The generic limit set of cellular automata

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The generic limit set of cellular automata

Saliha Djenaoui, Pierre Guillon

To cite this version:

Saliha Djenaoui, Pierre Guillon. The generic limit set of cellular automata. 2017. �hal-01861590v2�

The generic limit set of cellular automata

Saliha Djenaoui

and Pierre Guillon

†?

Universit´ e d’Aix-Marseille CNRS, Centrale Marseille, I2M, UMR 7373,, 13453 Marseille, France

?

D´ epartement de Math´ ematiques, Universit´ e Badji Mokhtar-Annaba, B.P. 12, Sidi Amar, 23220 Annaba, Alg´ erie, [email protected]

†

Interdisciplinary Scientific Center J.-V. Poncelet (ISCP), Independent University of Moscow, CNRS, UMI 2615, Russia, [email protected]

April 24, 2019

Abstract

In topological dynamics, the generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions.

Keywords : cellular automata, basin of attraction, limit set, attractor, directional dynamics, Baire category, symbolic dynamics.

1 Introduction

In a topological dynamical system (DS), the limit set is the set of points that appear arbitrarily late during the evolution (see [4]). But it may include points which look transient, because they do not appear arbitrarily late around any orbit.

J. Milnor, interested in the dynamics on the space of measures, introduced in [23] the notion of likely limit set, that provides a useful tool for studying asymptotic behavior for almost all orbits. He also implicitly defined a topological version of the same intuitive idea, that he calls thegeneric limit set. The goal of our article is to formalize this concept. In other words, we focus on the asymptotic behavior for almost all orbits in the sense of Baire category theory. We study some topological properties of the generic limit set, which is the smallest closed set that has a comeager realm.

We show that the generic limit set is actually equal to the limit set if the DS is semi-nonwandering (Propo- sition 4.5), a broad property that is implied by nonwanderingness, or equicontinuous (Proposition 4.19). We also prove that the generic limit set is the closure of the asymptotic set of the equicontinuity points if the DS is almost equicontinuous (Proposition 4.16).

For cellular automata (CA), we know that all subshift attractors have a dense open realm of attraction (see [15, 9, 17, 16]). We prove that the generic limit set is a subshift (Proposition 4.9), which is included in all subshift attractors (Corollary 3.15).

We emphasize directional dynamics of cellular automata, which is devoted to their qualitative behaviour (equicontinuity, sensitivity, expansiveness) when composed with shifts (see [24, 7]). First, in oblique directions (the shifts are bigger than the radius), a weak semi-mixingness property makes the generic limit set equal to the limit set. If the generic limit set is finite, it is shown that it consists of only one periodic orbit of a

monochrome configuration (Proposition 4.15) and the DS is almost equicontinuous (Proposition 4.13). We show that it is the case if the cellular automaton is almost equicontinuous in two directions of opposite sign;

moreover the realm of this periodic orbit then contains a dense open set (Proposition 5.9). We mention a nontrivial example where the period is nontrivial (Example 5.18). We give a classification of generic limit sets in the context of directional dynamics (Theorem 5.10).

We formulate most topological-dynamical results in a very general framework of sequences of continuous functions, which correspond to nonuniform dynamical systems. The purpose is double: to be able to apply our results to directional dynamics of cellular automata in a smoother way than previous works (which often had to introduce several ad-hoc definitions), and hopefully to propose a large setting in which different kinds of attractor properties can be studied, that could be useful in other subcases.

The paper is structured as follows : In Section 2, we provide the basic background on the subject of topological dynamical systems and cellular automata. In Section 3, we show some preliminary results about attractors, limit sets, realms. In Section 4, we define the generic limit set and prove the main results about it. In Section 5, we show some consequences on directional dynamics of cellular automata, and provide a list of examples. In Section 6, we compare the generic limit set with the likely limit set.

2 Preliminaries

2.1 Topology

In this article, (X, d) is a compact metric space. We put B_δ(x) ={y∈X|d(x, y)< δ}and call it theopen ball with center x∈X and radiusδ >0. For U, V ⊆X, wenoted(U, V) = inf{d(x, y)|x∈U, y∈V}. We may writed(U, y) =d(U,{y}) fory∈Y. We also writeB_δ(U) ={y∈X|d(U, y)< δ}andB_δ(U) its closure.

A subsetU ⊆X is calledcomeager in X if it includes a countable intersection of dense open sets. A subsetU ismeagerifX\U is comeager inX. By the Baire category theorem, the intersection of countably many dense open sets inX is dense inX. Hence, a comeager set is dense inX.

We also say that a set U ⊆X is comeager in some setV ⊆X ifU ∩V is comeager in the induced topological spaceV.

A subsetA of X is said to have the Baire property if there is an open setU such that the symmetric difference A∆U = (A\U)∪(U\A) is meager in X. The family of sets with the Baire property forms a σ-algebra. Every Borel subset has the Baire property (for more details, see for instance [25]).

We recall the following folklore remark, further used several times.

Remark 2.1. A set W ⊆X with the Baire property is not comeager if and only if there exists a nonempty open setU in whichW∩U is meager.

Proof. W^C∆U is meager for some open setU, andW ∩U ⊆W^C∆U. Just remark that W is comeager if and only ifU =∅.

2.2 Topological dynamics

Now we introduce some key concepts of the topological dynamics. A (time-nonuniform)dynamical system (DS) is any sequenceF = (Ft)t∈N of continuous self-maps of some compact metric spaceX. This general formalism will be useful when studying directional dynamics of cellular automata, but the reader should keep in mind the following specific, more classical, case. F isuniform ifF_t=G^t, for all t∈ Nand some continuous self-mapGofX. We may then write the uniform DS simply asG.

We are interested in the orbits O_F(x) = {F_t(x)|t∈N} of points x ∈ X. We say that a set U is F-invariant (resp. strongly) ifF_t+1(U)⊆F_t(U) (resp. F_t⁻¹(U) =U =F_t(U)), for everyt∈N.

Equicontinuity. For ε > 0, a point x∈X is ε-stable if there exists δ > 0 such that ∀y ∈Bδ(x),∀t ∈ N, d(Ft(x), Ft(y)) < ε. The set EF ⊆ X of equicontinuous points for F is the set of points which are ε-stable for everyε >0. IfEFis comeager, then we say thatFisalmost equicontinuous. IfEF=X, then

we say thatF isequicontinuous. Equivalently by compactness, for everyε > 0, there is a uniformδ >0 such that∀x∈X,∀y∈Bδ(x),∀t∈N, d(Ft(x), Ft(y))< ε.

