María Anguiano
email: anguiano@us.es

Alain Haraux
email: haraux@ann.jussieu.fr

The ε-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

Keywords: AMS classification numbers: 37L30, 35B41 Fractal dimension, Attractors, Entropy

, an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.

Introduction

Let M be a precompact set in a metric space X. We recall the definition of the fractal dimension of M (see, for instance, Temam [START_REF] Temam | Infinite dimensional dynamical systems in mechanics and physics[END_REF]). According to Hausdorff criteria the set M can be covered by a finite number of ε-balls in X for every ε > 0. Denote by N ε (M, X) the minimal number of ε-balls in X which cover M . Then the Kolmogorov ε-entropy of the set M in X is defined to be the following number

H ε (M, X) ≡ log 2 N ε (M, X),
and the fractal dimension of M can be defined in the following way

dim F (M ) = dim F (M, X) = lim sup ε→0 + H ε (M, X) log 2 1 ε
.

In the present paper, we shall be dealing with estimates of the fractal dimension of the invariant sets (attractors) of the semigroups generated by infinite-dimensional dynamical systems. The usual way of estimating the fractal dimension of invariant sets involving the Liapunov exponents and k-contraction maps (see, for instance, Temam [START_REF] Temam | Infinite dimensional dynamical systems in mechanics and physics[END_REF]) requires the semigroup to be quasidifferentiable with respect to the initial data on the attractor. It is well known that the Hausdorff dimension is less than or equal to the fractal dimension. In this sense, in [START_REF] Chepyzhov | A note on the fractal dimension of attractors of dissipative dynamical systems[END_REF], Chepyzhov and Ilyin show that the Hausdorff and fractal dimension have the same upper bound generalizing to the infinite-dimensional case the method of Chen [START_REF] Chen | A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors[END_REF].

To avoid the differentiability hypothesis, Zelik, in [START_REF] Zelik | The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension[END_REF], presents a new approach to estimate the dimension of invariant sets. The basic tool of his method is the following very general property.

Theorem 1 (Zelik) Let V and H be Banach spaces, V be compactly embedded in H and let K be a compact subset of H. Assume that there exists a map L : K → K such that L(K) = K and the following 'smoothing' property is valid

L(k 1 ) -L(k 2 ) V ≤ C |k 1 -k 2 | H , ∀k 1 , k 2 ∈ K, C > 0.
(1)

Then, the fractal dimension of K in H is finite and can be estimated in the following way:

dim F (K, H) ≤ H 1 4C (B V (0, 1), H) ,
where C is the same as in (1) and B V (0, 1) means the unit ball centered at 0 in the space V .

In the present work, we show (see Theorem 2) an estimation of the Kolmogorov ε-entropy of B V (0, 1) in H where H is a Hilbert space and V is a Sobolev-like subspace of H. Then we deduce from Zelik's result an estimate of the fractal dimension of the attractor of some nonlinear parabolic equations in terms of the physical parameters. This result is quite explicit and rather close from the estimate obtained in [START_REF] Chepyzhov | A note on the fractal dimension of attractors of dissipative dynamical systems[END_REF] under slightly different but quite related assumptions.

Main results

Let H be a separable Hilbert space with scalar product (•, •) H and norm |•| H . Let V be a dense subspace of H, endowed with a Hilbert structure such that the inclusion map of V into H is compact. Then H is included in V ′ with continuous imbedding. By • V and (•, •) V we denote the norm and the scalar product in V , respectively. We will denote by •, • the duality product between V ′ and V .

Let A ∈ L(V, V ′ ) be the duality map: V → V ′ . It is a self-adjoint monotone operator such that A -1 ∈ L(V ′ , V ) ⊂ L(H, H) is a compact, positive, self-adjoint operator from H to itself.
As a consequence of the Hilbert-Schmidt Theorem there exists a nondecreasing sequence of positive real numbers, 0 < λ 1 ≤ λ 2 ≤ ... ≤ λ j ≤ ......, with lim j→∞ λ j = +∞ and there exists an orthonormal basis {w j : j ≥ 1} of H with A w j = λ j w j for all j ≥ 1.

The sequence (λ j ) is the sequence of eigenvalues repeated according to their multiplicity.

We now assume that (λ j ) satisfies the following growth assumption:

(H1) There exist positive constants c and α such that

λ j ≥ cj α .
Under the last assumption, the first goal in this section is to prove an estimate of the Kolmogorov ε-entropy of B V (0, 1) := {u ∈ V, u V ≤ 1}. In order to do that, we shall identify H with l 2 through the identification u -→ (u j ) j∈N * where u = j u j w j .

Theorem 2 Assume the assumption (H1). Then, the Kolmogorov ε-entropy of B V (0, 1) in H satisfies

∀ε > 0, H ε (B V (0, 1), H) < log 3 + α log 2 2 cε 2 1/α . (2)
Proof. Let u ∈ B V (0, 1). We observe that

u ∈ B V (0, 1) ⇐⇒ ∞ j=1 λ j u 2 j ≤ 1.
Let W ⊂ H be the Hilbert space of vectors u for which j cj α u 2 j < ∞ with the norm

u W = c j j α u 2 j 1/2 . Then u ∈ B W (0, 1) ⇐⇒ ∞ j=1 cj α u 2 j ≤ 1.
Using (H1), we have that B V (0, 1) ⊂ B W (0, 1) and therefore

H ε (B V (0, 1), H) ≤ H ε (B W (0, 1), H). (3)
If we denote µ j = c -1 j -α , we can write B W (0, 1) as an ellipsoid given by

E = {(u j ) ∞ j=1 | ∞ j=1 u 2 j µ j ≤ 1}.
For a given ε > 0, let us give first an upper bound for N √ 2ε (E, H). Let d be the smallest integer such that µ d+1 ≤ ε 2 . We consider the truncated ellipsoid

E = {u ∈ E | u j = 0 for all j ≥ d + 1}.
Given any ε-cover {u 1 , ..., u N } of E, i.e. for each u ∈ E, there exists some i ∈ {1, ..., N } such that

d j=1 (u j -u i j ) 2 ≤ ε 2 .
For any u ∈ E, we have

∞ j=d+1 u 2 j ≤ µ d+1 ∞ j=d+1 u 2 j µ j ≤ ε 2 ,
and hence for some i ∈ {1, ..., N },

∞ j=1 (u j -u i j ) 2 = d j=1 (u j -u i j ) 2 + ∞ j=d+1 u 2 j ≤ 2ε 2 .
Therefore, {u 1 , ..., u N } forms a √ 2ε-cover of the full ellipsoid E. We now view E as a subset of R d , i.e.

