

The current literature on gender inequality in the organizational context has focused on practices that help to create inequality; thus hiring, role allocation, promotion, compensation, and structuring are factors which create an unequal environment [START_REF] Amis | The organizational reproduction of inequality[END_REF]. Despite some progress, organizations remain places of male-dominance [START_REF] O'neil | Women's careers at the start of the 21st century: Patterns and paradoxes[END_REF].

To explain the challenges that women face in organizations, the literature often adopts the concept of metaphors. In psychology, management, leadership, and gender literature, some predominant metaphors are being explored by scholars, such as Glass Ceiling [START_REF] Dozier | Breaking Public Relations' Glass Ceiling[END_REF], Queen Bee Phenomenon [START_REF] Staines | The queen bee syndrome[END_REF], and Glass Cliff [START_REF] Ryan | The glass cliff: Evidence that women are overrepresented in precarious leadership positions[END_REF]. The metaphors improve the comprehension of individual and organizational behavior in creating a hostile environment and culture toward female workers at the same time as focussing attention toward these pressing issues [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF].

Women's job evaluation is still based on male patterns and thus, the theories and models to analyze their careers are derived from a male's career [START_REF] O'neil | Women's careers at the start of the 21st century: Patterns and paradoxes[END_REF].

Therefore, it is important to understand the female career path, challenges, and aspirations. Furthermore, research on female careers has been frequently fragmented through a myriad of knowledge fields and research traditions [START_REF] O'neil | Women's careers at the start of the 21st century: Patterns and paradoxes[END_REF].

Hence, integration and summarization of this research field would help to create organizational and public policies that could change the landscape for women in the workplace. In this sense, a literature review [START_REF] Snyder | Literature review as a research methodology: An overview and guidelines[END_REF][START_REF] Tranfield | Towards a methodology for developing evidence-informed management knowledge by means of systematic review[END_REF] is an appropriate solution as it provides a relevant evaluation of the available literature on a topic and it can offer some insights based on evidence to the relevant stakeholders.

Despite the presence of extensive works on these metaphors, there is a gap concerning a more systematic and holistic view of them. No systematic review that fully investigated the myriad of metaphors was found, and the publications that reviewed this literature have often focused on the comparison between two or three metaphors (see. Bendl andSchmidt, 2010, Carli and[START_REF] Carli | Women face a labyrinth: An examination of metaphors for women leaders[END_REF] or proposed a non-systematic review (see. [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF]. Thus, the research is often fragmented, and a systematic review of these concepts is needed.

To fill this research gap, this paper aims to identify and systematically summarize the relevant research on metaphors that are used to explain gender inequalities in the organizational context. To do so, a systematic literature review was conducted through bibliometrics and content analysis methods using data from the Scopus and Web of Science databases.

This paper provides both theoretical and practical implications. By organizing the fragmented literature, there is an opportunity to identify how it is evolving, and what the main results are. This information can provide insights into the development of new theories and a further understanding of the ones available. Besides this, gender disparity is a worldwide issue, and communicating the elements that constrain the development of women's careers can trigger important discussions for the development of public policies and new organizational practices to change the current unequal landscape.

How Metaphors Shape Women's Careers

The metaphors are useful to explain a social phenomenon as they contribute to structure the basic understanding of the world, providing new insights, catching and leading the attention to important issues [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF]. The most popular is the Glass Ceiling [START_REF] Carli | Women face a labyrinth: An examination of metaphors for women leaders[END_REF], which was first used in a publication in The Wall Street Journal in 1986. The glass ceiling metaphor is used to illustrate the invisible obstacles that women face and that explains the low representativeness of women in senior positions [START_REF] Dozier | Breaking Public Relations' Glass Ceiling[END_REF][START_REF] Morrison | Women and minorities in management[END_REF].

