INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY

Nonlinearity 17 (2004) 105–116 PII: S0951-7715(04)53178-X

On the density of directional entropy in latticedynamical systems

V Afraimovich, A Morante and E Ugalde

Facultad de Ciencias and Instituto de Investigacion en Comunicacion Optica, UniversidadAutonoma de San Luis Potosı, Alvaro Obregon 64, C.P. 78000, San Luis Potosı, Mexico

Received 31 August 2002, in final form 11 September 2003Published 6 October 2003Online at stacks.iop.org/Non/17/105 (DOI: 10.1088/0951-7715/17/1/007)

Recommended by L Bunimovich

AbstractWe study the direction dependence of the density of directional entropy inlattice dynamical systems. We show that if the dynamics is homogeneous andcontinuous, then this density does not depend on the direction in space–time.By using symbolic dynamics we derive formulae for the density for weaklycoupled hyperbolic maps. As a corollary, we present examples where thisdensity actually depends on the direction, provided that individual subsystemsare sufficiently different.

Mathematics Subject Classification: 37L60

1. Introduction

It is natural to assume that the complexity of behaviour of extended systems in differentdirections in space–time can be different. A characteristic of such a complexity has beenintroduced in [9,10] for the case where an extended system is modelled by a cellular automaton(CA) and it was called the directional entropy. Roughly speaking, it was shown that thenumber of different pictures in a window in the form of a parallelogram with two sides oflength T parallel to a prescribed direction, θ , in space–time and the two other ‘horizontal’sides of length L parallel to the space-axis behaves asymptotically as CeT Hθ , T � 1, wherethe constant C depends on L. The quantity Hθ is said to be the topological entropy in thedirection θ (see also [4] for a modified definition). It was shown in [13] that, as a function ofθ , the directional entropy may be discontinuous. In [11, 12], some properties of the measure-theoretical directional entropy have been studied. In [6], explicit formulae for the directionalentropy of permutative CA were obtained.

If one deals with the case where a function of state takes continuous values (unlike CA), oneshould replace cylinders (which label different pictures) by ε-separated pieces of orbits. Thiswas done in [2], where directional topological entropy was introduced for lattice dynamical

0951-7715/04/010105+12$30.00 © 2004 IOP Publishing Ltd and LMS Publishing Ltd Printed in the UK 105

106 V Afraimovich et al

systems (LDSs). Unlike the CA-case, the directional entropy could be infinity here, so a notionof ‘directional entropy per unit length’ (as in [5]) or density, hθ , was introduced also in [2].Now, the number of different pictures in the window described above behaves as CeLT hθ ,T � 1.

In [2] the main properties of these quantities were studied in the case when the evolutionoperator commutes with spatial translations. It was found for weakly coupled hyperbolicsubsystems that the density of directional entropy is in fact independent of direction in space–time. In this work we show that the same is true for an arbitrary translation-invariant LDS. Thedensity of the entropy could depend on a direction in space–time only in the case when coupledsubsystems are sufficiently different. Making use of symbolic dynamics, we derive formulaefor the density in the case of weakly coupled different hyperbolic maps. As a corollary weshow that the density can take different values depending on the direction in space–time.

2. Definitions

We recall the definitions of one-dimensional LDSs and of density of directional entropy forsuch systems (see [2] for more details).

Let I be a metric space and let d(·, ·) be the corresponding distance. Consider the directproduct, IZ, endowed with the product topology. Assume the existence of a compact setM ⊂ IZ and the existence of a map F from M into itself. The pair (M, F ) is called anLDS. In the following, we shall frequently use the spatial shift σ� : IZ → IZ defined by(σ�u)s = us+1, u = {us}s∈Z. An LDS (M, F ) is translation-invariant if F ◦ σ� = σ� ◦ F .

An orbit of (M, F ) is a sequence {u(t)}t∈Z+ = {us(t)}s∈Z,t∈Z+ where u(t) ∈ M andu(t + 1) = Fu(t) for all t ∈ Z

+ (here Z+ denotes the set of non-negative integers).