E = {u ∈ R d | d j=1 u 2 j µ j ≤ 1},
and we prove the inequality

N ε ( E, R d ) ≤ vol( E + ε 2 B d (1)) vol( ε 2 B d (1)) , (4)
where

B d (1) = {u ∈ R d | d j=1 |u j | 2 ≤ 1}.
The proof of ( 4) is actually simple: first of all let us consider any finite family of points

A = {(a i ) i∈J } ⊂ E for which all balls B(a i , ε 2 ) are pairwise disjoint. Then we have i B(a i , ε 2 ) ⊂ E + ε 2 B d (1), hence vol( ε 2 B d (1))card(A) = i∈J vol(B(a i , ε 2
)

) ≤ vol( E + ε 2 B d (1)).
To conclude, it is sufficient to remark that since the cardinality of such finite sets is bounded, we can consider such a set A with maximal cardinality. Then for any a ∈ A in E , the ball B(a, ε 2 ) intersects at least one of the balls B(a k , ε

2 ), implying ||a -a k || ≤ ε. It follows that the balls B(a i , ε) with a i ∈ A give an ε-covering of E. The result follows immediately. Since ε 2 < µ j for all j ∈ {1, ..., d}, we can see that E contains the ball

εB d (1) = {u ∈ R d | d j=1 |u j | 2 ≤ ε 2 }, hence vol( E + ε 2 B d (1)) ≤ vol( 3 2 E). (5)
From ( 4) and ( 5), we deduce

N ε ( E, H) ≤ 3 d ε d vol( E) vol(B d (1)) .
Since the ellipsoid E is the image of the the unit ball by the linear transform

(x 1 , ..., x d ) -→ ( √ µ 1 x 1 , ..., √ µ d x d ) it follows classically that vol( E) vol(B d (1)) = d j=1 √ µ j
and we can deduce that

N ε ( E, H) ≤ 3 d ε d d j=1 (cj α ) -1/2 = 3 d ε d 1 √ c d (d!) -α/2 ≤ 3 d ε d 1 √ c d d e -α 2 d
, using the fact that µ j = c -1 j -α and the elementary inequality

d! ≥ d e d .
Since µ d+1 ≤ ε 2 , we deduce that

N ε ( E, H) ≤ 3 d d + 1 d α 2 d e α 2 d ≤ 3 d e αd ,
and therefore

H ε ( E, H) ≤ d log 3 + α log 2 . Since ε 2 < µ d = c -1 d -α , we have d < c -1/α ε -2/α
, and we obtain

H ε ( E, H) < log 3 + α log 2 1 cε 2 1/α .
Then we deduce

H ε √ 2 (E, H) < log 3 + α log 2 1 cε 2 1/α .
So that by an obvious change of notation

H ε (E, H) < log 3 + α log 2 2 cε 2 1/α
. and (3) completes the proof.

Remark 3 This upper bound is rather sharp: for a lower bound of the entropy, we observe that the ellipsoid E contains the truncated ellipsoid E, which contains the ball εB d [START_REF] Chen | A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors[END_REF]. Then, we have

N ε 2 (E, H) ≥ N ε 2 (εB d (1), H) ≥ 2 d , (6)
as a consequence of the obvious inequality

card(A)vol( ε 2 B d (1)) ≥ vol(εB d (1)) = 2 d vol( ε 2 B d (1))
valid for any ε 2 -covering of εB d (1) in H with centers forming the set A. Indeed the orthogonal projections in H of the covering balls on the d-dimensional space are covering balls of the projection (equal to εB d (1)) with centers in the d-dimensional space and the matter is reduced to d dimensions. When λ j ≤ Cj α , we can deduce

H ε (B V (0, 1), H) ≥ H ε (E, H). (7) Since ε 2 ≥ µ d+1 = C -1 (d + 1) -α , we have d ≥ C -1/α ε -2/α -1,
and from (6), we obtain

H ε 2 (E, H) ≥ 1 Cε 2 1/α -1,
and by an obvious change of notation

H ε (E, H) ≥ 1 4Cε 2 1/α -1.
Therefore, from (7), we obtain

H ε (B V (0, 1), H) ≥ 1 4Cε 2 1/α -1.
which is not so far from (2).

Now, consider

Au + g(u) = λu, (8)
where g : V → V ′ is a continuous nondecreasing function and λ is a positive constant.

We define by C the set of equilibria of (8) and we consider the identity map I : C → C.

The second goal in this section is to estimate the fractal dimension of C. First, we prove the following result.

Proposition 4 For all u, v ∈ C,

u -v V ≤ √ λ |u -v| H .
Proof. Let u and v belong to C and set u -v, where u and v are solutions to (8). Then, we obtain

Au -Av, u -v + g(u) -g(v), u -v = λ |u -v| 2 H .
Since g is non-decreasing, the conclusion follows easily.

Finally, using Theorems 1 and 2 together with Proposition 4, we deduce the following result.

Theorem 5 Assume the assumption (H1). Then, any compact subset K ⊂ C has a finite fractal dimension with

dim F K < log 3 + α log 2 32λ c 1/α .
Remark 6 As we shall see in the next section, in the applications to concrete elliptic equations, the function N (λ) = min {n ∈ N, λ n+1 ≥ λ} behaves like some positive power of λ for large values of λ . The following example now shows that for general monotonic maps g, the estimate given by Theorem 5 is optimal up to a multiplicative constant in such a case, therefore essentially optimal as far as the growth as a function of λ is concerned and a general monotone map g is allowed. Let us consider λ > 0, n ∈ N such that λ n < λ ≤ λ n+1 and set

g(u) := n j=1
(λ -λ j )P j (u), where P j is the orthogonal projection from H to the eigenspace of A corresponding to the eigenvalue λ j . Now the equation Au + g(u) = λu reduces to

Au = n j=1 λ j P j (u) + λ(u - n j=1 P j (u)),
so that

X n = n j=1 P j (H) = n j=1 ker(A -λ j I) ⊂ C.
Consequently, in this case C contains a vector space of dimension

d = card {j ∈ N * , λ > λ j } = N (λ).
In particular for the unit ball K of this finite dimensional space, which is a compact subset of C we find

dim F K ≥ N (λ). When λ k ≤ Ck α , then N (λ) ≥ λ C 1/α -1.
This confirms the optimality of the upper estimate up to a constant for λ large.