Recently, this phenomenon has been investigated in different contexts, such as academia [START_REF] Roberto | The academic "glass-ceiling": investigating the increase of female academicians in Italy[END_REF], journalism [START_REF] Rao | Gazing past the glass ceiling: Indian and South African female journalists' perceptions of their role and power in the newsroom[END_REF], and mathematical applications to identify the gender imbalance (Ó Náraigh, 2019). The result of the glass ceiling is that women are kept in middle or low-rank jobs, resulting in stereotypes that women are more suitable for wifely and maternal roles [START_REF] Rudman | Backlash effects for disconfirming gender stereotypes in organizations[END_REF] or not ambitious enough [START_REF] Marry | Le "plafond de verre" en France: le public plus égalitaire que le privé?[END_REF] to get into leadership positions.

More recently, [START_REF] Carli | Women face a labyrinth: An examination of metaphors for women leaders[END_REF] propose that the glass ceiling does not fully represent the current conditions of women, and state that Labyrinth is a more useful metaphor as it is more subtle and complex, being valid with a varying number of obstacles, and even being effective when the leadership conditions for women change.

Regardless of the discussion, it is noted that these metaphors can help to better understand social phenomena and to shape social perception [START_REF] Carli | Women face a labyrinth: An examination of metaphors for women leaders[END_REF].

Another phenomenon widely discussed in the literature is the Queen Bee [START_REF] Staines | The queen bee syndrome[END_REF]. This metaphor indicates that in a context where there is male dominance, women who achieve leadership positions become a barrier to the progression of other women [START_REF] Derks | Gender-bias primes elicit Queen-Bee responses among senior policewomen[END_REF]. There are other metaphors concerning the inequality in the gendered organization, such as Glass Cliff, which indicates that women are more prone to be promoted to leadership positions when the company is in a crisis (Ryan andHaslam, 2005, 2007). However, female CEOs are still more likely to be fired even when the company is going well [START_REF] Gupta | You're fired! Gender disparities in CEO dismissal[END_REF].

Although these metaphors are useful to communicate the challenges that women face, some critiques have been made concerning them. [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF] state that the metaphors are not capable of shattering the glass ceiling, as evidence shows that female participation in leadership positions is still small. Nonetheless, it is relevant to notice that these discussions around the metaphors help to take the challenges that women face into mainstream research, and raise awareness that both public and management policies may reduce their impact.

Method

Planning the Review

The use of the literature review as a research method provides evidence-based results and insights into the analysis field [START_REF] Tranfield | Towards a methodology for developing evidence-informed management knowledge by means of systematic review[END_REF]. Therefore, as the objective of the paper is to identify and systematically summarize the relevant research on metaphors that are used to explain gender inequalities in the organizational context, the use of the systematic literature method is appropriate and can contribute to further develop the body of knowledge available [START_REF] Snyder | Literature review as a research methodology: An overview and guidelines[END_REF][START_REF] Tranfield | Towards a methodology for developing evidence-informed management knowledge by means of systematic review[END_REF].

Bibliometric analysis allows the assessment of a large number of scientific papers from different scientific databases. Hence, it was considered relevant in investigating the extensive literature on metaphors. Furthermore, content analysis was used to deepen the understanding of the research and increase the data from the quantitative analysis provided by the bibliometrics. This research followed the three steps recommended by [START_REF] Tranfield | Towards a methodology for developing evidence-informed management knowledge by means of systematic review[END_REF]: planning the review, conducting a review, and reporting and dissemination. The first step was to establish the research questions of the study. They were defined to achieve the objective of the paper. The questions, their quantitative or qualitative nature, and the methodology used to answer them are presented in Figure 1.

Research Questions

Note. Figure elaborated by the researchers to establish the research questions and the information that would be collected from the papers.

After the proposition of the research questions, the search terms were defined, considering the literature that discussed the metaphors which are used to explain gender inequalities in the organizational context [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF], they were: Glass Ceiling, Glass Cliff, Glass Escalator, Glass Floor, Glass Walls, Velvet Ghettos, Concrete Ceiling, Glass Doors, Tokenism, Homosocial Reproduction, Homophily, Old Boys Network, Backlash Effects, Queen Bee Phenomenon, Career Tournaments, Matilda Effect, Leaky Pipeline, Firewall, Scissors Effect, and Labyrinth. Aiming to identify the reliability of the chosen search terms and to certify that the scrutiny would be as holistic as possible, the researchers sent them via email to two other researchers specialized in women's careers and they evaluated whether the terms were sufficient to cover this knowledge field of metaphors. Thus, as the peers indicated the adequacy of the search terms, the research proceeded to the next step.