Given a finite window W ⊂ Z × Z+ and a number ε > 0, a set K ⊂ M is called (ε, W)-

separated if for every pair of distinct elements u, v ∈ K , there exists (s, t) ∈ W such that

d((F tu)s, (Ftv)s) � ε.

The number of distinct orbits with accuracy ε in the window W is defined by

N(ε, W) = max{#K : K is a (ε, W)-separated set}.Consider an interval of the form [k, k +L] with k ∈ Z and L ∈ N. Given a direction θ ∈ (0, π)

and T ∈ R+, let [k, k + L]θ,T denote the following window (see figure 1):

[k, k + L]θ,T = {(x + ρ cos θ, ρ sin θ) : x ∈ [k, k + L], ρ ∈ [0, T ]} ∩ (Z × Z+).

The density of topological entropy of (M, F ) in the direction θ is defined by

hθ(M, F ) = limε→0

(lim

L→∞

(supk∈Z

(lim

T →∞1

LT sin θlog N(ε, [k, k + L]θ,T )

)))

(the normalization LT sin θ corresponds to the area of the window [k, k +L]θ,T ). The quantityhθ(M, F ) exists and belongs to [0, +∞]. The present definition differs slightly from theone in [2]. In particular, we have divided the previous quantity by sin θ in order to simplifycomputations.

A topological invariance-like property has been proved in [2] for the density of directionalentropy. To be more precise, let (MF , F ) and (MG, G) be two translation-invariant LDSthat are topologically conjugated (the conjugacy map is a homeomorphism in the producttopology). Then for any θ ∈ (0, π), the following equality holds:

hθ(MF , F ) = hθ(MG, G).

On the density of directional entropy in LDSs 107

Figure 1. An example of the window [k, k + L]θ,T .

3. Density of directional entropy for translation-invariant systems

Let (�, σ ) be a topological Markov chain (TMC) (see [8]). Consider the LDS (�Z, στ ), whereστ is the temporal shift acting on each w = {wt

s}s∈Z,t∈Z+ in �Z as follows: (στw)ts = wt+1s for

all s and t . If h is the topological entropy of (�, σ ), a theorem in [2] claims that

hθ(�Z, στ ) = h, θ ∈ (0, π).

The independence from the direction θ of the density of directional entropy holds in moregeneral situations.

Theorem 3.1. Assume that the map F is continuous and commutes with the spatial shift, σ� ,and assume that F(M) ⊂ M and σ�(M) = M. Then the quantity hθ(M, F ) does notdepend on θ . Moreover, this quantity is equal to the topological entropy of the Z × Z

+-actiongenerated by (σ�, F ) on M.

Before proving this theorem, let us give a definition of the topological entropy inspired by (andequivalent to) the one in [1]. For an open finite cover A of M, and for each finite ‘space–timewindow’ W ⊂ Z × Z

+, let us define the W -refinement of A as

AW =∨

(s,t)∈W

F−t (σ−s� (A)).

Then we define the W -complexity of A as

C(A, W) = min{#B : B is a sub-cover of AW }.It is easy to verify that for any W, W ′ ∈ Z × Z

+ one has

C(A, W ∪ W ′) � C(A, W)C(A, W ′). (1)

Let W = (W1 ⊂ W2 ⊂ · · ·) be a sequence of finite space–time windows satisfying thefollowing two conditions.

(a) The sequence W converges to the plane in the sense of van Hove, i.e.-

⋃∞n=1 Wn = Z × Z

+ and- limn→∞(#∂Wn/#Wn) = 0, where ∂Wn = {(s, t) ∈ Wn : (s + s ′, t + t ′) ∈ Wn

for some (s ′, t ′) ∈ {−1, 0, 1}2}.

108 V Afraimovich et al

(b) For every Wn ∈ W there exists a tiling lattice Mn ⊂ Z × Z+ such that:

- Z × Z+ = ⋃

(s,t)∈Mn(Wn + (s, t)) and

- (Wn + (s, t)) ∩ (Wn + (s ′, t ′)) = ∅ whenever (s, t) = (s ′, t ′).