F is sensitive if there existsε >0 such that ∀x∈X,∀δ >0,∃y ∈B_δ(x),∃t ∈ N, d(F_t(x), F_t(y))≥ε.

This implies thatEF=∅.

Remark 2.2. EFis comeager in some nonempty open set U ⊆X if and only if, for everyε >0, the set of ε-stable points is.

Proof. EFcan be written as the decreasing countable intersection of the sets of 1/n-stable points.

A class which is between nonuniform and uniform DS is the following: F is semi-uniform if F_t = G_t· · ·G₁, for all t ∈ N and some equicontinuous sequence (G_t)_t≥1 of self-maps of X. This is trivially satisfied when {G_t|t≥1} is finite; in particular, in the case of uniform DS: G_t =G₁ for every t≥1. We will sometimes denoteG_T+

J1,tK the compositionGT+t· · ·GT+2GT+1, so thatFt+T =G_T+

J1,tKFT.

Transitivity. Classical notions from topological dynamics can be adapted in our framework of nonuniform DS. Here is an example, that will actually be used especially for a uniform DS, the shift map from the next section, but we give the nonuniform version for completeness, and for the user to get used to the little differences that hold in this setting, in the perspective of better understanding related notions, like the ones introduced in Subsection 5.1. A DSFis said to betransitive(resp. weakly mixing) if for any nonempty open subsetsU, V (resp. andU⁰, V⁰) ofX and for everyT ∈N, there existst≥T such thatF_t⁻¹(U)∩V 6=∅ (resp. andF_t⁻¹(U⁰)∩V⁰6=∅). One can prove that, if the space is perfect, then it is enough to suppose this forT = 0. For any uniform DS, it is classical that it is enough to suppose this forT = 1 (and forT = 0 if the uniform DS is surjective).

The following lemma will be used in the context of shift maps: it can be interpreted as the fact that a transitive DSmixes the space, in the sense that it transforms alocal topological property into a global one.

Lemma 2.3. Let F= (F_t)_t∈N be a transitive DS, where allF_t are homeomorphisms, and letW ⊆X be a stronglyF-invariant subset.

1. W is either dense or nowhere dense.

2. W either has empty interior or includes a dense open set.

3. IfW has the Baire property, then it is either meager or comeager.

4. IfS

i∈NWi has nonempty interior (resp. is not meager), where eachWi is stronglyF-invariant (resp.

and has the Baire property), then there exists i∈N such thatWi includes a dense open set (resp. is comeager).

Proof.

1. SupposeW is dense in some nonempty open setU. SinceFis transitive, for every nonempty open set V, there existst such thatF_t⁻¹(V)∩U is a nonempty open set. Moreover,W is dense inU, so that F_t⁻¹(V)∩U∩W 6=∅. SinceFt(W)⊆W,V ∩Ft(U)∩W ⊇Ft(F_t⁻¹(V)∩U∩W)6=∅. So,W is dense in X.

2. Suppose that W includes a nonempty open set and is strongly F-invariant; then W includes the (nonempty open) orbit of this open set. From Point 1, this orbit is a dense open set.

3. Suppose that W is not meager and consider any nonempty open subset V ⊆ X. By Remark 2.1, there exists an open set U such that W is comeager in U. By transitivity, there exists t ∈ N such that F_t⁻¹(U)∩V is a nonempty open subset. By assumption,W is not meager in its nonempty open (because Ft is a homeomorphism) imageU ∩Ft(V), and sinceF_t⁻¹ is a homeomorphism,F_t⁻¹(W) is not meager in F_t⁻¹(U)∩V. By reverse invariance, we get thatW is not meager inF_t⁻¹(U)∩V. By Remark 2.1, sinceW is not meager in any nonempty open set, it is comeager.

4. One of theWi has to have nonempty interior (resp. to not be meager). We conclude by the previous point.

2.3 Symbolic dynamics

Configurations. Let A be a finite set called the alphabet. A word over A is any finite sequence of elements of A. DenoteA^∗ =S

n∈NAⁿ the set of all finite wordsu=u0. . . u_n−1;|u|=nis the length ofu.

We say that v is a subwordof uand write v <u, if there are k, l <|u| such that v =u

Jk,lJ. A^Z is the space of configurations, equipped with the following metric:

d(x, y) := 2⁻ⁿ, wheren= min{i∈N|xi6=yi or x_−i 6=y_−i}. A^Z is a Cantor space. Thecylinderofu∈A^∗in positioniis [u]i=

x∈A^Z x

Ji,i+|u|J=u . Cylinders are clopen (closed and open). The (full)shiftis the dynamical systemσover spaceA^Z defined byσ(x)i =xi+1

fori∈Zandx∈A^Z. It is 2-Lipschitz.

The (spatially) periodicconfiguration^∞u^∞ is defined by (^∞u^∞)_k|u|+i =ui fork∈Z,0 ≤i <|u| and u∈A⁺. Amonochromeconfiguration is one with only one symbol: ^∞0^∞, for some 0∈A.

Subshifts. A subshiftis any subsystemσ_|Σ of the full shift; we usually simply denote it by Σ, which is then simply a closed stronglyσ-invariant subset ofA^Z. Equivalently, there exists a forbidden language F ⊆A^∗ such that Σ = Σ_F =

x∈A^Z

∀u<x, u /∈ F . IfF can be taken finite, then one says that Σ_F is a subshift of finite type(SFT); in that caseFcan be taken included inA^kfor somek∈N, which is anorder for the SFT. We write that Σ_Fis ak-SFT. Let Σ⊆A^Zbe a subshift. ThenL(Σ) ={u∈A^∗| ∃x∈Σ, u<x}

is thelanguageof Σ.

Remark 2.4. It is clear that (A^Z, σ) is transitive. Point 3 of Lemma 2.3 hence applies, and gives that nonfull subshifts are all nowhere dense (hence meager).

We shall use the following lemma to show results concerning the dynamics of cellular automata.

Lemma 2.5. Let ε >0 andV ⊆A^Z. 1. σ^j(B_ε(V))⊆B₂jε(σ^j(V))for allj∈Z.

2. If V is strongly σ-invariant, x∈A^Z, and p >0 are such that for all n ∈Z, σ^pn(x)∈ B_ε(V), then

∀i∈Z, σⁱ(x)∈B₂pε(V).

3. IfV is a2k+ 1-SFT, thenT

i∈Zσⁱ(B₂−k(V)) =V. Proof.

1. Letx∈σ^j(Bε(V)). Thenσ^−j(x)∈Bε(V), which means thatd(σ^−j(x), V)< ε. Moreover,d(x, σ^j(V)) = d(σ^jσ^−j(x), σ^j(V))<2^jd(σ^−j(x), V)<2^jε. So,x∈B₂jε(σ^j(V)).