3 Application to some elliptic equations (Ω) we denote the norm in H 1 0 (Ω), which is associated to the inner product ((•, •)) := (∇•, ∇•) . We will denote by •, • the duality product between H -1 (Ω) and H 1 0 (Ω). By • L ∞ (Ω) we denote the norm in L ∞ (Ω).

Let Ω ⊂ R N , N ≥ 1,
Let A = -∆ D with ∆ D the Dirichlet Laplacian on Ω. We denote by λ 1 the first eigenvalue of the A. Let λ j denote the j th eigenvalue of Ω for the Dirichlet boundary problem. We use the estimate (see Li and Yau [START_REF] Li | On the Schrödinger equation and the eigenvalue problem[END_REF] for more details)

λ j ≥ N C N N + 2 j 2/N V -2/N , ( 9
)
where V is the volume of Ω, and

C N = (2π) 2 B -2/N N
, with B N = volume of the unit N -ball.

Taking into account (9), in particular, we have (H1) with α = 2 N and c = 4πN

N + 2 (Γ(1 + N 2 )) 2/N |Ω| -2/N ,
where |Ω| denotes the N -dimensional measure of Ω. As a consequence we find the following result.

Corollary 7 Let g be any nondecreasing continuous function of the real variable s with super-linear growth at infinity. Let C be the set of solutions of the equation

u ∈ H 1 0 (Ω) ∪ L ∞ (Ω); -∆u + g(u) = λu. Then C is compact with a finite fractal dimension such that dim F C < 32 N/2 log 3 + 2 N log 2 N + 2 4πN N/2 1 Γ (1 + N/2) |Ω| λ N/2 .
Proof. Compactness is an immediate consequence of super-linear growth at infinity. Then it is sufficient to apply Theorem 5 with K = C

Application to parabolic equations

Now, we consider the following problem

u t -∆u + g(u) = λu in Ω, (10)
with the zero Dirichlet boundary condition,

u = 0 on ∂Ω, (11)
and the initial condition

u(x, 0) = u 0 (x), for x ∈ Ω, ( 12
)
where λ is a positive constant and g ∈ C 1 (R) is a non-decreasing function. We assume that the non-linear term g satisfies a dissipativity assumption of the form

g(s)s ≥ β |s| p ∀s ∈ R, (13)
and the following growth restriction of the derivative

|g(s 1 ) -g(s 2 )| ≤ γ M p-2 |s 1 -s 2 | for |s 1 |, |s 2 | ≤ M, (14)
for some β > 0, γ > 0, M > 0 and p > 2. A typical example of a function satisfying the previous conditions is g(s) = β |s| p-2 s, with p > 2. In this case we may take γ = β(p -1).

We define a semigroup {S(t), t ≥ 0} in L 2 (Ω) by

S(t)u 0 = u(t; 0, u 0 ) ∀u 0 ∈ L 2 (Ω), ∀t ≥ 0, (15)
where u(t; 0, u 0 ) is the unique solution of (10)-( 12). We denote by A the global attractor associated with the semigroup S defined by (15).

Our aim is to estimate the fractal dimension of A. First, we need the following results.

Proposition 8 Assume (13). Then the attractor A associated with (10)-( 12) is bounded in L 2 (Ω). More concretely, there exists a positive constant C(p, β, λ, Ω) such that

|a| L 2 (Ω) ≤ C, ∀a ∈ A.
Proof. Multiplying (10) by u,

1 2 d dt |u| 2 L 2 (Ω) + |∇u| 2 L 2 (Ω) = (λu -g(u), u) .
Using (13) and Young's inequality applied with the conjugate exponents p 2 and p p-2 , we have

(λu -g(u), u) ≤ Ω λu 2 -β |u| p dx ≤ 2(p -2) p 2 1 β |Ω| λ p p-2 ,
then, using the Poincaré inequality, we obtain

d dt |u| 2 L 2 (Ω) + 2λ 1 |u| 2 L 2 (Ω) ≤ 4(p -2) p 2 1 β |Ω| λ p p-2 .
Multiplying by e 2λ1t and integrating between 0 and t, we obtain

|u(t)| 2 L 2 (Ω) ≤ e -2λ1t |u 0 | 2 L 2 (Ω) + 2(p -2) p 2 1 βλ 1 |Ω| λ p p-2 1 -e -2λ1t .
We observe that if u(t) ∈ A, then there exists u 0 ∈ A such that u(t) = S(t)u 0 . Then, we have

|S(t)u 0 | 2 L 2 (Ω) ≤ e -2λ1t |u 0 | 2 L 2 (Ω) + 2(p -2) p 2 1 βλ 1 |Ω| λ p p-2 1 -e -2λ1t .
Fix t > 0, and consider a ∈ A. Then, there exists u 0 ∈ A such that a = S(t)u 0 , and we have

|a| 2 L 2 (Ω) ≤ e -2λ1t |u 0 | 2 L 2 (Ω) + 2(p -2) p 2 1 βλ 1 |Ω| λ p p-2 1 -e -2λ1t .
If t tends to +∞, we obtain

|a| 2 L 2 (Ω) ≤ 2(p -2) p 2 1 βλ 1 |Ω| λ p p-2 ∀a ∈ A.
Taking into account Proposition 8, we prove the following result Proposition 9 Assume (13). Then the attractor A associated with (10)-( 12) is uniformly bounded in L ∞ (Ω). More precisely,

a L ∞ (Ω) ≤ λ β 1 p-2 , ∀a ∈ A. (16)
Proof. Using (13), we observe

g(s) -λs ≥ 0 if s ≥ M := λ β 1 p-2 . ( 17
)
Let u ∈ L 2 (Ω), we define u + (x) = u(x), if u(x) > 0, 0, in other case, and

u -(x) = u(x), if u(x) < 0, 0, in other case.
Multiplying (10) by (u(x) -M ) + , taking into account (17) and using the Poincaré inequality, we have

d dt (u -M ) + 2 L 2 (Ω) ≤ -2λ 1 (u -M ) + 2 L 2 (Ω) .
Multiplying by e 2λ1t and integrating between 0 and t, we obtain

