Equivariant embedding of finite-dimensional dynamical systems

We prove an equivariant version of the classical Menger-Nobeling theorem regarding topological embeddings: Whenever a group $G$ acts on a finite-dimensional compact metric space $X$, a generic continuous equivariant function from $X$ into $([0,1]^r)^G$ is a topological embedding, provided that for every positive integer $N$ the space of points in $X$ with orbit size at most $N$ has topological dimension strictly less than $\frac{rN}{2}$. We emphasize that the result imposes no restrictions whatsoever on the acting group $G$ (beyond the existence of an action on a finite-dimensional space). Moreover, if $G$ is finitely generated then there exists a finite subset $F\subset G$ so that for a generic continuous map $h:X\to [0,1]^{r}$, the map $h^{F}:X\to ([0,1]^{r})^{F}$ given by $x\mapsto (f(gx))_{g\in F}$ is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.


Introduction
Various mathematical problems in topology involve instances of the following fundamental question : Given a topological space X and another topological space Y , when does X (topologically) embed in Y ?According to the classical Menger-Nöbeling theorem, a compact metric space X of (Lebesgue covering) dimension less than r 2 admits a topological embedding into [0, 1] r (see [HW41,Theorem 5.2]).
In topological dynamics, the analogous fundamental embedding questions take the following form: Given a group G that acts on two topological spaces X and Y , when does there exist an equivariant embedding of X into Y , namely a continuous function f : X → Y that is a homeomorphism from X onto the image f (X) ⊆ Y and so that f (g(x)) = g(f (x)) for every g ∈ G and every f ∈ X.In this paper, the topological spaces involved are always assumed to be compact and metrizable.So a continuous function f : X → Y defines a homeomorphism from X onto the image f (X) ⊆ Y if and only if it is injective.In this paper, by a topological dynamical system we mean a pair (G, X), where G is a topological group that acts by homeomorphisms on a compact metrizable space X.When Y = ([0, 1] r ) G for some group G is a Tychonoff cube and the action of the group G on Y = ([0, 1] r ) G is the G-shift (see Section 2.4) for a precise definition), the existence of an equivariant embedding of X into Y is equivalent to the existence of a continuous injective mapping f : X → ([0, 1] d ) G for which it holds f (gx) h = f (x) hg for all x ∈ X and g, h ∈ G.The problem of equivariantly embedding into such a space Y , known as a (G−) r-cubical shift has quite a long history and there is a fair amount of literature on this problem, mostly for the case where the group G is generated by a single homeomorphism (that is, G = Z or G = Z/mZ for some m ∈ N) or by finitely many commuting homeomorphisms (so that G is a finitely generated abelian group).We review some of this history in the next paragraph.
In [Jaw74] Jaworski showed that for any action of the group G = Z on any finitedimensional compact metric space X there exists an equivariant embedding of X into [0, 1] G , under the assumption that the generator of Z corrsponds to an aperiodic homeomoprhism of X, where a homeomorphism T : X → X is called aperiodic if T n x = x for all x ∈ X and nonzero integer n.Later Nerurkar [Ner91] showed that the aperiodicity assumption in Jaworski's result can be weakened to the assumption that there are at most finitely many periodic points with the same period.Gutman [Gut16] showed that the aperiodicity assumption in Jaworski's result can be further weakened to the assumption that the set of periodic point in X of period N has dimension strictly less that N r  2 for all N.An extension of this result for actions of finitely generated abelian groups was achieved by Gutman, Qiao and Szabó in [GQS18].
Our main result is a generalization of the above results where the acting group is arbitrary.Moreover as elucidated by Theorem 1.4, our result is sharp in a strong sense.
Theorem 1.1.Let (G, X) be a topological dynamical system where X is a finitedimensional compact metric space.Let r ∈ N. Suppose that for every N ∈ N it holds where Then a generic continuous function f : For the definition of generic, see Subsection 2.5.Theorem 1.1 implies in particular that if a finite group G acts freely on X and dim X < r 2 |G| then X embeds equivariantly into (([0, 1]) r ) G .The case where G is the trivial group coincides with the Menger-Nöbeling theorem.
Remark 1.2.Let G be a group which does not have finite index subgroups (e.g. an infinite simple group), then Condition (1) holds for any t.d.s (G, X), r, N ∈ N. Indeed for x ∈ X, let Fix G (x) = {g ∈ G| gx = x}.It is easy to see that there is a 1-1 correspondence between (left) cosets of Fix G (x) and Gx.