2. From the previous point,∀i∈Z, σⁱ(x) =σ^imodpσ^bi/pcp(x)∈σ^imodp(Bε(V))⊆B₂imodpε(σ^imodp(V)).

Since V is strongly σ-invariant, we also have that B₂imodpε(σ^imodp(V)) ⊆ B2^pε(V). Hence, ∀i ∈ Z, σⁱ(x)∈B2^pε(V).

3. Leti∈Z.

σ⁻ⁱ(x)∈B₂−k(V) ⇐⇒ d(σ⁻ⁱ(x), V)<2^−k

⇐⇒ ∃y∈V, σ⁻ⁱ(x)

J−k,kK=y

J−k,kK

⇐⇒ ∃y∈V, x

J−i−k,−i+kK=y

J−k,kK

⇐⇒ x

J−i−k,−i+kK∈ L(V). If this is true for everyi∈Zand V is (2k+ 1)-SFT, then x∈V.

2.4 Cellular automata

A map F : A^Z → A^Z is a cellular automaton (CA) if there exist integers r₋ ≤ r₊ (memory and anticipation) and a local rule f : A^r+−r−+1 → A such that for any x ∈ A^Z and any i ∈ Z, F(x)i = f(xi+r₋, . . . , xi+r₊). d=r+−r₋+ 1∈Nis sometimes called the diameterofF. Sometimes we assume that −r₋ = r+, which is then called the radius of F (it is always possible to obtain this, by taking r=max{|r₋|,|r+|} ∈N). By Curtis, Hedlund and Lyndon [14], a map F :A^Z→A^Z is a CA if and only if it is continuous and commutes with the shift. In particular, CA induce uniform DS overA^Z.

Directional dynamics. We call curve a map h : N → Z with bounded variation, that is: M_h = sup_t∈N|h(t+ 1)−h(t)|is finite. The map is meant to give a position in space for each time step. Following [7], theCA F in directionhwill refer to the sequence (F^tσ^h(t))_t∈N. We will use all notations for DS with F, hinstead ofF, when dealing with it (for instanceE_F,h is its set of equicontinuous points).

In a first reading, the reader can understand the next definitions and results by considering the classical case: hconstantly 0. In general, the directional dynamics of a CA can be read on its space-time diagram, by following has a curve when going in the time direction. An example of curve is given by the (possibly irrational) lines: α∈ Rwill stand for the directiont 7→ btαc. The dynamics along αthen corresponds to that studied in [24, 1].

Equicontinuity. A wordu∈A^∗ is (strongly)blockingfor a CAF along curvehif there exists an offset s ∈Z such that for everyx, y ∈ [u]_s, ∀t ∈ N, F^tσ^h(t)(x)

J0,MJ =F^tσ^h(t)(y)

J0,MJ, where M = max(−r₋+ max_t(h(t)−h(t+ 1)), r₊+ max_t(h(t+ 1)−h(t))), andr₋andr₊ are the (minimal) memory and anticipation forF. The terminology comes from the fact that in that case,uis both left- and right-blocking (with the same offset), which is taken as a definition in [7]: A wordu∈A^∗ isright-blocking for a CAF in direction hif there exists an offsets∈Zsuch that:

∀x, y∈[u]s, x

K−∞,sK=y

K−∞,sK =⇒ ∀t∈N, F^t(x)

K−∞,h(t)K=F^t(y)

K−∞,h(t)K. We defineleft-blockingwords similarly.

The following proposition explains how equicontinuity in cellular automata can be rephrased in terms of blocking words. The vertical case dates back from [17, 18], the linear directions from [24], the directions with bounded variations can be found in the proofs of [7, Prop 2.1]; a version with unbounded variations of the first point can even be found in [6, Prop 3.1.3, Cor 3.1.4].

Proposition 2.6. Let F be a CA andha curve.

1. If there is a left- and right-blocking word uforF in direction h, then EF,h includes the comeager set of configurations where uappears infinitely many times on both sides.

2. Otherwise,F is sensitive in direction h.

In particular, EF,h is either empty or comeager. The question is open whether this remains true in the unbounded-variation case (see [6, Rem 3.1.1]).

Definition 2.7 ([7, Def 2.5]). Let us denoteB the set of curves (recall that they are mapsh:N→Zwith bounded variation).

Forh, h⁰∈ B, we puthh⁰ if there existsM >0 such thath(t)≤h⁰(t) +M for all t∈N. We puth≺h⁰ if, besides h⁰ 6h. is a preorder relation onB, and we note∼the corresponding equivalence relation.

We also noteh≺≺h⁰ if limt→∞h⁰(t)−h(t) = +∞. ≺≺is a transitive relation which is finer than≺.

The preorder induces a notion of (closed, open, semi-open) curve interval, with some bounds h⁰ h⁰⁰, noted [h⁰, h⁰⁰], ]h⁰, h⁰⁰[, [h⁰, h⁰⁰[, ]h⁰, h⁰⁰]. We say that the interval is nondegenerate if h⁰ ≺ h⁰⁰. For an interval S⊆ Bwith bounds h⁰ andh⁰⁰, we also noteI(S) ={h∈ B|h⁰≺≺h≺≺h⁰⁰} ⊂]h⁰, h⁰⁰[.

A directionwill implicitly refer to an equivalence class for∼ (sometimes abusively confused with one representative). It is not so hard to get convinced that equicontinuity properties are preserved by∼.

Remark 2.8. LetF be a CA over A^Z andh, h⁰ ∈ B. Ifh∼h⁰, thenEF,h=EF,h⁰.

In particular, F is almost equicontinuous (resp. equicontinuous) along h if and only if F is almost equicontinuous (resp. equicontinuous) alongh⁰.

Proof. By assumption, there existsM ∈Nsuch that −M +h(x)≤h⁰(t)≤h(t) +M,∀t ∈ N. Let l ∈N and x ∈ A^Z. Let us show that if x ∈ A^Z is 2^−M−l-stable along h, then it is 2^−l-stable along h⁰. So, assume that there existsk∈Nsuch that∀y∈A^Z, x

J−k,kK=y

J−k,kK =⇒ ∀t∈N, F^tσ^h(t)(x)

J−M−l,M+lK= F^tσ^h(t)(x)

J−M−l,M+lK .In other words,

∀y∈A^Z, x

J−k,kK=y

J−k,kK =⇒ ∀t∈N, F^t(x)

J−M−l+h(x),M+l+h(x)K=F^t(x)

J−M−l+h(x),M+l+h(x)K. By assumption, we get thatJ−l+h⁰(t), l+h⁰(t)K⊆J−M −l+h(t), M+l+h(t)K,∀t∈N.Thus,

∀y∈A^Z, x

J−k,kK=y

J−k,kK =⇒ ∀t∈N, F^tσ^h0(t)(x)

J−l,lK=F^tσ^h0(t)(x)

J−l,lK , which is exactly 2^−l-stability ofx.