(u(t) -M ) + 2 L 2 (Ω) ≤ e -2λ1t (u 0 -M ) + 2 L 2 (Ω)
. We observe that if u(t) ∈ A, then there exists u 0 ∈ A such that u(t) = S(t)u 0 . Then, we have

Ω (S(t)u 0 -M ) 2 + dx ≤ e -2λ1t Ω (u 0 -M ) 2 + dx.
As A is bounded in L 2 (Ω), then we can deduce that there exists a positive constant C(p, β, λ, Ω), which is independent of u 0 , such that

Ω (u 0 -M ) 2 + dx ≤ Ω (u 0 -M ) 2 dx ≤ 2 |u 0 | 2 L 2 (Ω) + M 2 |Ω| ≤ C,
and, we have

Ω (S(t)u 0 -M ) 2 + dx ≤ Ce -2λ1t .
Fix t > 0, and consider a ∈ A. Then, there exists u 0 ∈ A such that a = S(t)u 0 , and we have

0 ≤ Ω (a -M ) 2 + dx ≤ Ce -2λ1t .
If t tends to +∞, we obtain

Ω (a -M ) 2 + dx = 0 ∀a ∈ A, then (a -M ) + = 0 =⇒ a ≤ M ∀a ∈ A.
We use a similar reasoning for (u + M ) -, and then we have

a L ∞ (Ω) ≤ M ∀a ∈ A. Now, we define C γ,β := 1 2 + γ β , (18)
and we consider the map S C γ,β λ : A → A. Taking into account Proposition 9, we prove the following result.

Proposition 10 Assume (13) and ( 14). Then, for all u 0 , v 0 ∈ A,

S C γ,β λ u 0 -S C γ,β λ v 0 H 1 0 (Ω) ≤ 2λ 1 + γ β |u 0 -v 0 | L 2 (Ω) ,
where C γ,β is given by (18).

Proof. Let u and v belong to A and set w = u -v and w 0 = u 0 -v 0 , where u and v are solutions to (10)-( 11) with initial data u 0 and v 0 , respectively. Then, we obtain

w t -∆w + g(u) -g(v) = λw in Ω, (19)
w = 0 on ∂Ω, w(x, 0) = u 0 (x) -v 0 (x), for x ∈ Ω.
We

(g(u) -g(v))wdx ≥ 0, we obtain 1 2 W ′ (t) + V (t) ≤ λW (t).
Multiplying by 2e -2λt , we obtain e -2λt W (t)

′ + 2e -2λt V (t) ≤ 0.

Integrating between 0 and t, we obtain

e -2λt W (t) + 2 t 0 e -2λs V (s)ds ≤ W (0), yielding W (t) + 2 t 0 V (s)ds ≤ e 2λt W (0). (20)
Now, multiplying (formally) (19) by ∂w ∂t , we obtain

∂w ∂t 2 L 2 (Ω) + 1 2 V ′ (t) + Ω (g(u) -g(v)) ∂w ∂t dx = λ(w, ∂w ∂t ).
We note that, owing to (14), ( 16) and Hölder's inequality,

Ω (g(u) -g(v)) ∂w ∂t dx ≤ γ β λ Ω |w| ∂w ∂t dx ≤ γ β λW (t) 1/2 ∂w ∂t L 2 (Ω) ,
and by Hölder's inequality

λ(w, ∂w ∂t ) ≤ λW (t) 1/2 ∂w ∂t L 2 (Ω) .
Then, by Young's inequality we have

1 2 V ′ (t) ≤ 1 4 λ 2 1 + γ β 2 W (t).
Then, using (20), we obtain

V ′ (t) ≤ e 2λt 1 2 λ 2 1 + γ β 2 W (0). (21)
Using the equality Finally, using Theorems 1 and 2 together with Proposition 10 and (9), we deduce the following result.

V (t) = 1 t t 0 V (
Proposition 11 Assume (13)-( 14). Then, the global attractor A associated with (10)-( 12) has finite fractal dimension in L 2 (Ω), and satisfies

dim F A < 8 N log 3 + 2 N log 2 N + 2 4πN N/2 1 Γ (1 + N/2) |Ω| 1 + γ β N/2 λ N/2 .
Remark 12 This result is substantially weaker than the estimate obtained in Theorem 3.1. in [START_REF] Chepyzhov | A note on the fractal dimension of attractors of dissipative dynamical systems[END_REF], but to obtain it we do not need any regularity hypothesis on g stronger than C 1 .

Remark 13 We presently do not know if ( 14) is really needed for our method to be employed. In particular the factor 1 + γ β N/2

does not appear in the estimate of [START_REF] Chepyzhov | A note on the fractal dimension of attractors of dissipative dynamical systems[END_REF] and the result of Theorem 5 even suggests that local compactness of the attractor might be a sufficient condition for its fractal dimension to be finite. This aspect seems to have been overlooked systematically in the literature until now and might be an interesting track of research for the future.

  be a bounded open domain with sufficiently smooth boundary. We denote by (•, •) the inner product in L 2 (Ω), and by |•| L 2 (Ω) the associated norm. By ||•|| H 1 0

  , where C γ,β is given by (18), we finally deduce from the above inequality an inequality of the form

									s)ds +	1 t	0	t	s V ′ (s)ds,
	and from (20) and (21), we deduce
				V (t) ≤ λW (0)	e 2λt 2λt	+	1 4	1 +	γ β	2 2λt e 2λt -(e 2λt -1) 2λt	.
	Taking t =	C γ,β λ
					V	C γ,β λ	≤ C γ,β λW (0),
	with
				C γ,β =	e 2C γ,β 2C γ,β	+	C γ,β 2	1 +	γ β	2	,
	where we have used that	(2C γ,β -1)e 2C γ,β + 1 2C γ,β	≤ 2C γ,β for C γ,β ≤ 1 2 .
	Finally, we estimate C γ,β . Taking into account that	2 3	1 +	γ β	-1	≤ C γ,β ≤	1 3	for the first term and using
	that C γ,β ≤ 1 1+ γ β	for the second one, we can deduce
					C γ,β ≤		3 4	e	2 3 +	1 2	1 +	γ β	.