Conducting the Review

To access the papers, the Scopus and the ISI Web of Sciences databases were chosen since they are the most important databases and cover a large amount of research [START_REF] Alryalat | Comparing bibliometric analysis using PubMed, Scopus, and Web of Science databases[END_REF]. The extraction process occurred on April 22, 2020. It included all papers that were published until this date. Three steps were established to select the papers. The first one was the research field. Papers were selected from business, economics, psychology, and social sciences fields, since those areas are historically more concerned with the issues of metaphors that address gender inequalities. Then, papers in English were selected to assess the international literature. Finally, only articles and reviews that passed through peer review were selected, thus attesting to the quality of the data. The extraction process is presented in Figure 2. Given the large amount of data the extraction provided, each search term was excerpted manually, as the Scopus limit for export and viewing was 2,000 documents.

The duplicated articles were excluded and the two databases were merged using the Bibliometrix Package in R (Aria and Cuccurllo, 2017). This resulted in 4,280 documents from Scopus, and 979 from ISI Web of Science. The merging of the two databases with the exclusion of the duplicates resulted in a total of 4,462 documents.

The research framework is shown in Figure 3. The 4,462 documents were exported into a .xlsx file to be opened using Microsoft Excel. They were then analyzed through a critical assessment of their titles, abstracts, and keywords. Papers that did not discuss issues of gender and female careers were excluded, resulting in a total of 1,269.

Bibliometric Analysis

Bibliometric and quantitative analyses were conducted of all papers to assess the evolution of the topic in question and to answer RQ1 and RQ2. The VOSviewer software was used to develop networks, and keyword network analyses were performed. Analyses were also carried out based on the year of the paper, leading researchers, and top journals.

Content Analysis

After the bibliometric analysis, a content analysis of the most relevant papers in the database was conducted. This analysis was chosen given its flexibility and applicability to a myriad of organizational phenomena. It followed the four steps presented by [START_REF] Duriau | A content analysis of the content analysis literature in organization studies: Research themes, data sources, and methodological refinements[END_REF]: data collection, text coding, content analysis, and interpretation of results. Given a large sum of articles on the database, the number of 200 citations was chosen as selection criteria, resulting in 27 documents.

The analysis was carried out as follows: first, the texts were read and codes were proposed for each paper based on the purpose of the research and its theoretical foundations. This step resulted in 9 specific codes: Backlash against women (1 paper); The content analysis process identified the topics that are being explored by these relevant papers; the work context observed; the metaphor(s) presented; the field of knowledge; the main results, and their implications. The analysis followed the frequency counts, and the papers were summarized according to their codes. To answer RQ3, the second level of analysis was used [START_REF] Duriau | A content analysis of the content analysis literature in organization studies: Research themes, data sources, and methodological refinements[END_REF], seeking to interpret the content on a deeper level.To answer RQ4, the first level was applied, using descriptive statistics to assess the trends and topics explored by the literature.

Findings Bibliometric Analysis

A total of 1,269 documents comprised the database. These papers were divided into 688 journals, by 2,441 authors and co-authors. The first paper was published in 1971 and was entitled "The Status of Women in the Profession: Tokenism" [START_REF] Jaquette | The Status of Women in the Profession: Tokenism[END_REF]. It discusses the metaphor of tokenism, which indicates the challenges that women face when entering a male-dominated sphere. Given the low portion of women, they gain a status of tokens, which causes the negative influences they face on the job and the difficulty to have equal treatment (Zimmer, 1998). [START_REF] Jaquette | The Status of Women in the Profession: Tokenism[END_REF] reviewed the status of women in the political sciences. In the first ten years of research, four papers were published.

RQ1: How has the research on metaphors that are used to explain gender inequalities in the organizational context literature has evolved over the years?