Hence, if xis an equicontinuity point alongh, then xis an equicontinuity point alongh⁰. The converse is symmetric.

3 Limit sets, asymptotic sets, realms

We will define notions that deal with asymptotic behavior of a DSF= (F_t)_t.

3.1 Limit sets

The (Ω-) limit setof U ⊆X is the set ΩF(U) = T

T∈NS

t≥TFt(U), and the asymptotic set ofU ⊆X is the set ωF(U) = S

x∈UΩF({x}). By compactness, these sets are nonempty (decreasing intersection of nonempty closed subsets). ΩF(U) is compact, but ωF(U) may not be, even for U =X (see Example 4.8).

Remark that ΩF(U)⊇T

t∈NFt(U), and this is an equality ifU is a closedF-invariant set.

We note ΩF= ΩF(X) andωF=ωF(X). For more about the asymptotic set of dynamical systems, one can refer to [12]. Note that it was calledaccessible set in [8], andultimate set in [11].

The following remark easily follows from the compactness of X, and can be understood as the fact that every set which is at positive distance from Ω_F is transient, that is, ultimately does not appear.

Remark 3.1. For everyU,lim_t→∞d(F_t(U),Ω_F(U)) = 0.

In the uniform case, it is clear that asymptotic sets are invariant. Here is a generalization of this fact.

Proposition 3.2. Let F= (G

J1,tK)t be a semi-uniform DS,U ⊆X, andj∈N. 1. Ify∈ΩF(U), then(G_t+

J1,jK(y))t admits a limit point inΩF(U).

2. Conversely, ifz∈ΩF(U), then it is a limit point of(Gt+J1,jK(y))t for somey∈ΩF(U).

Of course, the corresponding statement is also true for theω, which is defined as a union of limit sets.

Proof.

1. By assumption, there are increasing times (tk)_k∈Nand points (xk)_k∈N∈U^Nsuch that lim_k→∞Ft_k(xk) = y. Let ε > 0. By equicontinuity of the (G_t+

J1,jK)t, there exists δ > 0 such that for all z, z⁰ with d(z, z⁰) < δ, we have ∀t ∈ N, d(G_t+

J1,jK(z), G_t+

J1,jK(z⁰)) < ε/2. Then there is K ∈ N such that for all k ≥ K, d(Ft_k(xk), y) < δ, so that d(Ft_k+j(xk), G_tk+

J1,jK(y)) < ε/2. If z is a limit point for (G_tk+

J1,jK(y))_k∈N, one sees that there exist infinitely manyk such thatd(Ft_k+j(xk), z)< ε, so thatz is also in ΩF({xk|k∈N})⊆ΩF(U).

2. Now let z∈ΩF(U), so that it is the limit point of (Ft_k(xk))k∈N for some (xk)k∈N∈U^N and (tk) an increasing sequence, that we can assume to be greater thanj. By compactness, (Ft_k−j(xk))k∈Nadmits a limit pointy∈Ω_F(U). By triangular inequality, we haved(G_tk−j+

J1,jK(y), z)≤d(G_tk−j+

J1,jK(y), F_tk(x_k))+

d(F_tk(x_k), z). When k goes to ∞, the second term of the sum converges to 0, and a subsequence of the first term converges to 0, thanks to equicontinuity of (G_tk−j+

J1,jK)_k∈N.

The following corollary is useless for the purpose of the present paper, but may help the reader to connect with the known uniform case.

Corollary 3.3. If Gis a uniform DS andU ⊆X, thenG(ΩG(U)) = ΩG(U).

Proof. G_t+

J1,1K=Gt=Gfor everyt∈N, so each point of Proposition 3.2 gives one inclusion.

3.2 Realms

Therealm(of attraction) of V is:

D_F(V) ={x∈X|ω_F(x)⊆V} .

The realm is sometimes called the basin of attraction; we prefer another name to recall that it is relevant even for setsV which have noattractive property.

The direct realm ofV is the set dF(V) =S

T∈NT

t≥TF_t⁻¹(V) of configurations whose orbits lie ulti- mately inV.

From the definition and some compactness arguments, the reader can be convinced of the following remarks. Note that the realm and the direct realm are related through opposite inclusions, depending on whether the set is open or closed.

Remark 3.4. LetV ⊆X, andVi⊆X for any iin some arbitrary set I.

1. DF(V)⊆T

ε>0d_F(B_ε(V)).

2. IfV is closed, then D_F(V) ={x∈X|lim_t→∞d(F_t(x), V) = 0}=T

ε>0d_F(B_ε(V))⊇d_F(V).

3. On the contrary, ifV is open, thenD_F(V)⊆d_F(V).

4. DF(S

i∈IVi)⊇S

i∈IDF(Vi).

5. DF(T

i∈IVi) =T

i∈IDF(Vi).

Conjugating the realm operator with complementation is also very relevant dynamically: as stated in the following remark.

Remark 3.5. For every DS F and subset V, the set of points whose orbits have a limit point in V is D_F(V^C)^C = {x∈X|ω_F(x)∩V 6=∅} ⊇ D_F(V), and the set of points whose orbits visit V infinitely many times isd_F(V^C)^C=T

T∈NS

t≥TF_t⁻¹(V)⊇d_F(V).

Note thatD_F(V^C)^C is nonempty if and only if V intersects ω_F.

From Remark 3.4, ifV is closed (resp. open), then DF(V^C)^C includes (resp. is included in)dF(V^C)^C.

3.3 Related concepts

Nonwanderingness. LetFbe a DS over spaceX. We say thatFisnonwanderingif for every nonempty open setU ⊆X,d_F(U^C)^C is not meager.

This definition does not give a specific role to time 0, unlike, seemingly, the classical definition, for uniform DS. Nevertheless, they are equivalent in the uniform case, which helps understand the essence of that concept.

Remark 3.6. LetF be a uniform DS. ThenF is nonwandering if and only if, for every nonempty open set V ⊆X, there existst≥1 such thatV ∩F^−t(V)6=∅.

Proof.