Here Wn + (i, j) denotes the set {(s + i, t + j) : (s, t) ∈ Wn}.Given a finite window W ⊂ Z × Z

+, we say that a tiling lattice Sn ⊂ Mn is W -optimalif W ⊂ ⋃

(s,t)∈Sn(Wn + (s, t)) and #(

⋃(s,t)∈Sn

(Wn + (s, t))\W) � #(∂W) × #Wn. It is notdifficult to see that W -optimal lattices exist.

Claim 3.2. The limit

h(A|W) = limn→∞

log C(A, Wn)

#Wn

exists and is called the topological entropy of the Z × Z+-action with respect to the cover A

and the sequence W .

Proof. Let h∗ = infn∈N(log C(A, Wn))/#Wn. Given δ > 0 let n∗ ∈ N be such that(log C(A, Wn∗))/#Wn∗ � h∗ + δ/2. For each m � n∗ fix a Wm-optimal lattice Sn∗ ⊂ Mn∗ .Since C(A, W) increases with W , and because of the sub-additivity property stated inequation (1), it follows that

log C(A, Wm)

#Wm

�log C(A,

⋃(s,t)∈Sn∗ (Wn∗ + (s, t)))

#Wm

� log C(A, Wn∗)

#Wn∗× #Sn∗ × #Wn∗

#Wm

�(

h∗ +δ

2

)×

(1 +

#(∂Wm)

#Wm

× #Wn∗

).

Finally, since #(∂Wm)/#Wm → 0 as m → ∞, then, for each m sufficiently large,

#(∂Wm)

#Wm

� δ

(2h∗ + δ) × #Wn∗

and the result follows with h(A|W) = h∗. �

Lemma 3.3. Let W = (W1 ⊂ W2 ⊂ · · ·) and W ′ = (W ′1 ⊂ W ′

2 ⊂ · · ·) be two sequences ofspace–time windows satisfying properties (a) and (b) stated above. Then h(A|W) = h(A|W ′).

Proof. Given ε > 0, let n∗ be such that (log C(A, Wn))/#Wn � h(A|W) + ε/2 for alln � n∗. Let Mn be a tiling lattice for the window Wn. Then, for each m � n∗

there exists a W ′m-optimal lattice S ′

n∗ ⊂ Mn∗ such that W ′m ⊂ ⋃

(s,t)∈S ′n∗ (Wn∗ + (s, t)), and

#(⋃

(s,t)∈S ′n∗ (Wn∗ + (s, t))\W ′

m) � #(∂W ′m) × #Wn∗ . Then, proceeding as in claim 3.2, we

obtain

h(A|W ′) � log C(A, W ′m)

#W ′m

�(h(A|W) +

ε

2

)×

(1 +

#(∂W ′m)

#W ′m

× #Wn∗

).

It follows for m large enough that

#(∂W ′m)

#W ′m

� ε

(2(h(A|W) + ε) × #Wn∗,

thus, h(A|W ′) � h(A|W) + ε and therefore h(A, W ′) � h(A|W). Interchanging W with W ′,we obtain the reciprocal inequality. �

On the density of directional entropy in LDSs 109

We are now able to define the topological entropy of the action {F t ◦ σ s� : (s, t) ∈ Z × Z

+} as

htop(M|F t ◦ σ s�) = sup

Ah(A|W),

where W is any sequence of space–time windows satisfying the properties (a) and (b) statedabove, and the supremum is taken over all open finite covers of M. It can be proved thatfor each ε > 0 there exists a finite cover of M by open balls of radius ε, say Aε, such thatsupA h(A|W) = limε→0 h(Aε|W).

In order to prove theorem 3.1, we need the following lemma.

Lemma 3.4. Let Aε be a finite cover of M by open balls of radius ε, and let W ⊂ Z × Z+ be

any space–time window. Then

N(2ε, W) � C(Aε, W) � N

(�

2, W

),

where � is the Lebesgue number of the cover Aε.