We conclude that for a group G which does not have finite index it holds (G, X) N = ∅ for all N ∈ N.
Remark 1.3.When G is an infinite sofic group (for instance G = Z), it is not possible to remove the assumption that X is finite dimensional in Theorem 1.1.Although in this case any compact metrizable space X embeds (topologically) in [0, 1]) G , there is a further obstruction to the existence of an equivariant embedding, namely the mean dimension of (G, X).Mean dimension is an isomorphism invariant of topological dynamical systems introduced by Gromov ([Gro99]).Heuristically, whereas topological entropy measures the number of bits per unit of time required to describe a point in a system, mean dimension measures the required number of parameters per unit of time.The initial systematic development of mean dimension theory was carried out by Lindenstrauss and Weiss in the seminar paper [LW00].We refrain from defining mean dimension here and refer the interested read to [LW00] and [Li13].Lindenstrauss and Tsukamoto formulated a conjecture in [LT14] regarding sufficient conditions for the existence of an equivariant embedding of a Z-dynamical system in the Z-shift on ([0, 1] r ) Z .The conjectured sufficient conditions involve mean dimension and dimensions of periodic points.For finite dimensional Z-systems, the Lindenstrauss and Tsukamoto conjecture reduces to Gutman's result [Gut16].Additional cases of the conjecture were established in [Gut15,GT20] but in full generality the conjecture is still open.In [GQT19] a general embedding conjecture that generalizes the Lindenstrauss and Tsukamoto conjecture to Z k -actions (X, Z k ) (k ∈ N) first appeared explicitly.In the finite dimensional case, the conjecture is known to hold ( [GQS18]).Some additional cases of the conjecture were established in [GQS18,GQT19] but in full generality the conjecture is still open.In contrast to Theorem 1.1, the existing embedding theorems for infinite dimensional systems seem to rely heavily on the group structure of Z or Z d .Even the formulation of the embedding conjecture in [GQT19] does not trivially extend to actions of non-commutative (say amenable) groups.For a free action (G, X), it is still unknown having infinite mean dimension (when it is well-defined) is the only additional obstruction for the existence of an equivariant embedding into ([0, 1] r ) G for some r ∈ N.
Complementing Remark 1.2, we have the following: Theorem 1.4.(Sharpness of Theorem 1.1) Let N, r ∈ N. Let G be a group which has a subgroup G ′ of index N, then there exists a faithful t.d.s.(G, X) so that dim(G, X) N = ⌈ rN 2 ⌉ (thus Condition 1 does not hold ) and for all continuous functions f : We deduce Theorem 1.1 from the following theorem, our main auxiliary theorem: Theorem 1.5.Let X, Y be compact metrizable spaces and let F = g 1 , . . ., g N be an N-tuple of continuous injective functions from X to Y .For every f : Y → [0, 1] r , let f F : X → ([0, 1] r ) N be given by For every partition P of F let Suppose that for every partition P of F it holds that (2) In addition to proving Theorem 1.1, our main auxiliary theorem has an important application related to the celebrated Takens embedding theorem.We now provide the necessary background.
Consider an experimentalist observing a physical system modeled by a Z-system (X, T ).It often happens that what is observed is the values of k measurements h(x), h(T x), . . ., h(T k−1 x), for a real-valued observable h : X → R. One is led to ask, to what extent the original system can be reconstructed from such sequences of measurements (possibly at different initial points) and what is the minimal number k of delay-coordinates, required for a reliable reconstruction.This question has been treated in the literature by what today is known as Takens-type delay embedding theorems, essentially stating that the reconstruction of (X, T ) is possible for certain observables h, as long as the measurements h(x), h(T x), . . ., h(T k−1 x) are known for all x ∈ X and large enough k.Indeed the first result obtained in this area is the Takens delay embedding theorem ([Tak81]) -for a compact manifold X of dimension d, it is a generic property (w.r.t.Whitney C 2 -topology) for pairs (h, T ), where T : X → X is a C 2 -diffeomorphism and h : X → R a C 2 -function, that the (2d + 1)-delay observation map is an embedding.Note Takens considered a setting where T : X → X and h : X → X are perturbed in order to achieve embedding.The paper [SYC91] introduced a setup where the dynamics is fixed and only the observable is perturbed.Thus in order to achieve embedding, some conditions on periodic points are necessary 1 .Whereas [SYC91] assumed some regularity conditions both on the dynamics and the observable 2 , the paper [Gut16] was the first to study the problem of delayed embedding for fixed dynamics in the purely topological setting.The main theorem of [Gut16] implies that for a is an embedding.Our last result is a generalization of the two above mentioned results to the context of finitely generated group actions.