• If (F^t) is nonwandering, then the setdF(V^C)^C of point whose orbits visitV infinitely many times is in particular nonempty. Let xbe such a point, and t1 < t2 be two time steps such that y =F^t1(x) andF^t2−t1(y) =F^t2(x) are both in V. It is then clear thaty∈V ∩F^t1−t2(V).

• Now suppose that for every nonempty open setU ⊆X, there existst≥1 such thatU∩F^−t(U)6=∅.

Let us show by induction on n ∈ N that there exist distinct time steps 0 = t₀ < t₁ < · · · < t_n at which the setW_{(t0,t1,···,tn)}(U) =Tn

i=0F^−ti(U) of points whose orbits visitU is nonempty. It is trivial for n = 0. Suppose that W_{(0,t1,···,tn)}(U) 6=∅. By assumption, we have that there exists t ≥1 such that W_{(0,t1,···,tn)}(U)∩F^−t(W_{(0,t1,···,tn)}(U))6=∅. In particular,W_{(0,t,t+t1,t+t2,···,t+tn)}(U) contains this intersection, so that it is nonempty.

Now consider the set

Wn(V) = [

(t₁,···,t_n)∈Nⁿ 0<t₁<···<tn

W_{(0,t1,···,tn)}(V)

of points of U whose orbits visitU at least nmore times. Note that it is an open set, which is dense in V because it includes the nonempty Wn(U)⊆U, for every open subsetU ⊆V. The setdF(V^C)^C of points whose orbits visit V infinitely many times can be written as the intersectionT

n∈NWn(V), and is thus comeager inV.

Moreover, let us mention, even if it will not be used later, that uniform DS are known to admit a nonempty largest nonwandering subsystem, containing, as a comeager set, the set ofrecurrentpoints, which are those pointsx∈X such that x∈ωF(x) (see for instance [5]). Clearly, the set of recurrent points is a subset of the asymptotic set.

Nilpotence. We say that F is nilpotent if there is a point z ∈ X such that ∃T ∈ N,∀x ∈ X,∀t ≥ T, Ft(x) =z. Fisasymptotically nilpotentifω_F is a singleton.

It is known that CA are nilpotent if and only if their limit set is finite (see for instance [4]). Also, it has been shown [12] that asymptotically nilpotent CA are actually nilpotent. In that case (see for instance [4, 12]),z=σ(z), so that the CA is actually nilpotent in every direction.

Asymptotic pairs. Two pointsx, y∈Xare said to beasymptoticto each other (or (x, y) is anasymp- totic pair) whenever limt→∞d(Ft(x), Ft(y)) = 0. The asymptotic class ofy is the set AF(y) of points asymptotic to it. Let us generalize the realm notations to every sequence (V_t)_t∈N of closed subsets ofX, by defining:DF((V_t)_t) ={x∈X|lim_t→∞d(F_t(x), V_t) = 0}andd_F((V_t)_t) ={x∈X| ∃T ∈N,∀t≥T, F_t(x)∈V_t}.

We may also noteDF((y_t)_t) andd_F((y_t)_t) ifV_tis a singleton{yt}. With this notation,AF(y) =DF(({Ft(y)})t).

One can observe from the definition thaty∈ A_F(y)⊆ D_F(ω_F(y)).

The following remark states that, in a finite space, asymptotic pairs correspond to ultimately equal orbits.

Remark 3.7. Let Gbe a uniform DS over a finite space X, andx, y∈X. Ifxandy are asymptotic, then

∃t∈N, G^t(x) =G^t(y). In particular, ifGis injective (or surjective), then x=y.

Proof. The first statement comes fromX being discrete. The second statement is clear because ifXis finite, then injectivity or surjectivity ofGare equivalent to bijectivity of anyG^t.

The following remark states that when an asymptotic class is big, then it should contain many equicon- tinuous points.

Remark 3.8. If AF(y) is comeager (resp. not meager) in some nonempty open subset U ⊆X, for some y∈X, thenEFis comeager (resp. not meager) inU.

In particular, note that EF∩ AF(y) is also comeager (resp. not meager) in U.

Proof. The assumption gives that for every n≥ 1, the unionS

T∈NT

t≥TF_t⁻¹(B_1/n(F_t(y))) of closed sets is comeager in U, as a superset of AF(y). Hence, S

T∈NT

t≥TF_t⁻¹(B_1/n(F_t(y))) is not meager in any nonempty open subset V ⊆U. This implies that there isT ∈N such thatT

t≥TF_t⁻¹(B_1/n(F_t(y))) is not meager in V; as a closed set, and by Remark 2.1, it must then include a nonempty open subset W ⊆V. For every x∈ W, by openness, there exists δ > 0 such that for every z ∈ B_δ(x), z ∈ W, which implies that ∀t ≥T, F_t(z)∈B_1/n(F_t(y)). In particular,F_t(x)∈B_1/n(F_t(y)) and, by triangular inequality, we get F_t(z)∈B_2/n(F_t(x))⊆B_3/n(F_t(x)). We deduce that xis 3/n-stable. In other words, the set of 3/n-stable points includes nonempty open subsets of every nonempty open subset of U. This means that this set is comeager inU for everyn∈N. We conclude by Remark 2.2.

The statement about non-meagerness can be obtained from the other one thanks to Remark 2.1.

3.4 Decomposition of realms

The following proposition can be compared partly to [23, Lem 3]: if a set is decomposable into invariant components, then its realm can be decomposed accordingly.

Proposition 3.9. Let F= (G

J1,tK)_t∈N be a semi-uniform DS. Suppose (V_i)_i is a finite collection of closed pairwise disjoint sets which are invariant by every Gt. ThenDF(F

iVi) =F

iDF(Vi).

Proof. Since the Vi are closed and pairwise disjoint, there are at positive pairwise distance. Let ε = min_i6=jd(Vi, Vj)/2>0. By equicontinuity of (Gt), there exists δ >0 such that∀t∈N,∀x, y∈X, d(x, y)<

δ =⇒ d(Gt(x), Gt(y))< ε. Letx∈ DF(F

iVi), so that there existsT ∈Nsuch that∀t≥T, d(Ft(x),F

iVi)<

min(δ, ε). In particular, there exists i such that d(FT(x), Vi) < min(δ, ε). Let us show by induction on t ≥ T that d(Ft(x), Vi) <min(δ, ε). Since Gt+1(Vi) ⊆Vi, we have d(Ft+1(x), Vi) ≤d(Ft+1(x), Gt+1(Vi)).