Proof. We follow the proof of the theorem 7.7 in [14]. Let dW(x, y) = max(s,t)∈W d(F t ◦σ s

�(x), F t ◦ σ s�(y)). Given x ∈ M let B(x) denote one of the elements in AW

ε

.=∨(s,t)∈W F−t (σ−s

� (Aε)) containing x.The first inequality in the statement of the lemma easily follows from the next observation.

If x, y ∈ M are such that dW(x, y) � 2ε, then B(x) = B(y).Now, let K be a maximal �/2 separated set with respect to the distance dW , i.e.

#K = N(�/2, W). Let B ∈ AWε be a minimal sub-cover, i.e. #B = C(Aε, W). Assume

that #B > #K . First, let us observe that the ‘tubes’ T (x) = {y ∈ M : dW(x, y) < �/2},x ∈ K form a cover of M because of the maximality of the set K . For definition, every tubebelongs to an element of AW

ε . Therefore there exists a sub-cover of AWε containing fewer

elements than #B. This contradiction completes the proof. �

Proof of theorem 3.1. Since F is shift-invariant, then

hθ(M, F ) = limε→0

(lim

L→∞

(lim

T →∞1

LT sin θlog N(ε, [0, L]θ,T )

))

= limε→0

(lim

L→∞

(lim

T →∞1

#[0, L]θ,T

log N(ε, [0, L]θ,T )

)).

Choose an increasing sequence {Lm}∞m=1 ⊂ N, and for each index m choose another increasingsequence {Tn}∞n=1 ⊂ R

+ such that

hθ(M, F ) = limε→0

(lim

m→∞

(lim

n→∞1

#[0, Lm]θ,Tn

log N(ε, [0, Lm]θ,Tn)

)).

Let

hθ,ε,m = limn→∞

1

#[0, Lm]θ,Tn

log N(ε, [0, Lm]θ,Tn),

and n(m) ∈ N be such that∣∣∣∣ 1

#[0, Lm]θ,Tn

log N(ε, [0, Lm]θ,Tn) − hθ,ε,m

∣∣∣∣ � e−m

for all n � n(m). Then

hθ(M, F ) = limε→0

(lim

m→∞1

#[0, Lm]θ,Tn

log N(ε, [0, Lm]θ,Tn)

).

110 V Afraimovich et al

We choose n(m) increasing with m.Let

hθ,ε = limm→∞

(lim

n→∞1

#[0, Lm]θ,Tn

log N(ε, [0, Lm]θ,Tn)

).

Consider the sequence of space–time windows Wθ.= (W1 ⊂ W2 ⊂ · · ·), with Wm

.=[0, Lm]θ,Tn(m)

, which clearly satisfies the conditions (a) and (b) stated above for each θ ∈ (0, π).Lemma 3.4 implies

hθ,2ε � h(Aε|Wθ ) � hθ,�/2,

where Aε is a finite cover of M by open balls of radius ε, and � is the Lebesgue numberof Aε. Note that hθ,ε depends monotonously on ε, so that limε→0 h(Aε|Wθ ) ∈ [0, ∞]exists. Since � → 0 as ε → 0, by using suitable covers Aε of M one has thathθ(M, F ) = limε→0 h(Aε|Wθ ) = htop(M|F t ◦ σ s

�). �

4. The density of directional entropy for perturbations of uncoupled systems

The notion of density of directional entropy is pertinent in the case of non-translation-invariantLDS. To see it, we consider a class of examples for which that quantity can be actuallycomputed. Those examples are perturbations of an LDS consisting of uncoupled hyperbolicmaps conjugated to (not necessarily identical with) TMC. The computations are performedfor uncoupled system, and theorem 4.2 is used to extend the validity of the formulae to thecorresponding weakly coupled LDS.