Theorem 1.6.Let G be a finitely generated group, let S ⊆ G be a finite generating set for G, and let r ≥ 1 be a natural number.Let (G, X) be a topological dynamical system with dim(X) < +∞ such that for every Let S •0 := {e G } and given a natural number n, let Let M be the smallest natural number which satisfies M > 2 dim(X) r , and let Structure of the paper: Section 2 contains basic definitions.In Subsection 3.1, Theorem 1.1 is proven assuming Theorem 1.5.In Subsection 3.2, Theorem 1.4 is proven.Section 4 contains the proof of Theorem 1.6 assuming Theorem 1.5 as well as Example 4.3 showing that if the group is not finitely generated then the conclusion of Theorem 1.6 does not necessarily hold.Section 5, where Theorem 1.5 is proven, is the main technical part of the article.

Preliminaries
2.2.Dimension.Let X be a metric space.Let α and β be finite open covers of X.We say that β refines α, denoted β ≻ α, if every member of β is contained in a member of α.The join of α and β is defined as α ∨ β = {A ∩ B| A ∈ α, B ∈ β}.Similarly, one may define the join n i=1 α i of any finite collection of open covers α i , i = 1, . . ., n, of X. Assume α consists of the open sets U 1 , U 2 . . ., U n .Define its order by ord(α) = max x∈X U ∈α 1 U (x) − 1 and let D(α) stand for the minimum order with respect to all covers β refining α, i.e., D(α) = min β≻α ord(β).The Lebesgue covering dimension is defined as where the supremum is over all finite open covers of X.In this article dimension always refers to Lebesgue covering dimension.Note that Lebesgue covering dimension can only take values in N ∪ {0, ∞}.

Dynamical systems.
A topological dynamical system (t.d.s.) is a pair (G, X) where G is a group equipped with the discrete topology 3 , (X, d) is a compact metric space4 and G acts on X such that the action map G × X → X given by (g, x) → gx is continuous.A t.d.s.(G, X) is also referred to as a G-system or a G-(group) action.
The orbit of x under G is denoted by The kernel of the action (G, X) is the set A morphism between two dynamical systems (G, X) and (G, Y ) is given by a continuous mapping ϕ : X → Y which is G-equivariant (ϕ(gx) = gϕ(x) for all x ∈ X and g ∈ G).If ϕ is an injective morphism, it is called an embedding.
2.4.Cubical shits and orbit maps.For any space Z, the group G acts on the space Z G by g(y) h = y hg for all y ∈ Z G and g, h ∈ G.For any space Z, the group G acts on the space Z G by g(y) h = y hg for all y ∈ Z G and g, h ∈ G.When Z = [0, 1] r , then this action is referred to as a G-shift.When Z = [0, 1] r , then the system given by x → f (gx) g∈G , known as the orbit-map.
2.5.Genericity.We denote the space of continuous functions from X to [0, 1] r by C(X, [0, 1] r ).equipped with the topology of uniform convergence.By the Baire category theorem ([Kec95, Theorem 8.4]) the space C(X, [0, 1] r ), is a Baire space, i.e., a topological space where any comeagre set is dense.We refer to a property that holds on a comeagre set of C(X, [0, 1] r ) as generic.
2.6.Partitions.Let P be a partition of set S. For every s ∈ S denote by P(s) the unique element P ∈ P such that s ∈ P .Let P and P be two partitions of the same set.One says P is finer than P, P P, if for every P ∈ P, there exists P ∈ P such that P ⊆ P .2.7.Partition compatible subsets.Definition 2.1.Let X, Y be compact metrizable spaces and let F = g 1 , . . ., g N be an N-tuple of continuous functions from X to Y .Suppose x ∈ X. Define an equivalence relation on F by We denote by x F the x-induced partition of F the partition of F generated by the equivalence classes of ∼ x .