They are less than ε by equicontinuity of (Gt), using the recurrence hypothesis. By definition of ε, we have minj6=id(Ft+1(x), Vj) ≥ minj6=i(d(Vj, Vi)−d(Ft+1(x), Vi)) ≥ ε. So min(δ, ε) ≥ d(Ft+1(x),F

jVj) = minj(d(Ft+1(x), Vj)≥min(ε, d(Ft+1(x), Vi)). It results thatd(Ft+1(x), Vi)≤min(δ, ε), as wanted.

Since for everyt≥Tandj6=i,d(Ft(x), Vj)≥ε, we deduced(Ft(x), Vi) = minjd(Ft(x), Vj) =d(Ft(x),F

jVj) converges to 0.

The other inclusion comes from Point 4 of Remark 3.4.

Realms of finite sets. Proposition 3.11 shows that the realm of a finite set contains a finite number of asymptotic classes. We shall use this to show Proposition 4.13. It uses the following lemma.

Lemma 3.10. Let F= (G

J1,tK)_t∈N be a semi-uniform DS over space X andV ⊆X be finite. Then there existsδ >0such that for allx, x⁰ ∈ DF(V)andT ∈Nwithd(FT(x), FT(x⁰))≤δ, and∀t≥T, d(Ft(x), V)≤δ andd(Ft(x⁰), V)≤δ,(x, x⁰)is an asymptotic pair.

Proposition 3.11. Let F= (G

J1,tK)_t∈N be a semi-uniform DS over space X and V ⊆ X be finite. Then there are at most|V| asymptotic classes inDF(V).

Proof. For 0≤i≤ |V|, let xi ∈ DF(V), andδ be as in Lemma 3.10. There exists T ∈Nsuch that for all i, ∀t≥T, d(Ft(xi), V)< δ/2, and in particular, ∃yi∈V, d(FT(xi), yi)< δ/2. By the pigeon-hole principle, there are distinct i, j such that yi = yj, so that d(FT(xi), FT(xj)) < δ by the triangular inequality. By Lemma 3.10, we then know that (x_i, x_j) is an asymptotic pair. Hence we can partitionDF(V) into at most

|V|asymptotic classes.

Proof of Lemma 3.10. Letε= ¹₃min{d(y, y⁰)|y, y⁰ ∈V, y6=y⁰}>0. By equicontinuity of (Gt), there exists δ > 0 such that ∀t ∈ N,∀x, x⁰ ∈ X, d(x, x⁰) ≤ δ =⇒ d(Gt(x), Gt(x⁰)) < ε. Without loss of generality, we can assume δ ≤ε. Let x, x⁰ be as in the statement of the lemma, and for t ∈N, lety(t)∈ V be such that d(Ft(x), y(t)) =d(Ft(x), V), and y⁰(t) be defined similarly. Let us show by induction ont ≥T that y(t) =y⁰(t), which by definition of ε, is equivalent tod(y(t), y⁰(t))<3ε. First, by the triangular inequality,

d(y(T), y⁰(T))≤d(y(T), FT(x)) +d(FT(x), FT(x⁰)) +d(FT(x⁰), y⁰(T))<3δ≤3ε.

Now suppose this is true fort≥T, and let us prove it fort+ 1. By the triangular inequality, we also have d(y(t+ 1), y⁰(t+ 1))≤d(y(t+ 1), F_t+1(x)) +d(G_t+1F_t(x), G_t+1F_t(x⁰)) +d(F_t+1(x⁰), y⁰(t+ 1)). The first and third terms are at mostδ by hypothesis, while the central one is at mostεby definition ofδ. All in all, we get thaty(t+ 1) =y⁰(t+ 1). We can conclude with, once again, the triangular inequality: d(F_t(x), F_t(x⁰))≤ d(F_t(x), y(t)) +d(y(t), y⁰(t)) +d(y⁰(t), F_t(x⁰)). If t≥T, this isd(F_t(x), V) + 0 +d(V, F_t(x⁰))→_t→∞0.

3.5 Realms for cellular automata

The following proposition is very important to show Proposition 5.9: transitivity of the shift brings some properties to realms and direct realms of shift-invariant sets through CA.

Proposition 3.12. Let F= (Ft)t∈N be a sequence of CA overX =A^Z, andV ⊆A^Z. 1. ωF,h(σ(V)) =σ(ωF,h(V))andDF(σ(V)) =σ(DF(V)).

2. IfV is strongly σ-invariant, thenD_F(V)either has empty interior or includes a dense open set; it is either nowhere dense or dense. If, moreover,V is closed, thenD_F(V)is either comeager or meager.

3. IfV is a2k+ 1-SFT andD_F(V)has nonempty interior, then d_F(V)is dense.

4. IfV is a subshift, thendF(V)is meager, unless F_T⁻¹(V) is full, for someT ∈N. Proof.

1. This is clear by definition thatω_F(σ(x)) =σ(ω_F(x)).

2. From the previous point,DF(V) is stronglyσ-invariant. Besides, one can see that, ifV is closed, then DF(V) =T

ε>0d_F(B_ε(V)) has the Baire property. The three statements then come from Lemma 2.3 (applied to the uniform DSσ).

3. Let us show that, for an arbitrary w ∈ A^∗, [w]∩dF(V) is nonempty. Since DF(V) has nonempty interior, there exists u∈A^∗ such that [u]⊆ DF(V).

Letx=^∞(uw)^∞∈[u] be the periodic configuration of periodp=|uw|and such thatx

J0,pJ=uw.

Since x∈[u]⊆ DF(V), there existsT ∈Nsuch that ∀t > T, d(Ft(x), V)<2^−k−p. Since ∀n∈Z, x= σ^np(x), we even have:

∀t > T,∀n∈Z, d(Ftσ^np(x), V)<2^−k−p .

By Point 2 of Lemma 2.5, for all such t > T, ∀i∈ Z, σⁱF_t(x) ∈B₂−k(V). Since V is a 2k+ 1-SFT, F_t(x) ∈ V, by Point 3 of Lemma 2.5, that is, x ∈ d_F(V). By shift-invariance, we also have that σ^|u|(x)∈[w]∩d_F(V).

4. By definition, dF(V)⊆S

T∈NF_T⁻¹(V). If for everyT ∈N, F_T⁻¹(V) is not full, Point 2 of Lemma 2.3 gives that it has empty interior (because it is closed and strongly σ-invariant). In the end, d_F(V) is meager.

Unsurprisingly, realms of CA behave well with respect to the shift.

Proposition 3.13. Let F be a CA, h a curve, and U be strongly σ-invariant. Then F(ω_F,h(U)) = σ(ωF,h(U)) =ωF,h(U).