Analyzing the decades, the period with the biggest number of papers is from 2011 to 2020, which indicates an increase in the interest of scholars to investigate inequality in the workplace. Figure 4 shows the research evolution. impacting, and being impacted by women's progress to the top of the organizational hierarchy in their careers.

RQ2: What and who are the current and most influential journals and authors?

From the 688 journals that published the articles from the sample, 8 of them published 12 articles or more, as presented in Figure 5. The eight journals publish papers about gender and inequality in the organizational context. Five of them have gender discussion as their core. However, it is important to notice that the discussion about inequality in the workplace has a fundamental impact in the organizational context, therefore, there is a need to widely explore these issues in business journals to present the result to managers and decisionmakers.

The leading researchers who are discussing these metaphors to explain gender inequalities the organizational context are presented in Figure 6. The main author, considering the number of papers and citations, is Professor Michelle Ryan from the University of Exeter; she is concerned with issues of the Glass Cliff metaphor. Among the other authors, it is possible to notice that Glass Cliff and Queen Bee Phenomenon are the metaphors discussed by them with a more solid research base. Concerning the topics that were explored by the literature, a co-occurrence of keyword network analysis was performed. This was done so using all the keywords, and resulted in five keyword clusters. The criteria to select the keywords were the ones with more than 10 mentions across any number of papers. This resulted in a total of 107 words, divided into five clusters, as shown in Figure 7. The first cluster (red) highlights the word gender with 314 occurrences, and summarizes the issues discussed in these metaphors, such as inequality and gender stereotypes. Three metaphors are also mentioned (Glass Cliff, Glass Escalator, and Queen Bee), along with some relevant terms, such as board, diversity, and governance. These topics are widely explored in the reviewed literature since diversity in the board can reduce inequality and increase the presence of women in leadership positions [START_REF] Cook | Women and top leadership positions: towards an institutional analysis[END_REF]. The second cluster (purple) contains the discussion on mentoring and careers, since women often have a lack of mentorship at the beginning of their career and this can be a key element to career progression. The third cluster (yellow) discusses the theoretical issues of gender, male and female, feminism, and gender identity.

The fourth cluster (green) reinforces issues of gender faced by women, such as the wage gap, inequality, and their status in the organization. The fifth and last cluster (blue) includes the discussion on the Glass Ceiling as a challenge that women face to progress in their career, therefore, equality is harder to reach since there are obstacles that keep women in middle and low positions in the organizational structure.

Content Analysis

The 27 papers were divided into 4 sections (Metaphors; Gender and Leadership;

Challenges, Stereotypes, and Toxic Environment; and Gender in Academia). The following metaphors were discussed in 27 papers: Glass Ceiling (15 papers), Backlash

Effects (3 papers), Glass Cliff (2 papers), Glass Escalator (2 papers), Tokenism (2 papers), Labyrinths (1 paper), Leaky Pipeline (1 paper) and Matilda Effect (1 paper).

Regarding the methodological procedures, the following methods were identified: Quantitative (14 papers); Literature Review (5 papers); Theoretical (4 papers); Qualitative (2 papers); and Quantitative and Qualitative (2 papers).

The suitability of women leadership roles was the focus of the papers' analyses [START_REF] Rudman | Prescriptive gender stereotypes and backlash toward agentic women[END_REF] and most of them presented the challenges that those women face in achieving a position of power, such as being assigned to a precarious position [START_REF] Ryan | The glass cliff: Evidence that women are overrepresented in precarious leadership positions[END_REF] and hurdles they must face throughout their career (Probert, 2000;[START_REF] Rossiter | The Matthew Matilda effect in science[END_REF][START_REF] Ragins | Gender gap in the executive suite: CEOs and female executives report on breaking the glass ceiling[END_REF][START_REF] Lyness | Climbing the corporate ladder: do female and male executives follow the same route?[END_REF].