Acknowledgments

María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466, and in part by European Commission, Excellent Science-European Research Council (ERC) H2020-EU.1.1.-639227.


Rodrigo De A Maués
email: rmaues@inf.puc-rio.br

Simone Junqueira Barbosa
email: simone@inf.puc-rio.br

Simone Diniz

Junqueira Barbosa

Cross-Communicability: Evaluating the Meta-communication of Cross-Platform Applications

Keywords: Cross-platform, user interface design, communicability, semiotic inspection method, semiotic engineering

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

People are increasingly using applications and services that span a wide variety of computing devices, such as PCs, smartphones, tablets and digital TVs [START_REF] Antila | Challenges in designing inter-usable systems[END_REF][START_REF] Denis | Inter-usability of multi-device systems -a conceptual framework[END_REF][START_REF] Paternò | Designing multi-device user interfaces: how to adapt to the changing device[END_REF][START_REF] Seffah | Multi-devices "Multiple" user interfaces: development models and research opportunities[END_REF][START_REF] Wäljas | Crossplatform service user experience: a field study and an initial framework[END_REF].

However, developing such systems can be a challenge because it is necessary to know not only how but also when to incorporate features available in one platform into the others, since each platform may differ in its capabilities and constraints (e.g., screen size and resolution, input mechanisms) [START_REF] Seffah | Multi-devices "Multiple" user interfaces: development models and research opportunities[END_REF][START_REF] Wäljas | Crossplatform service user experience: a field study and an initial framework[END_REF][START_REF] Florins | Graceful degradation of user interfaces as a design method for multiplatform systems[END_REF][START_REF] Lin | Employing patterns and layers for early-stage design and prototyping of cross-device user interfaces[END_REF]. Hence, it can be difficult and costly to maintain the quality of the system across each platform. Evaluating cross-platform systems is as challenging as designing them [START_REF] Antila | Challenges in designing inter-usable systems[END_REF][START_REF] Seffah | Multi-devices "Multiple" user interfaces: development models and research opportunities[END_REF]. Traditional methods for assessing quality attributes of systems need to be adapted for the cross-platform context. The evaluated quality of separate parts (i.e., system versions on each platform) does not necessarily correspond to the quality of the whole crossplatform system [START_REF] Denis | Inter-usability of multi-device systems -a conceptual framework[END_REF][START_REF] Seffah | Multi-devices "Multiple" user interfaces: development models and research opportunities[END_REF]. People alternately use the "same" application in different plat-2 Background

Semiotic Engineering Basics

Semiotic Engineering [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF] is a reflective and explanatory (as opposed to predictive) theory of human-computer interaction (HCI) that focuses on communicative rather than cognitive aspects of HCI analysis and design. It views the user interfaces of interactive systems as meta-communication artifacts, i.e., through the user interface, designers 1 convey to the users their understanding of who the users are, what they know the users want or need to do, in which preferred ways, and why. Users then unfold and interpret this message as they interact with the system. This metacommunication message can be paraphrased as [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF]: "Here is my understanding of who you are, what I have learned you want or need to do, in which preferred ways, and why. This is the system I have therefore designed for you, and this is the way you can or should use it in order to fulfill a range of purposes that fall within this vision. " (p.25) The designer-to-user message is comprised of signs [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | Semiotic Engineering Methods for Scientific Research in HCI[END_REF]. A sign is anything that stands for something else, to somebody in some respect or capacity [START_REF] Peirce | The essential Peirce (Vols. I and II)[END_REF]. Signs compose one or more signification systems that arise from culturally (and, in the case of HCI, also artificially) encoded associations between content and expressions [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | A semiotic framing for end-user development[END_REF]. For example, words and images typically come from signification systems that exist in a culture outside the specific context of HCI, whereas mouse pointers belong to signification systems that are native to computer applications.

Semiotic Engineering classifies the signs in three different types [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | Semiotic Engineering Methods for Scientific Research in HCI[END_REF][START_REF] De Souza | Can inspection methods generate valid new knowledge in HCI? The case of semiotic inspection[END_REF]: static, dynamic and meta-linguistic. Static signs express and mean the system's state. They are motionless and persistent when no interaction is taking place (e.g., icons, text areas, buttons at a given moment, menus). Dynamic signs express and mean the system behavior and emerge during interaction. They represent the transitions between the system states (e.g., a save button becomes enabled after entering the name of a client in a registration form). Meta-linguistic signs refer to other interface signs and are used by the designer to explicitly communicate to users the meanings encoded in the user interface and how they can be used (e.g., instructions and explanations, error and warning messages, hints). They are most commonly present in help systems, online/offline documentation, user manuals and promotional materials.

Based on this theoretical framework, the quality of a user interface is given by its communicability, which is "the system's property to convey effectively and efficiently to users its underlying design and interactive principles" [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] Prates | Methods and tools: a method for evaluating the communicability of user interfaces[END_REF]. On the one hand, when a user can comprehend how the system works because the designer expressed himself properly through the user interface (communicability), it becomes easier to learn how to use it (usability) [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF]. On the other hand, when the user fails to understand the communication intended by the designer, a communication breakdown takes place that may hinder or even preclude the use of the system [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF]. Semiotic Engineering has two qualitative and interpretive methods to assess the communicability of a user interface [START_REF] Barbosa | Interação Humano-Computador[END_REF][START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | Semiotic Engineering Methods for Scientific Research in HCI[END_REF]: the Communicability Evaluation Method (CEM) [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] Prates | Methods and tools: a method for evaluating the communicability of user interfaces[END_REF], which involves user observation, and the Semiotic Inspection Method (SIM) [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | The Semiotic Inspection Method[END_REF][START_REF] De Souza | Semiotic Engineering Methods for Scientific Research in HCI[END_REF][START_REF] De Souza | Can inspection methods generate valid new knowledge in HCI? The case of semiotic inspection[END_REF], which involves user interface inspection. While CEM focuses on the reception of the meta-communication message by the users, SIM focuses on its emission by the designer. Our paper extends SIM to cover the first part of the communication processthe emission of the messagein cross-platform systems.