4.1. Invariance of the density of directional entropy

Here we consider coupled maps of the closed intervals I s ⊂ I = [a, b] for s ∈ Z. Define themap F :

⊗s∈Z

I s → RZ by

(Fu)s = fs(us) + γFs(us−r , . . . , us+r ), u = {us}s∈Z ∈⊗s∈Z

I s. (2)

Here γ > 0 is the coupling parameter, each local map fs : I s → R is continuous, andeach coupling map Fs :

⊗−r�j�r I s+j → R is assumed to be C1-smooth. Moreover, we

assume that

β.= sup

s∈Z

(max

1�j�2r+1

(max

(z1,...,z2r+1)

∣∣∣∣∂ Fs(z1, . . . , z2r+1)

∂zj

∣∣∣∣))

< +∞,

where the first maximum is taken over all possible vectors (z1, . . . , z2r+1) ∈ ⊗−r�j�r I s+j .

Suppose now that for every s ∈ Z the map fs satisfies the following hypothesis.

H. There exists a finite collection of pairwise disjoint closed intervals {I si }ps

i=1 in I s , such thatfor each i, 1 � i � ps , one has(i) fs is differentiable on I s

i and 1 < α.= infs∈Z(minI s

i|f ′

s |) < +∞, and(ii) there exists j , 1 � j � ps , such that I s

j ⊂ Intfs(Isi ).

Under these conditions the map fs has a closed invariant subset �s ⊂ Cs = ⋃ps

i=1 I si ⊂ I s ,

such that fs |�sis topologically conjugated to a TMC (�As

, σ ). Here the set of admissiblesequences �As

⊂ {1, . . . , ps}Z+

is endowed with the metric d(ωs, ωs) = exp(−k), ωs ={ωt

s}, ωs = {ωts} ∈ �As

, where k = min{t ∈ Z+ : ωt

s = ωts}.

Consider now the LDS (�, στ ) having as phase space the set �.= ⊗

s∈Z�As

. Thefollowing result gives us a symbolic description of an invariant set of (2) in the set C .= ⊗

s∈ZCs .

On the density of directional entropy in LDSs 111

Theorem 4.1. If fs satisfies the condition (H) and ps < p < ∞ for all s, then there existsγ ∗ > 0 such that, for any coupling parameter γ < γ ∗, there exists an F-invariant closed set�F ⊂ C and a homeomorphism, in the product topologies, � : � → �F so that

� ◦ στ = F ◦ �.

(in fact, �−1 is the usual itinerary map).

The theorem can be proved by using the method applied in the proof of lemma 4.4 below andarguments similar to the ones in [3]. In fact, a more general result will be published elsewheresoon. This proof can be made also by using methods of [7]. We exploit this theorem to showthe invariance of the density.

Theorem 4.2. Under the conditions of theorem 4.1, and assuming thatδ

.= infs∈Z(min1�i<j�psdist(I s

i , I sj )) > 0 and that γ ∗ � (α − 1)/(β(8r + 4)), one has

hθ(�F , F) = hθ(�, στ ).

The proof of this theorem is based on the next two lemmas. We will compare the numbers ofseparated orbits in different systems and we will use a sub-index to indicate the system we areconsidering.

Lemma 4.3. Let S = [k, k + L] with fixed k ∈ Z, L ∈ N, θ ∈ (0, π), T ∈ R+ and

ε ∈ (0, min{1, δ}). There exists a positive integer N depending only on ε, such that

N�(ε, Sθ,T ) � N�F (ε, Sθ,T ),

where S = [k − N cot θ, k + L + N cot θ ] and T = T + N csc θ .

Proof. It is enough to prove that, if ω and ω are (ε, Sθ,T )-separated points in �, thecorresponding points u

.= �ω and u.= �ω in �F are (ε, Sθ,T )-separated . Let ω, ω be

(ε, Sθ,T )-separated. There exists an integer pair (s, t) ∈ Sθ,T , such that

d(ωs(t), ωs(t)) � ε.

Hence, there is an integer 0 � N � log(1/ε) such that ωNs (t) = ωN

s (t). In other words,ω0

s (t + N) = ω0s (t + N). Therefore,

| us(t + N) − us(t + N)| � δ � ε.