Proof. By Point 1 of Proposition 3.12, ωF,h(U) is strongly σ-invariant, and since F commutes with σ, we have that ∀t ≥ 1, F σh(t)−h(t−1)(ω_F,h(U)) = F(ω_F,h(U)). By Proposition 3.2 (applied to j = 1 and G_t=F σh(t)−h(t−1), so that{G_t|t∈N} is finite), we obtain that for anyy, z ∈ω_F,h(U),F σ^k(y)∈ω_F,h(U) for somek, andz=F σ^l(x) for somel and somex∈ω_F,h(U).

Corollary 3.14. Let F be a CA, ha curve and U such that V =ωF,h(U) is finite and U = DF,h(V) is stronglyσ-invariant. ThenF induces a self-bijection of V, andU =F

y∈V AF,h(y).

Proof. By Proposition 3.13, we see thatF(V) =V, so thatF induces a surjection, hence a bijection ofV. By Proposition 3.11, there are at most|V| asymptotic classes inU. By the first point,V ⊆U, so that each y ∈V should be in one of these classes. By Remark 3.7, they are all in distinct classes, so that we obtain U ⊇F

y∈V AF,h(y) (the converse inclusion being trivial).

Attractors. In a DS, an attractor is the limit set of an inward set, that is an open set U such that Ft+1(U) ⊆Ft(U), for all t ∈ N. There are other definitions of attractors in the literature, but this one, found for example in [15, 17], is particularly relevant in totally disconnected spaces, whereU can equivalently simply be assumed to be an invariant clopen set. References [19, 9] focus on subshift attractors of CA: in that case the attractor enjoys a definition as the limit set of a so-calledspreading cylinder. ΩF= ΩF(X) is then the (unique) maximal attractor. A quasi-attractor is an intersection of attractors (possibly empty, in our setting). The minimal quasi-attractor is thus the intersection of all attractors.

Directly from Point 2 of Proposition 3.12, we recover the following (recall that every attractor has a nonempty open realm).

Corollary 3.15([22]). For any CA in any direction, the realm of any subshift attractor is a dense open set.

The following example will be described more deeply in Example 5.13, but gives here a first illustration of the concept of limit set and attractor.

Example 3.16. Let Min be defined over {0,1}^Z by Min(x)i = min(xi, xi+1). One has {^∞0^∞} =T

k≥0Vk, where Vk = Ω_Min([0]k) ={x∈Ω_Min| ∀i≤k, xi= 0} is an attractor but not a subshift, for every k ∈Z (see [18]). {^∞0^∞}is the unique minimal quasi-attractor, and its realmD_Min(^∞0^∞) =T

k∈Z

x∈ {0,1}^Z

∃i≥k, xi= 0 is comeager.

This property of having a comeager realm motivates the next definition.

4 The generic limit set

Milnor [23] suggests the following definition, which is the purpose of the present section.

Definition 4.1. Being given a DS F, thegeneric limit set ω˜F is the intersection of all the closed subsets of X which have a comeager realm of attraction.

The generic limit set ˜ω_F can actually be defined as the smallest closed subset of X with a comeager realm, thanks to the following proposition.

Proposition 4.2. Let Fbe a DS. The realm of the generic limit set is comeager.

In particular, it is nonempty! But much more thant this: it is the smallest closed set which includes all limit points of allgeneric orbits.

Proof. Any compact metric space admits a countable basis: there exists a countable set{Ui|i∈N}of closed subsets such that every closed setU can be written asT

i∈IUUi for someIU ⊆N. In particular, ˜ωF is the intersectionT

U

T

i∈IUUi, whereU ranges over closed sets with comeager realm; that is, ˜ωF=T

i∈IUi, where I is the union of IU, for every closed U with comeager realm. If i is in I, then it is in someIU, so that U ⊆Ui, whereU has comeager realm, so thatUi has comeager realm, too.

By Point 5 of Remark 3.4,DF(˜ωF) =DF(T

i∈IUi) =T

i∈IDF(Ui). We know that an intersection of countably many comeager sets is comeager. ThenDF(˜ωF) is comeager.

Note that the generic limit set is the closure of the asymptotic set of some comeager set (it is exactly the closure of the asymptotic set of its realm), but it may not be the asymptotic set of any set: see for example Example 5.14, where the generic limit set is full, but the asymptotic set is not.

Remark 4.3. LetF be a DS over spaceX andV ⊆X.

1. If V does not intersect ω˜F, then DF(V^C)^C is meager. In particular, if V is closed, then dF(V^C)^C is meager, and there is no nonempty open set U in which S

t≥TF_t⁻¹(V) is dense for allT ∈N.

2. IfV is open and intersects ω˜_F, then DF(V^C)^C is not meager. In particular, dF(V^C)^C is not meager, and there exists a nonempty open set U in whichS

t≥TF_t⁻¹(V)is dense for all T ∈N.

There are counter-examples when this remark cannot be stated as an equivalence: take for instance open setV =]0,1[ in the uniform DS defined byF_t(x) =x/2^tforx∈[0,1].

Proof. We prove the first statement of each point.

• IfV^C ⊇ω˜F, thenDF(V^C)⊇ DF(˜ωF) is comeager.

• If V is open and intersects ˜ωF, then ˜ωF\V is closed; by the minimality of the generic limit set, DF(˜ωF\V) is not comeager. This set is equal toDF(˜ωF)∩ DF(V^C). Since the first one is comeager, we deduce that the second one is not.

The second statement of each point comes from the inclusions between realm and direct realm in Remark 3.5.

The third statement comes from Remark 2.1 and the definition ofdF(V^C)^C.

A consequence of this is the following proposition: the generic limit set intersects any closed set with dense realm.

Proposition 4.4. SupposeF is a DS andV a closed set with dense realmDF(V). ThenV intersects ω˜F. Proof. Point 2 of Remark 3.4 gives that DF(V) =T

n>0dF(B1/n(V)). It results that each dF(B1/n(V)) is dense. Its open supersetS

t≥TF_t⁻¹(B1/n(V)) should also be dense. Hence,dF(B1/n(V)^C)^C is comeager. In particular, it is not meager. Then Point 1 of Remark 4.3 gives thatB_1/n(V) intersects ˜ωF. Since this is true for everyn >0, andB_1/n(V) is closed,V should intersect ˜ωF.

4.1 Nonwandering systems

We will see that, for nonwandering dynamical systems, the generic limit set is the full space. Let us prove a more general result, which will be also useful for Corollary 5.6. We say that a DSFissemi-nonwandering if for every open subset U which intersects Ω_F, dF(U^C)^C is not meager. It is clear that a DS over some spaceX is nonwandering if and only if it is semi-nonwandering and its limit set isX (note that this second property happens, in the uniform case, exactly for surjective systems).