The Semiotic Inspection Method

The Semiotic Inspection Method (SIM) [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF][START_REF] De Souza | The Semiotic Inspection Method[END_REF][START_REF] De Souza | Semiotic Engineering Methods for Scientific Research in HCI[END_REF][START_REF] De Souza | Can inspection methods generate valid new knowledge in HCI? The case of semiotic inspection[END_REF] is a qualitative inspection method grounded in Semiotic Engineering that allows the evaluation of the communicability of a computer system through the analysis of its static, dynamic and metalinguistic signs. The goal is to identify communication breakdowns that may take place during the user-system interaction and to reconstruct the designers' metamessage conveyed by the user interface. Like other inspection methods (e.g. Heuristic Evaluation [START_REF] Nielsen | Usability Inspection Methods[END_REF] and Cognitive Walkthrough [START_REF] Wharton | The Cognitive Walkthrough Method: a Practitioner's Guide[END_REF]), SIM does not involve the participation of users [START_REF] Barbosa | Interação Humano-Computador[END_REF]; it is the evaluator who examines the interface in search of ambiguities and inconsistencies in the signs chosen by the designer. Thus, as in any theorybased method, the more knowledge the evaluator has, the better the evaluation results will be.

Heuristic evaluation and cognitive walkthroughs focus on identifying usability issues, mostly related to the ease of use and learning of the system operations, while the strategic aspects of the system (the logic underlying the design decisions) conveyed by the user interface are not covered. Instead of assuming users always adopt a planbased problem-solving method, semiotic engineering is more aligned with Suchman's work on situated action [START_REF] Suchman | Plans and situated actions: The Problem of Human-Machine Communication[END_REF], which suggests that users are constantly reevaluating their current circumstances and adapting their "plan sketches" via communicative actions performed at/through the user interface. By inspecting the user interface language, we are able to predict some of these breakdowns, which may lead to user interaction problems. Rather than relying on a cognitive plan-based analysis, our inspection draws on Linguistics and Semiotics, which provide us with clues on how people make sense of and negotiate meaning.

SIM requires a preparation phase, in which the evaluator defines the purpose of the inspection, performs an informal inspection by navigating through the system to define the focus of the evaluation, and finally elaborates the inspection scenarios. Next, the evaluator proceeds to the execution of the inspection. To properly execute the method, the evaluator must assume a "user advocate" position. The execution of the method is carried out in five distinct steps: (1) a meta-linguistic signs inspection; (2) a static signs inspection; (3) a dynamic signs inspection; (4) a contrastive comparison of designer-to-user meta-communications identified in steps (1), ( 2), (3); and, finally, (5) a conclusive appreciation of the overall system communicability. In the first three steps, the evaluator inspects the signs within the scope of the evaluation and reconstructs the meta-messages being conveyed by them at each level, filling out the template of the designer-to-user meta-communication and identifying potential communicability problems at each level. Because each sign level has distinct expressive possibilities, the generated messages may not be identical. Thus, in step (4) the evaluator contrasts the meta-messages generated in the previous steps and checks for real or potential inconsistencies or ambiguities among them. Finally, in step [START_REF] Wäljas | Crossplatform service user experience: a field study and an initial framework[END_REF], the evaluator qualitatively assesses the system communicability by unifying the metacommunication messages and then generates an evaluation report.

Impermeability and Computer Manipulations of Signs

Before we proceed with the extended application of SIM to cross-platform systems, it is important to briefly present two semiotic concepts that will be used in our analysis: impermeability and computer manipulations of signs.

In [START_REF] De Souza | A semiotic framing for end-user development[END_REF] those concepts were used to provide a semiotic framing for End-User Development (EUD). Both are related to the identity of an application, which is an important matter for both EUD and cross-platform development. While for EUD the main concern is how to empower users to customize or extend a system without losing the designed system identity, the main concern of the designer of a cross-platform system is how to design a different version for each platform while preserving the system identity across them, so as not to hinder the alternate use of the various versions. By assuming that traversing between platforms bears resemblance to traversing between customized versions of a system, we propose to apply these concepts beyond the scope of EUD for the first time, to help in the semiotic inspection of crossplatform systems. It should be made clear that the designer did not necessarily make these modifications, but based on the differences between supposedly equivalent signs we can assume which manipulations are capable of transforming a system version into another. From the nature of the manipulation, we can identify potential crosscommunication breakdowns.

The concept of impermeability [START_REF] De Souza | A semiotic framing for end-user development[END_REF] is related to the encapsulation of signs, i.e., the sign meaning is in an atomic capsule and therefore it cannot be altered. Thus, the originally encoded meaning of impermeable signs is always preserved. Impermeable signs can be essential or accidental. Essential impermeable signs can only be used in monotonic manipulations, i.e., those that do not destroy the basic meanings (identity) of the application. Accidental impermeable signs may not be changed either, but they may be subtracted from the application by means of an operation we call pruning. For instance, let us consider a "skip to next track" button. It is an impermeable sign and thus its meaning of skipping to the next track on a playlist should not be changed, otherwise it may confuse the user (e.g., if the next track button is used to skip to the next playlist instead). If this sign is essential (i.e., part of the system identity), it should also be present in any other platform. However, if it is not part of the system identity (i.e., it is an accidental sign), it may be pruned in any platform.

From a computer symbol-processing point of view, two different kinds of modifications can be made to signs [START_REF] De Souza | A semiotic framing for end-user development[END_REF]: meaning-preserving modifications (types I, II, and III) and meaning-changing ones (types IV through VII).

Meaning-preserving manipulations affect only impermeable signs and preserve the application identity. Type I changes correspond to renaming and aliasing operations that affect only lexical components. Changing the label of a button and changing an icon appearance are examples of renaming. Renaming should be limited to lexical items that are not part of the application's identity to avoid confusing the user about its meaning. Type II manipulations involve making changes only to syntactic components, such as reordering: changing the layout sequence of user interface elements or the order in which operations are performed. However, this reordering cannot change the meaning expressed by the individual reordered components. Finally, a type III manipulation (also known as pruning) corresponds to the possibility of selecting nonessential (accidental) components to be included in or excluded from the application. Pruning must preserve not only the identity of the application, but also the impermeability of the pruned components. Thus, an important design decision is to select which of the impermeable signs can be pruned if the application identity is to be preserved. When porting the system to a more constrained platform, the designer of crossplatform systems usually has to leave features out (type IIIpruning) or adjust them according to the space available in the screen (type IIreordering). Also due to space constraints or specific platform patterns, the designer may have to change the appearance of some components (type Irenaming and aliasing).