Let P be a partition of F .For W ⊂ X define the (P, F )-compatible subset by Definition 2.1 has a natural generalization to functions of two variables which we now present.Definition 2.2.Let X, Y be compact metrizable spaces and let g : X → Y be a function.For j ∈ [2], define the function g Definition 2.3.Let X, Y be compact metrizable spaces and let F = g 1 , . . ., g N be a finite ordered set of continuous functions from X to Y .Define the induced 2N-tuple of continuous functions from X × X to Y by N , g (2) N Let P be a partition of F and Ŵ ⊂ X ∆ .Following (4), we may define the ( P, F)compatible subset by Ŵ P = (x, y) ∈ Ŵ : (x, y) F = P .
(5) 2.8.Generating sets.Given a subset S of a group G and n ∈ N we denote by S −1 the set of inverses of elements of S and We denote the identity element of G by e G , and 3. Embedding finite-dimensional systems into cubical shifts 3.1.Proof of the main theorem.
Proof of Theorem 1.1, assuming Theorem 1.5.Note one may assume w.l.o.g. that (G, X) is faithful by considering the induced t.d.s.(G/ Ker(G, X), X).If G is a finite group, then one may directly apply Theorem 1.5 with X = Y and F = G to deduce the conclusion.Assume G is infinite.Note one may assume w.l.o.g. that G is countable.Indeed as (G, X) is faithful, there is an injective group homomorphism i : G → Homeo(X), where Homeo(X) is the group of homeomorphisms of X, which is Polish when equipped with the supremum (uniform) metric ([Kec95, Subsection 9.B (8)]).Thus one may find a dense (w.r.t. the supremum metric) subgroup G ′ < i(G), where G ′ is countable.In particular for every x ∈ X, |G ′ • x| = |G • x|, and so for every N ∈ N, Hence, by possibly replacing G by G ′ , one may assume that G is at most countable.
Choose N ∈ N such that 2 dim(X) < rN.Let ǫ > 0 and Using the definition of sep ǫ (•), it is not difficult to see that X (F,ǫ) is a closed subset of X. Recall notation (4).We claim that for every partition P of F it holds that dim(X Indeed, if |P| < rN then for every x ∈ X (F,ǫ) P it holds that G • x = F • x, as the condition sep ǫ (F • x) ≥ rN implies that |F • x)| ≥ rN so the cardinality of the partition x F is strictly bigger than |P|, in particular x F = P. Thus in this case X (F,ǫ) P ⊆ X |P| , and so dim(X Otherwise, |P| ≥ rN and so dim(X For any finite set F ⊂ G and ǫ > 0, by applying Theorem 1.5 with F = F , Y = X and X = X (F,ǫ) , we conclude that the function f F : 3.2.Sharpness of the main theorem.
Proof of Theorem 1.4.Flores (see [Flo35], a more accessible source is [Eng95, Eng78, §1.11H]) proved that C n , the union of all faces of dimension less than or equal to n of the (2n + 2)-simplex (the convex hull of 2n + 3 points in R 2n+2 being affinely independent) does not embed into R 2n .Note dim with G acting trivially on Z, by multiplication on G/G ′ and by shift on {0, 1} G .Clearly (G, X) is faithful.Denote by eG ′ the coset of G/G ′ containing the identity of G. Denote by 0 the element of {0, 1} G consisting only of zeroes.Note that for any continuous function f : As Z does not embed into [0, 1] rN , the map F is not injective and as a consequence

Takens Embedding Theorem for finitely generated groups
Proof of Theorem 1.6 assuming Theorem 1.5.Let r ≥ 1, (G, X) and M be as in the statement of the theorem.Note that by Theorem 1.5 (used for the case X = Y ) it is enough to show that with F = S ≤•(M −1) , for every partition P of F it holds We thus have as desired.