It is simple to see that if (s, t) ∈ Sθ,T then (s, t + N) ∈ Sθ,T . �

In what follows we use the ceiling function ·� and the floor function �·� (i.e. x�, �x� ∈ Z,x � x� < x + 1 and x − 1 � �x� < x). Let D = sups∈Z

(max1�i�psdiam(I s

i )) andM = min{2, (α + 1)/2}.

Lemma 4.4. Let S = [k, k + L] with a fixed k ∈ Z, L ∈ N, θ ∈ (0, π), T ∈ R+, and

ε ∈ (0, min{1, D/M}). For n.= (log D − log ε)/log M� the following inequality holds:

N�F (ε, Sθ,T ) � N�(ε, Sθ,T ),

where S = [k − n(r + cot θ), k + L + n(r + cot θ)] and T = T + n csc θ .

Proof. It is enough to prove that, if u and v are (ε, Sθ,T )-separated points in �F , thecorresponding points �−1u and �−1v in � are (ε, Sθ,T )-separated. Let u, v be (ε, Sθ,T )-separated. There exists an integer pair (s0, t0) ∈ Sθ,T for which | us0(t0) − vs0(t0) | � ε. If the

112 V Afraimovich et al

points us0(t0) and vs0(t0) already belong to different sub-intervals of I , then the configurations(�−1u)s0(t0) and (�−1v)s0(t0) must differ in their first symbol and therefore

d((�−1u)s0(t0), (�−1v)s0(t0)) = 1,

i.e. �−1u and �−1v are (ε, Sθ,T )-separated configurations.If it is not so, we will construct a collection of pairs (si, ti), i = 1, . . . , n =

(log D − log ε)/log M�, such that

|usi+1(ti+1) − vsi+1(ti+1)| � M|usi(ti) − vsi

(ti)|,hence, |usn

(tn) − vsn(tn)| � Mn|us0(t0) − vs0(t0)| � Mnε > D. Given i, 0 � i � n − 1, let

δi.= max

−r�j�r{|usi+j (ti) − vsi+j (ti)|}.

If δi > 2 |usi(ti) − vsi

(ti)|, we set ti+1 = ti and choose si+1 ∈ {si − r, . . . , si + r} in such a waythat | usi+1(ti) − vsi+1(ti) | = δi . So,

|usi+1(ti+1) − vsi+1(ti+1)| � 2|usi(ti) − vsi

(ti)|.If δi � 2 |usi

(ti) − vsi(ti)|, we let the system evolve, writing for suitable yi ∈ I si and

(z1i , . . . , z

2r+1i ) ∈ ⊗

−r�j�r I si+j ,

|usi(ti + 1) − vsi

(ti + 1)|

=∣∣∣∣∣∣f ′

si(yi) (usi

(ti)− vsi(ti)) + γ

r∑j=−r

∂Fsi(z1

i , . . . , z2r+1i )

∂zr+j+1(usi+j (ti) − vsi+j (ti))

∣∣∣∣∣∣� |f ′

si(yi)||usi

(ti) − vsi(ti)| − γ

r∑j=−r

∣∣∣∣∂Fsi(z1

i , . . . , z2r+1i )

∂zr+j+1

∣∣∣∣|usi+j (ti) − vsi+j (ti)|

� [α − γβ(4r + 2)] |usi(ti) − vsi

(ti)|� α + 1

2|usi

(ti) − vsi(ti)|.

It follows that

|usi+1(ti+1) − vsi+1(ti+1)| � α + 1

2|usi

(ti) − vsi(ti)|,

provided that si+1 = si and ti+1 = ti + 1.Thus, |usn

(tn)−vsn(tn)| > D, i.e. the points usn

(tn) and vsn(tn) lie in different sub-intervals

of I . It follows from the definition of si and ti that |si − s0| � nr for any 1 � i � n and that|tn − t0| � n. Therefore, (si, ti) belongs to the window Sθ,T . �

Proof of theorem 4.2. Since limT →∞ T /T = 1, lemma 4.3 implies that

limT →∞

1

LT sin θlog N�(ε, Sθ,T ) � lim

T →∞

1

LT sin θlog N�F (ε, Sθ,T ).