Proposition 4.5. A DS Fis semi-nonwandering if and only ifω˜_F= Ω_F. Proof.

• Suppose F is not semi-nonwandering. This means that there exists an open set U which intersects ΩF, and such thatdF(U^C) is comeager. SinceU^C is closed,DF(U^C)⊇dF(U^C) is also comeager. By definition, ˜ωFis then included in U^C, and thus cannot include ΩF.

• Conversely, suppose that F is semi-nonwandering,x∈ΩF and ε >0. By definition, dF(Bε(x)^C)^C is not meager. By inclusion, neither is d_F(B_ε(x)^C)^C. By Point 1 of Remark 4.3, we deduce thatB_ε(x) intersects ˜ω_F. Since this is true for everyε >0, we get that x∈ω˜_F. The inclusions ˜ω_F⊆ω_F⊆Ω_Fare always true.

The following is a direct corollary of Proposition 4.5.

Corollary 4.6. A DSF over some spaceX is nonwandering if and only ifω˜F=X.

It is clear that the two properties in Corollary 4.6 imply surjectivity. Since it is known that surjective CA are all nonwandering (see for instance [18, Prop 5.23]), we get the following.

Corollary 4.7. A CAF overA^Z is surjective if and only ifω˜F =A^Z, if and only ifωF is comeager.

The second statement is also true for uniform DS [12, Cor 26].

Proof. The first statement is a direct corollary of Corollary 4.6.

For the second statement, nonwandering uniform DS are known to admit a comeager set of recurrent points, which are all inωF (see Subsection 3.3).

Figure 1: Lonely Gliders: space is horizontal and time goes upwards;← (resp. >) are represented by black (resp. white) squares, and→ (resp. <) are represented by dark (resp. light) grey squares.

The following example answers a question left open in [12]: the asymptotic set of a surjective CA is comeager, but not always full.

Example 4.8 (Lonely gliders). Let A={>, <,→,←}, andF the CA, defined by the following local rule:

f : (x₋₁, x₀, x₁)7→

→ if x₋₁=→ and x₀=<

→ if x₋₁6=> andx₀=←

< if x₋₁=> andx0=←

> if x0=→ andx1=<

← if x0=→ andx16=<

← if x0=> andx1=←

x0 otherwise.

A typical space-time diagram of this CA is shown in Figure 1. Intuitively, each configuration can be decom- posed into valid zones, which contain at most one arrow, towards with chevrons <and > are supposed to point. The arrow moves in the direction to which it points, until it reaches the end of the zone (noticed by an invalid pattern of the formab, wherea 6=> and b ∈ {>,→,←}, or symmetric), in which case it turns back. With this in mind, it is not difficult to understand thatF is reversible (hence surjective) and that any invalid pattern is a blocking word. From Corollary 4.7, the asymptotic set is comeager. Yet, it is not full.

Proof. Let us prove that some configurationxwith an infinite valid zone which contains one arrow cannot be the limit point of any orbit. Indeed, any configuration whose orbit comes arbitrarily close to x should also have an infinite valid zone (because the zones are invariant), and hence at most one arrow in it (like the configuration whose orbit is illustrated in Figure 1. Any limit point of such an orbit has no arrow in its infinite valid zone (the arrowgoes to infinity).

4.2 First property for CA

It is rather clear that the generic limit set of a uniform DS induces a subsystem. Let us see that this is true also for directional dynamics CA, and that it is also invariant by shift.

Proposition 4.9. LetF= (Ft)tbe a sequence of CA. Thenω˜Fis a subshift. Its realm is stronglyσ-invariant.

Proof. By definition, ˜ωF is closed. LetU =DF(˜ωF). Sinceσis a homeomorphism,σ^k(U) is also comeager for allk∈Z. ThenW =T

k∈Zσ^k(U) is still comeager, as an intersection of countably many comeager sets.

One hasW ⊆U, so thatωF(W)⊆ωF(U) = ˜ωF. Conversely, the definition of ˜ωFgives that it is included in ωF(W). Overall,ωF(W) = ˜ωF. Since W is stronglyσ-invariant, ˜ωF=ωF(W) is also stronglyσ-invariant, by Proposition 3.13.

Moreover, Corollary 3.15 directly gives that ˜ωFis included in all subshift attractors.

Proposition 4.10. Consider the CAF in some directionh. Then ω˜_F,h is an F-invariant subshift.

Proof. We just apply Proposition 3.13 to ˜ω_F,h=ω_F,h(D_F,h(˜ω_F,h)).

4.3 Indecomposability

Now we prove that the generic limit set of a cellular automaton is indecomposable in some sense.

Proposition 4.11. Let V =Fn−1

i=0 Vi, where n∈ N and the Vi are closed subsets which are invariant by some CA F in some direction h, and, strongly, by σ^p, for some p >0. IfDF,h(V) has nonempty interior (resp. is not meager), then there exists i∈J0, nJsuch that DF,h(V_i)is dense (resp. comeager).

Proof. One hasDF,h(V) =Fn−1

i=0 DF,h(V_i) by Proposition 3.9.

By Point 4 of Lemma 2.3, there existsi∈J0, nJsuch thatDF,h(V_i) is dense (resp. comeager).

Corollary 4.12. LetF be a CA andha curve. ω˜_F,h cannot be decomposed as a disjoint union of non-trivial subshift subsystems (or even non-trivial strongly σ^p-invariant subsystems, for some p >0).

In other words, we can say that ˜ωF,his connected, when considering the dynamical pseudo-metric related to the action of (F, σ): ˜d(x, y) = inf_{i,j∈Z,s,t∈N}d(F^sσⁱ(x), F^tσ^j(y)).

Proof. We assume that ˜ωF,h =Fn−1

i=0 Vi, where the Vi are closed, invariant, strongly σ^p-invariant sets. By Proposition 4.11, there existsi∈J0, nJsuch thatDF,h(Vi) is comeager. By definition, ˜ωF,h is then included inVi, and hence equal.

4.4 Finite generic limit set

If the generic limit set is finite, then the DS is almost equicontinuous.

Proposition 4.13. Let Fbe a semi-uniform DS and V =ω(U)be finite, for some setU which is comeager in some nonempty open subset W ⊆X. Then EFis comeager in W.

This of course implies thatEF∩U is comeager inW.

Before proving the proposition, we immediately deduce the following.

Corollary 4.14.

• If ω˜_F is finite, thenF is almost equicontinuous.

• IfFhas no equicontinuous point, then the asymptotic (and limit) sets of all non-meager sets are infinite.