Lemma 4.1.Let G be a group that acts on a set X, and let S ⊆ G be a finite generating set, and M ∈ N. Given a natural number n, recall the definition of S ≤•n by (3), and let Proof.Since S ≤•(n−1) ⊆ S ≤•n for every n, it follows that It follows by the pigeonhole principle that there exists 1 ≤ n ≤ M so that As S ≤•n = s∈S sS ≤•(n−1) , it follows that for every s ∈ S it holds Note that the map s• : G → G given by g → s • g is an injective map for every s ∈ S. By Equation (9), when restricted to S ≤•(n−1) • x it is a self-map of a finite set.Thus by Equation ( 8), it follows that s(S ≤•(n−1) • x) = S ≤•(n−1) • x for all s ∈ S. Hence we also have Remark 4.2.From Theorem 1.6 it follows that for any natural numbers d, r ∈ N and finitely generated group G there exists an integer N d,r = N d,r (G) ∈ N and a finite set F d,r = F d,r (G) ⊆ G of size at most N d,r so that for any topological dynamical system (G, X) with dim X ≤ d the set of continuous functions f : ).Indeed, Theorem 1.6 shows that for a group G generated by a finite set S, one may take r ⌋ + 1, defined after (3).When r = 1, we have that G is a cyclic group so up to group isomorphism either G = Z or G = Z/mZ for some natural number m.In this case Theorem 1.6 recovers the result of [Gut16], where it was proven that F = {0, 1, 2, . . ., 2d} is sufficient.The lower bound r • N d,r (G) ≥ 2d + 1 holds for any d, r ∈ N and any group G, as evidenced by considering a d-dimensional space X which does not embed in R 2d (see the proof of Theorem 1.4 in Subsection 3.2), thus we conclude that the optimal (minimal) value of N d,r (Z) is ⌊ 2d r ⌋+1.It is interesting to find the minimal value possible for N d,r (G) for other finitely generated groups G beyond Z .For example for G = Z 2 and S = {(1, 0), (0, 1)}, the proof . One wonders how far this is from the minimal value possible.
We present an example showing that if the group is not finitely generated then the conclusion of Theorem Theorem 1.6 does not necessarily hold.
Example 4.3.Let 1 = r 3 > r 4 > r 5 . . .be a sequence of positive numbers with lim i→∞ r i = 0. Let S i be the circle of radius r i around the origin in the plane.Let X be the compact (one-dimensional) metric space Let G = F ∞ , the free group with a non finite countable number of generators.Denote G = g i ∞ i=1 .We define the action of G on X by specifying the action of each generator: the element g i rotates S i by 2π/i and acts as identity otherwise.Note that (G, X) 1 = (G, X) 2 = {(0, 0)}.Note that for N ≥ 3, (G, X) N is a finite union of circles S i and the origin.Thus for every N ∈ N it holds However for every finite F ⊂ G, for every continuous functions f : X → [0, 1] the map f F : X → [0, 1] F , x → (f (gx)) g∈F is not an embedding as one may find a circle S i on which it equals f |S i .

The main auxiliary theorem
5.1.Overview of the proof.Given a set Ẑ ⊆ X ∆ , denote Since Ẑ is compact, it follows that there exists ǫ > 0 such that for all (x 1 , x 2 ) ∈ Ẑ, the distance between f F (x 1 ) and Lemma 5.2.(cf.[Gut16, Lemma 3.3] Suppose there exists a countable collection of compact sets Ẑ1 , . . ., Ẑn , . . .⊂ X, such that X = ∞ n=1 Ẑn and so that for every n ∈ N the set G F ( Ẑn ) is dense.Then the set of continuous functions By assumption G F ( Ẑn ) is dense for every n.
Given Lemma 5.2, to conclude the proof of Theorem 1.5, it suffices to find a countable cover of X ∆ by compact sets Ẑ1 , Ẑ2 , . . .⊂ X ∆ , so that for every n ∈ N the set G F ( Ẑn ) is dense.5.2.Coherent sets.Let (X, d), (Y, d ′ ) be compact metric spaces and let F = g 1 , . . ., g N be a finite ordered set of continuous injective functions from X to Y .Observe that X ∆ = P X ∆ P , where the union is a finite union over partitions of F.
Definition 5.3.A set Ẑ ⊆ X ∆ is said to be F -coherent if there exists a partition F such that Ẑ ⊆ X ∆ P and so that for every (i 1 , j 1 ), (i 2 , j 2 ) ∈ [N] × [2] such that P(g Lemma 5.4.Let P be a partition of F Then the set X ∆ P is a countable union of compact F -coherent sets. Proof.