Every application requires that users learn a new and unique signification system that is used in HCI [START_REF] De Souza | A semiotic framing for end-user development[END_REF]. Therefore, the application's identity and the notion of impermeability are important to keep the user's semiosis sufficiently tied to the designer's vision, so that productive interpretations can be motivated, and unproductive ones discouraged [START_REF] De Souza | A semiotic framing for end-user development[END_REF]. This is especially important when it comes to cross-platform systems, since you may have many different versions of the same system.

Meaning-changing manipulations, however, can threaten the application identity, and therefore should be avoided whenever possible in order to minimize conflicts between system versions. A type IV manipulation, for example, involves using existing signs to mean something different from what they were designed to mean (e.g., the same button triggers different actions on each platform). Hence, when transitioning from one version to another, users may feel frustrated because of this conflict of meanings. A type V manipulation makes changes to meanings and lexical items but not to grammatical structures. Type VI may involve reordering components or even eliminating components that change the conveyed meaning. And finally, type VII makes changes to meanings, grammatical structures and lexical items, consisting of the biggest threat to the application identity.

In our analytical study, we give examples of some of the manipulations we found and discuss about their potential consequences concerning cross-platform systems.

Using SIM to Evaluate Cross-Communicability

When evaluating each version of a cross-platform system separately, each one may have high communicability, i.e., it may efficiently and effectively convey to the users its underlying design intent and interactive principles. However, when traversing between platforms, the user carry understandings from one version that may not be applicable to another, creating conflicts and communication breakdowns. Besides, the meta-communication message of each version should also convey the composition of the cross-platform system, i.e., how functionalities are organized in and across platforms [START_REF] Wäljas | Crossplatform service user experience: a field study and an initial framework[END_REF]. The quality of the meta-communication conveyed by the cross-platform system as a whole, including all versions, is what we call cross-communicability.

The traditional SIM alone is not suitable for evaluating cross-communicability, since we must also systematically contrast the meta-communication messages across system versions. Hence, after the preparation phase required to the traditional SIM described previously, we divide the execution of the method applied in two phases: vertical (within-platform) analysis and horizontal (between-platform) analysis.

Vertical Analysis

The first phase, called vertical analysis, is designed to evaluate the communicability of the system in each platform individually. Hence, it consists of conducting the same five steps of the traditional SIM, with one difference: the evaluator should also highlight the signs that denote any compositional aspects of the cross-platform system (e.g., a sign that acknowledges the existence of another system version).

Because the inspections must be conducted individually, different evaluators may conduct the inspection of each version. There are three main benefits of this approach: it distributes and reduces the cost of the inspection; it avoids contaminating the inspection of one version with another; and it allows designating each system version to the most appropriate evaluator, a professional with solid technical HCI knowledge in that platform.

Horizontal Analysis

As stated before, when evaluating a cross-platform system it is important to contrast the meta-communication messages of each system version. Thus, after completing the communicability assessment of each version, the next phase consists of performing a horizontal analysis, which is divided in three steps: (1) identification and collation of sign manipulations between the evaluated platforms; (2) an analysis of the impact of each manipulation on the meta-communication of each version; and finally (3) a conclusive appreciation of the quality of the overall system cross-communicability. The signs collated in the vertical analysis may undergo modifications across the evaluated system versions due to the constraints, advantages or patterns of each platform. Thus, the evaluator should (1) identify the meaning-preserving and meaningchanging manipulations in each pairwise combination of the evaluated platforms;

(2) examine the manipulations collated and categorized in the previous step to assess how they could negatively affect the horizontal meta-communication messages by intensifying already identified vertical communication breakdowns (e.g., causing ambiguities) or even by creating new ones; and (3) qualitatively assess the system cross-communicability by unifying the meta-communication message obtained in each previous step, judging the costs and benefits of the identified manipulations made between platforms, and then he generates an evaluation report.

Analytical Study

For the application of SIM on a cross-platform system, we chose an Internet radio and social networking website named 8tracks, 2 which revolves around the concept of streaming user-curated playlists composed of at least 8 tracks. Besides being a popular cross-platform system, 8tracks was chosen because it has different versions available on different devices (desktop and mobile) and also has different versions available on the same device (mobile native apps 3 and mobile web app), therefore covering a broad range of combinations to analyze in our study.

Preparation Phase

The goal of this analysis is to identify different communication breakdowns that may occur individually on each platform version of a cross-platform system and their relation to other breakdowns that arise when these versions are put together. We defined the context of use for the evaluation through the following scenario:

2 http://8tracks.com/ 3 8tracks has two different mobile native apps available, an iOS and an Android version, but only the Android version was evaluated in our study.

"Rodrigo will throw a party and needs to set a good playlist, but has no idea how to start. Gabriel, Rodrigo's friend, suggested he use 8tracks system, where he can discover and listen to several different mixes. Rodrigo had already created an account in the past, but he has never really used the system. He thinks this might be the perfect opportunity to give it a try. He then searches 8tracks for mixes with the latest hits. He saves his favorite mixes, to be able to play them later at the party, if he so wants. He also creates his own mix only with the favorite songs from the mixes he has played. So as not to forget which ones they are, he will use another feature that allows saving only individual favorite tracks to be bought later, instead of saving full mixes." Although it was observed by a previous informal inspection that not all the functionalities in the scenario would be available in all the evaluated system versions, we decided to let it be this way to evaluate how each version conveys information about the system composition as well. Next, we briefly present the main results from the vertical and horizontal analyses.