The samples investigated were mainly from the United States (10 papers), Sweden (2 papers), Norway (1 paper), United Kingdom (1 paper), and Australia (1 paper). Some publications investigated more than one country at once, such as [START_REF] Arulampalam | Is there a glass ceiling over Europe? Exploring the gender pay gap across the wage distribution[END_REF] which investigated 11 countries from the European Community, and [START_REF] Reynolds | Women in the legislatures and executives of the world: Knocking at the highest glass ceiling[END_REF] which examined 180 nations with autonomous governments from the 5 continents.

1990). The results of the papers about the glass ceiling showed that women often have less authority, receiving fewer stock options, less international mobility [START_REF] Lyness | Above the glass ceiling? A comparison of matched samples of female and male executives[END_REF], and lower wages than men [START_REF] Albrecht | Is there a glass ceiling in Sweden?[END_REF][START_REF] Arulampalam | Is there a glass ceiling over Europe? Exploring the gender pay gap across the wage distribution[END_REF]. This phenomenon was also analyzed in the political landscape [START_REF] Reynolds | Women in the legislatures and executives of the world: Knocking at the highest glass ceiling[END_REF]. Thus, the Glass Ceiling was identified as a gendered phenomenon and couldn't be observed with racial inequalities between men [START_REF] Cotter | The glass ceiling effect[END_REF].

In female-dominated fields, men, especially white men, hold an advantage in promotion, a phenomenon called Glass Escalator [START_REF] Williams | The glass escalator: Hidden advantages for men in the "female" professions[END_REF][START_REF] Maume | Glass ceilings and glass escalators: Occupational segregation and race and sex differences in managerial promotions[END_REF]. To [START_REF] Eagly | Women and the labyrinth of leadership[END_REF], the Labyrinth is a metaphor more appropriate than Glass

Ceiling to explain the challenges that women face to take a leadership role.

The presence of women on boards is relevant because it brings new values and outcomes to the organization [START_REF] Torchia | Women directors on corporate boards: From tokenism to critical mass[END_REF][START_REF] Terjesen | Women directors on corporate boards: A review and research agenda[END_REF]. However, women are often hired for precarious leadership positions, reinforcing the women's image as disqualified for the position (Ryan andHaslam, 2005, 2007;[START_REF] Rudman | Backlash effects for disconfirming gender stereotypes in organizations[END_REF].

Gender and Leadership

Gender is relevant to leadership as it can shape the assertiveness of men and women, their job evaluation, their influence, and their probability of becoming leaders [START_REF] Ridgeway | Gender, status, and leadership[END_REF]. Regarding this probability, men and women face different paths to achieve leadership positions, the latter facing more barriers [START_REF] Lyness | Climbing the corporate ladder: do female and male executives follow the same route?[END_REF].

Previous research showed the increase of women in the corporate board, whereas the increase in CEO positions was not highlighted [START_REF] Daily | A decade of corporate women: Some progress in the boardroom, none in the executive suite[END_REF].

Although diversity in the board is important [START_REF] Terjesen | Women directors on corporate boards: A review and research agenda[END_REF] and can increase innovation [START_REF] Torchia | Women directors on corporate boards: From tokenism to critical mass[END_REF], the organization needs to be held accountable for the corporate practices that help to reinforce the inequality present in CEO roles, such as promotion and compensation practices [START_REF] Oakley | Gender-based barriers to senior management positions: Understanding the scarcity of female CEOs[END_REF].

Despite the prejudice that female leaders face for not being considered as having "what it takes" to be a leader [START_REF] Rudman | Status incongruity and backlash effects: Defending the gender hierarchy motivates prejudice against female leaders[END_REF], some women achieve this status and highlighted some strategies to do so: constantly exceed performance expectations; develop a style that men are comfortable with; seek difficult or high visibility assignments; have an influential mentor; and network with influential colleges. The barriers they faced to get to the top were as follows: lack of significant general management experience; women not in the pipeline long enough; men stereotyping and holding preconceptions; exclusion from informal networks; and inhospitable corporate culture [START_REF] Ragins | Gender gap in the executive suite: CEOs and female executives report on breaking the glass ceiling[END_REF].