By the triangle inequality, for every z = (z 1 , z 2 ), z = (z 1 , z2 ) ∈ U (x 1 ,x 2 ) and every (i 1 , j 1 ), (i 2 , j 2 ) ∈ [N] × [2] such that P(i 1 , j 1 ) = P(i 2 , j 2 ) it holds d ′ (g 3. Main auxiliary lemma.In view of Lemma 5.2 and Lemma 5.4, in order to prove Theorem 1.5 it is enough to prove the following lemma. Lemma 5.5.Let (X, d), (Y, d ′ ) be compact metrizable spaces and let F = g 1 , . . ., g N be a finite ordered set of continuous injective functions from X to Y .Assume that for every partition P of F the inequality (2) holds.Let P be a partition of F and let To prove Lemma 5.5 we distinguish between two different types of partitions P as follows: Definition 5.6.A partition P of [N] × [2] is said to be intersective if there exists i 1 , i 2 ∈ [N] such that P(i 1 , 1) = P(i 2 , 2).Otherwise, P is said to be nonintersective.
(2) If P is an intersective partition, then there exists a homeomorphism T : Proof.
We use the following ad-hoc combinatorial lemma:

Then at least one of the following holds: (A) There exists
2 , and by the inequality follows that the restriction of F 2 to W * is not injective.This implies that there exist w, w ′ ∈ W * such that w = w ′ and F 2 (w) = F 2 (w ′ ).Since F 1 is injective on W * by construction, and F 1 (W * ) = V * 1 , it follows that F 1 (w) = F 1 (w ′ ) and F 1 (w), F 1 (w ′ ) ∈ V * 1 , so we are in case (B).
Lemma 5.12.Let V 1 , V 2 , W be finite sets, let Y be an arbitrary set, and let Further, suppose that the restrictions of φ 1 and φ 2 to V * 1 and V * 2 respectively, are both injective and that φ , and that at least one of the statements (A) and (B) from Lemma 5.11 hold.Then Proof.Statement (A) from Lemma 5.11 implies that that there exists w 2 ), so by assumption for such w it holds φ 1 (F 1 (w)) = φ 2 (F 2 (w)), and so in this case φ Now suppose statement (B) from Lemma 5.11 holds.Namely, we assume that there exists w, w We are now ready to prove Lemma 5.5.
Proof of Lemma 5.5.Let P be a partition of F identified with By the compactness of X and the continuity of the maps g i : X → Y one may find η > 0 such that for all i ∈ [N] and x, x ′ ∈ X satisfying d(x, x ′ ) < η it holds d ′ (g i (x), g i (x ′ )) < δ.For j ∈ [2], let Z j := π j ( Ẑ) denote the projection of Ẑ ⊆ X × X into the j'th copy of X.For j ∈ [2] let P j denote the partition of [N] defined by P j (i 1 ) = P j (i 2 ) iff P(i 1 , j) = P(i 2 , j), and let Then it holds Z j ⊆ X P j and so by the inequality (2), dim(Z j ) < r 2 M j for j ∈ [2].For j ∈ [2], write P j = {P (j) 1 , . . ., P (j) M j }.Then for every j ∈ [2] and t ∈ [M j ] there exists a function gt,j ∈ C(Z j , Y ) such that g i | Z j = gt,j for every i ∈ P (j) t .By Ostrand's theorem and the condition dim(Z 2 ) < r 2 M 2 , one may find families of sets C t,ℓ is a family of pairwise disjoint closed subsets of Z 2 having diameter smaller than η, and so that every x ∈ Z 2 is covered by at least (M C(2) t,ℓ is a collection of pairwise disjoint closed subsets of Y .By the choice of η, the diameter of each of these sets is less than δ.
The next step of the proof splits into two cases: • Case 1: The partition P is non-intersective.In this case, by Ostrand's theorem and the condition dim(Z 1 ) < r 2 M 1 , one may find families of sets C (1) t,ℓ is a family of pairwise disjoint closed subsets of Z 1 having diameter smaller than η, and so that every x ∈ Z 1 is covered by at least .
Then by the discussion above Cℓ is a collection of pairwise disjoint compact subsets of Y having diameter less than δ.
To complete the proof, we show that f ∈ G F ( Ẑ).To prove that f ∈ G F ( Ẑ), we need to show that for every (x 1 , x 2 ) ∈ Ẑ there exists i ∈ [N] and ℓ ∈ [r] such that fℓ (g i (x 1 )) = fℓ (g i (x 2 )).