Hence,

supk∈Z

(lim

T →∞1

LT sin θlog N�(ε, Sθ,T )

)� sup

k∈Z

(lim

T →∞

1

LT sin θlog N�F (ε, Sθ,T )

).

Let L.= (L + 2N cot θ). Since limL→∞ L/L = 1, then

limL→∞

(supk∈Z

(lim

T →∞1

LT sin θlog N�(ε, Sθ,T )

))

� limL→∞

(supk∈Z

(lim

T →∞

1

LT sin θlog N�F (ε, Sθ,T )

))

On the density of directional entropy in LDSs 113

and the inequality hθ(�, στ ) � hθ(�f , F) follows.The inequality

limL→∞

(supk∈Z

(lim

T →∞1

LT sin θlog N�F (ε, Sθ,T )

))

� limL→∞

(supk∈Z

(lim

T →∞

1

LT sin θlog N�(ε, Sθ,T )

))can be obtained in a similar way by using lemma 4.4. �

4.2. Density of directional entropy for direct products

We now provide examples of systems for which the density of entropy depends on θ . In viewof the results of the previous subsection, we will compute the density of directional entropyfor the direct product of TMC. Then, as stated in theorem 4.2, the obtained formulae will bestill valid for the coupled case if the coupling is weak enough.

We denote by hs the topological entropy of the TMC (�As, σ ) with the transition matrix As .

The following result holds for less restrictive conditions than the ones imposed below, butto avoid some technicalities we state it in the form of theorem 4.5.

Theorem 4.5. Assume that (�As, σ ) is topologically transitive, and there exists p such that

�As⊂ �p, the full shift on p symbols. Then,

hθ(�, στ ) =

h+ if θ ∈(

0,π

2

),

h0 if θ = π

2,

h− if θ ∈(π

2, π

),

where

h+ = limT →∞

1

T

T −1∑s=0

hs, h− = limT →∞

1

T

0∑s=1−T

hs, h0 = limL→∞

(supk∈Z

1

L

L+k−1∑s=k

hs

).

Proof. We first assume that 0 < θ < π/2. For k ∈ Z and L ∈ N, the set [k, k + L]θ,T is theintersection of a parallelogram with Z × Z

+. Hence,

[k, k + L]θ,T = {(s, t) ∈ Z × Z+ : k � s � k + L + �T cos θ�, t s � t � t s},

for suitable t s and t s .Let 0 < ε < 1 be fixed. Because of the structure of �, N(ε, [k, k + L]θ,T ) =∏k+L+�T cos θ�

s=k ns(ε, t s, t s), where ns(ε, t s, t s) is the maximal cardinality of (ε, [t s , t s])-separated sets for the system (�As

, σ ).The computation of t s and t s gives us the equality

log N(ε, [k, k + L]θ,T ) = �1 + �2 + �3,

valid for T large enough, where

�1 =k+L−1∑s=k

log ns(ε, 0, �s tan θ�),

�2 =�k+T cos θ�∑

s=k+L

log ns(ε, (s − L) tan θ�, �s tan θ�),

114 V Afraimovich et al

and

�3 =k+L+�T cos θ�∑

s=�k+T cos θ�+1

log ns(ε, (s − L) tan θ�, �T sin θ�).

We have

limT →∞

1

T�1 = lim

T →∞1

T�3 = 0 log N(ε, [k, k + L]θ,T ) � �2.

Therefore, for every fixed L,

limT →∞

1

LT sin θlog N(ε, [k, k + L]θ,T ) = lim

T →∞1

LT sin θ�2.

Because of the transitivity of (�As, σ ), ns(ε, k, k′) = ns(ε, 0, k′ − k). It implies that

limT →∞

1

T sin θ

�k+T cos θ�∑s=k+L

log ns(ε, 0, �L tan θ� − 1) � limT →∞

1

T sin θ�2

� limT →∞

1

T sin θ

�k+T cos θ�∑s=k+L

log ns(ε, 0, L tan θ� + 1). (3)

It is easy to see that ns(ε, 0, N) = E′(As)N+kεE, where E′ = (1, 1, . . . , 1) and kε ≈ log(1/ε).