Challenges, Stereotypes, and Toxic Environment

The male-dominated environment can propose challenges for women's performance [START_REF] Inzlicht | A threatening intellectual environment: Why females are susceptible to experiencing problem-solving deficits in the presence of males[END_REF] and the stereotypes imposed on women help to perpetuate the gender issues in the organization. The lack of women in high leadership roles and the image of men as agentic poses a challenge for women to negotiate salaries and activities. The outcomes of negotiating are different for women because they have less access to power and status, thus reinforcing the glass ceiling in the organization [START_REF] Stuhlmacher | Gender differences in negotiation outcome: A meta-analysis[END_REF]. Even when women show leadership agentic behavior, they suffer a backlash, because women are not supposed to be agentic [START_REF] Rudman | Prescriptive gender stereotypes and backlash toward agentic women[END_REF].

Gender in academia

The academic community is a place where knowledge is created and reproduced, thus building citizens that think critically and contribute to build a better society. However, the underrepresentation of women in science is an issue that needs to be addressed [START_REF] Rossiter | The Matthew Matilda effect in science[END_REF]. This systematic undervaluation of women in science (also in literature, history, and medicine) helps to perpetuate those fields as maledominated and the accomplishments of those women are often thought of as male ones [START_REF] Rossiter | The Matthew Matilda effect in science[END_REF]. The undervaluation of women in science can also trigger an underrepresentation of women in STEM [START_REF] Blickenstaff | Women and science careers: leaky pipeline or gender filter?[END_REF]. The lack of role models and representativeness [START_REF] Rossiter | The Matthew Matilda effect in science[END_REF] contributes to women's scarcity in science [START_REF] Blickenstaff | Women and science careers: leaky pipeline or gender filter?[END_REF].

When it comes to gender disparities in the academic context, research shows how it is important to investigate the household impact on women's life. Besides developing equality initiatives, organizations need to identify how this domain affects the life of women and help to perpetuate those disparities [START_REF] Probert | I just couldn't fit it in': Gender and unequal outcomes in academic careers[END_REF].

Discussions and Conclusions

Research Implications

Notwithstanding some changes in the organizations, the workplace shows reminiscences of the gendered organization [START_REF] Williams | Gendered organizations in the new economy[END_REF]. The investigation of those social and organizational-constructed barriers that women face on a daily basis has resulted in the research stream of the metaphors [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF]. The relevance of this research lies in the fact that they can better explain, represent, and show the reality of women in the workplace [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF].

This paper extends the literature available on those metaphors that show how gender disparities are perpetuated in the organization, and how women's careers are in other countries with higher indicators of inequality, especially developing ones, gender could play a bigger role and thus be a useful context to bring new insights to this discussion (see. [START_REF] Alhadhrami | Women's enactment of leadership in a heavily gender-marked Islamic context[END_REF]. The effects of culture in gender issues have to be considered when investigating these metaphors, and further research could, for instance, investigate the existence of these metaphors in overlooked countries and thus advance the understanding of the factors and consequences that these discriminatory practices bring to the lives of women. Finally, the scholars concerned with those metaphors need to explore intersectionality by expanding the discussion on race, sexuality, and class.

Practical Implications

This research contributes by critically assessing the available research on metaphors that are used to explain gender inequalities in the organizational context and providing a guideline of journals and authors who are advancing the field. Furthermore, the content analysis supports the presentation of the main topics and of the discussions that the authors are introducing.

Still, the inspection of those metaphors and barriers enables companies and women to identify the practices that create and reproduce gender inequalities. It is important to revisit the organizational practices and assess what is being a hindrance to female progression in the organizational structure. Diversity is relevant to the company and can add more value to the business, thus it should be considered a priority.

Suggestions for Future Studies

Despite being able to answer the research questions, this study has limitations.

The citation criteria were chosen to select papers for content analysis, so further research could use metrics that privilege both experienced and early career researchers and select papers considering other factors, such as the sector investigated or specific metaphors to enhance the understanding of this phenomenon. Another limitation concerns the use of one language to search, so using more than English could extend the reach of the study. Research must advance in analyzing different cultures, exploring metaphors other than Glass Ceiling, and other professional careers, such as women in arts, politics, or health jobs. Future research could also propose the integration of the studies in meta-analyses.