The Perron–Frobenius theorem ensures that E′(As)N+kεE = O(λN

s ), where λs is the maximaleigenvalue of As . Since there are only finitely many TMC satisfying our assumptions, thenthere exists a constant Cp,ε such that

C−1p,ε(λs)

N � ns(ε, 0, N) � Cp,ε(λs)N . (4)

Inequalities (3) and (4) imply that

limT →∞

1

LT sin θ

�k+T cos θ�∑s=k+L

((L tan θ − 2) log λs − log Cp,ε

)� lim

T →∞1

LT sin θ�2.

Therefore (lim

T →∞L tan θ − 2

LT sin θ

�k+T cos θ�∑s=k

hs

)− cot θ log Cp,ε

L� lim

T →∞1

LT sin θ�2.

Let us remark that the limit on the left-hand side of this inequality is independent of k anddepends only on the sign of cos θ . The term on the right-hand side can be upper bounded in asimilar way, so that for θ ∈ (0, π/2) we have

limL→∞

(supk∈Z

(lim

T →∞1

LT sin θ�2

))= lim

T →∞1

T cos θ

�T cos θ�∑s=0

hs = h+.

Since this formula holds for any ε ∈ (0, 1), we have the desired result for θ ∈ (0, π/2). Theproof of the statement of the theorem for θ ∈ (π/2, π) can be done in a similar way.

Consider now the case θ = π/2. For 0 < ε < 1, log N(ε, [k, k + L]π/2,T ) =∑k+Ls=k log ns(ε, 0, T ), and by using inequalities (4), we show that

T

L+k−1∑s=k

hs − L log Cp,ε � log N(ε, [k, k + L]π/2,T ) � T

T +k−1∑s=k

hs + L log Cp,ε

and

limT →∞

1

LTlog N(ε, [k, k + L]π/2,T ) = 1

L

L+k−1∑s=k

hs.

On the density of directional entropy in LDSs 115

Thus,

limL→∞

(supk∈Z

(lim

T →∞1

LT sin θN(ε, [k, k + L]π/2,T )

))= lim

L→∞

(supk∈Z

1

L

L+k−1∑s=k

hs

)= h0. �

Corollary 4.6. Under the conditions of theorem 4.2, we have

hθ(�F , F) =

h+ if θ ∈(

0,π

2

),

h0 if θ = π

2,

h− if θ ∈(π

2, π

),

where h+, h− and h0 are defined as above, and hs = htop(fs |�s).

Corollary 4.7. Let θ ∈ (0, π)\{π/2}, then there exists ε > 0 such that

hθ(�F , F) = limT →∞

(lim

L→∞1

LT sin θlog N(ε, [0, L]θ,T )

).

Example 4.8. Let hs = a for s � 0, and hs = b < a for s > 0; then h− = h0 = a andh+ = b.

Remark 4.9. The results of this paper could be generalized for the case of the d-dimensionallattice, d > 1. For that, one must prove an analogue of theorem 4.1 first. Then, one has toextend the definition of density of directional entropy to the d-dimensional case. Finally, onehas to check that the necessary calculations are bearable and perform them.

Acknowledgments

AM acknowledges support from the ANUIES-ECOS-Nord programme M99P01. AM is aCONACyT fellow No 128036 at IICO-UASLP. VA was partially supported by NSF-CONACyTgrant No E120933 and by CONACyT grant 485100-5-36445-E, and EU was supported byCONACyT grant No J32389E. The authors greatly appreciate B Fernandez for his involvementin the study that allowed them to improve the manuscript essentially. They wish to thankA Katok and S Zelik for pointing out the importance of the problem of the relation betweenthe density and the entropy of the Z

2-action. They also wish to thank the referee for carefulreading of the manuscript and very useful suggestions.

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