  Figure 1.

Figure 2 .

 2 Figure 2. Extraction Process

Figure 3 .

 3 Figure 3. Research Framework

  Gender and Leadership Roles (10 papers); Gender in The Negotiation Process (1 paper); Gender in Academy (3 papers); Gender Stereotypes (1 paper); Glass Ceiling (7 papers); Glass Cliff (2 papers); Glass Escalator (1 paper); Threatening Environment for Women (1 paper). After this first coding process, the codes were analyzed to find similarities between them. This resulted in the following codes: Gender and Leadership (10 papers); Challenges, Stereotypes, and Toxic Environment (3 papers); Metaphors (11 papers); Gender in Academia (3 papers).

  Figure 4. Research Evolution

  Figure 5. Main Journals

  Figure 6. Main Authors

  Figure 7. Keywords Network

RQ3: What are the barriers to women taking leadership positions in the organizations?

Reading the papers, 22 barriers were found regarding the challenges that women face in achieving a leadership post in an organization. These were: Systemic Barriers;

RQ4: What are the current trends and topics being explored in the research on metaphors that explain the low representativeness of women in the organizational context?

Metaphors

The papers revolve around the Glass Ceiling (see. [START_REF] Morrison | Women and minorities in management[END_REF][START_REF] Lyness | Above the glass ceiling? A comparison of matched samples of female and male executives[END_REF][START_REF] Reynolds | Women in the legislatures and executives of the world: Knocking at the highest glass ceiling[END_REF][START_REF] Cotter | The glass ceiling effect[END_REF][START_REF] Albrecht | Is there a glass ceiling in Sweden?[END_REF], Glass Cliff (Ryan andHaslam, 2005, 2007), Glass Escalator [START_REF] Williams | The glass escalator: Hidden advantages for men in the "female" professions[END_REF][START_REF] Cotter | The glass ceiling effect[END_REF], the Backlash Effect [START_REF] Rudman | Backlash effects for disconfirming gender stereotypes in organizations[END_REF], and the Labyrinth [START_REF] Eagly | Women and the labyrinth of leadership[END_REF] The Glass Ceiling metaphor has been explained by theories postulating differences, discrimination explanations, and systematic barriers (Morrison and Von, shaped by summarizing the main findings and bringing together the results that this field is presenting. The research was conducted using two of the largest databases of scientific research and included 20 metaphors, increasing the coverage of the research and capturing more of the available literature. The joint application of some bibliometric techniques and content analysis strengthen this research's results. To the extent of the researchers' knowledge, no other research introduced a systematic overview of those metaphors.

The results showed that there is still a great focus on the Glass Ceiling metaphor.

However, critique has been explicitly focused on the fact that the glass ceiling may not be the best to explain the current status of women in the workplace (Carli andEagly, 2016, Bendl andSchmidt, 2010) or does not fully work in reducing the gap of women in senior positions [START_REF] Smith | A maze of metaphors around glass ceilings[END_REF]. This overuse of the Glass Ceiling metaphor can prevent the reader from fully understanding other issues that can improve the understanding of gender, such as its intersectionality with race, sexuality, and class [START_REF] Williams | Gendered organizations in the new economy[END_REF], since the Glass Ceiling is considered a gendered phenomenon [START_REF] Cotter | The glass ceiling effect[END_REF]. Furthermore, new configurations of organizations such as new business and startups might have new features that support women to achieve leadership positions, but imposing new barriers in other ways, thus, the use of other metaphors may be a better fit to understand the female career.

The corporate context prevails in the studies, except for in some scientific (see. [START_REF] Rossiter | The Matthew Matilda effect in science[END_REF][START_REF] Probert | I just couldn't fit it in': Gender and unequal outcomes in academic careers[END_REF] and political environments (see. [START_REF] Reynolds | Women in the legislatures and executives of the world: Knocking at the highest glass ceiling[END_REF]. All other papers focus on women in middle and senior positions in the companies. The authors of the main papers still focused their research on the USA and Europe, however