Vertical Analysis

A single evaluator conducted the study. Due to space constraints, in this paper, we do not describe the whole vertical analysis of the 8tracks versions, which consists of a typical application of SIM. Instead we present synthesized versions of the metamessage, focusing on some communication issues found when analyzing the system communicability on each platform. Although the vertical analysis was conducted separately for each platform, some references will be made to similar characteristics across the versions to avoid repetition and also due to space constraints. Desktop Web App. Besides the hints and tips throughout the system, there are many internal pages (e.g., about, licensing, FAQ) and a blog available where some meta-linguistic signs can be found. This documentation mainly explains the system purpose and its underlying concepts, instead of how to use it. Therefore, it is implied that the system is in general thought to be easy to use and to learn. The available explanations describe many aspects of the legal rules that the service must follow to allow the free streaming of music. These rules limit some functionalities of the system, but expecting that users will probably not want to read the documentation before using the system, the main rules are only briefly explained through dynamic messages to let users become aware of them by trial and error when interacting with the system. Some aspects of trial and error, however, can be frustrating due to the lack of appropriate communication. When attempting to skip to a track one too many times, for instance, the following message is shown: "Apologies for the inconvenience, but our music license requires us to limit the number of tracks you may skip each hour." Since there is no static or dynamic sign indicating how many tracks you have already skipped or may still skip, or how long it has been since you exceeded the number of "skips," and also because the skip button is always enabled, the user may often try to skip a track, fail, and then receive the infamous message over and over again. The meta-communication message derived from the documentation and help content tells us that the service is designed for music-lovers that want to create, share and discover new mixes in a radio-style way, i.e., like a non-interactive music player. Static signs on the mix page (Figure 1) appropriately convey this radio-style music discovery experience depicted in the meta-linguistic signs: the music progress is represented by a very thin and discreet line, indicating that it is not so important and that users should not worry about it, just like on the radio; there is no bar over this thin line to control the music progress; and the only control buttons available are a button to play/pause the music, a button to skip to the next track or to the next mix.

Regarding the search for mixes, a communication breakdown was found. Once a search has been done, the user may sort the results by hot, new or popular. Although it is easy to infer the outcome of sorting by newest mix, the same cannot be said about the other two criteria, and there is no explanation on the system about the difference between popular and hot mixes.

Meta-linguistic signs on some of the internal web pages of the system and on the 8tracks blog almost exclusively express the information regarding the system composition. References are made to both the mobile web and the mobile native versions. There is also a static sign at the bottom of each system page, a button, referring to the mobile native version.

Mobile Web App. Besides the fact that the meta-linguistic signs available on documentation of the desktop web version discussed previously are also available to any other system version, in the mobile web app there is a direct link to the desktop web app's "FAQ" and "About" pages, as if they were its own. Because of this and the fact that very little information about the mobile web version can be found in the system documentation, it can be assumed that this version has the same features of the desktop version. However, during the analysis of the static signs it became clear that not all the functions depicted in the meta-linguistic signs are available in this version. Hence, if a user consults the documentation he may be misled and frustrated after searching the whole app for a function that does not even exist in it.

Although it is harder to present hints on a mobile web system because it is not possible to hover over user interface components, in general there are really few messages and instructions in this platform, possibly meaning that the designer believes that this interface is also fairly easy to use and no further explanation is necessary.

In addition to the same meta-linguistic signs available at the desktop version, in this version two static signs can be found on the bottom of the dashboard page referring to both the mobile native and the desktop web versions.

Another communication breakdown was found on the system search mechanism. Although there is a tip on the kind of text the user should type to perform a search, in the results page there is no indication of which criteria were used to search the mixes. Hence, the user does not know whether the text he typed was matched with the mixes tags, with the mixes names, with another criterion or with all of them. Another potential breakdown found on the results page is the lack of static or meta-linguistic signs evidencing what happens when the user selects one out of the three options (most recent, hot this week or popular this month) on the search results page. Even with the aid of the dynamic signs, there is no clear evidence as to whether you are sorting or filtering the search results by one of these criteria.

Users can also directly search mixes by selecting tags in the dashboard. Although the static signs indicate that you can search only by one tag at a time, it is actually also possible to search by a combination of tags if you perform a long press on each desired tag at a time, and then you press a go button that will appear next to the tags (Figure 2). However, since this is not explained to users, it is a communicability breakdown, and users may never perceive this search possibility even if they intensely interact with the system through trial and error. Mobile Native App. Although there is no static sign referring to the other versions, this version contains an interesting new meta-linguistic sign regarding the system composition on its own internal FAQ: it clearly states that, if you want to create a mix or preview the songs you have added to your favorites, you need to use the desktop version. Hence, this sign not only recognizes the existence of another version but it also indicates the system distribution of functionalities, revealing more clearly the redundancy across these system versions. Nevertheless, there is no reference about the mobile web version on this platform.

There are additional meta-linguistic signs available on the app's Google Play page and also on an Update Log page internal to the app, clearly explaining the purpose, functionalities and limitations of this version. Both pages also display the changes made to each release of the app, which is important because it helps the user to interpret new features or modifications made to previously known signs.

The static signs indicate that the user can filter the mix search results by tags, but since this label is also next to the hot, new and popular buttons, the user may also wrongly infer that these buttons can filter the results (Figure 3). However, when inspecting the dynamic signs we notice that the result count is not affected; therefore these buttons only sort the results. Still, a distracted user may never notice this dynamic sign, since it is quite subtle.

This version also introduces a new sign: a hidden left-hand menu, used to hide navigation options until someone taps a button to expose them (Figure 3). This off-canvas pattern is very common and useful in mobile applications, since it avoids occupying the small screen of the device all the time. However, the way this menu is accessed in this platform introduces a temporary failure [START_REF] De Souza | The Semiotic Engineering of Human-Computer Interaction[END_REF]: it uses the sign of a home button, which naturally would make the user infer that he is going to a home or dashboard page of the system, which does not exist in this version. Some other problems previously discussed on the inspection of the desktop version were also found on this version, such as the hot/popular ambiguity. Fig. 3. Search results filtering page before (on the left) and after (on the right) pressing the home button, which actually triggers the hidden left-hand menu to appear on the screen. The highlighted signs are responsible for different vertical and horizontal breakdowns.

Horizontal Analysis

Having concluded the vertical analysis, we conducted the horizontal analysis to assess 8tracks's cross-communicability. First, we contrasted each evaluated version with the two others, resulting in six tables containing the modifications that could have been made at the user interface signs in going from one version to the other, classified by the nature of the manipulation. Manipulations from type I to V were identified, but most of them were meaning-preserving ones. Due to space constraints, only a sample of them is presented, classified by type to discuss the main communication breakdowns found across versions and to illustrate the benefits of this analysis.

It is worth mentioning that, in a cross-platform system such as 8tracks, i.e., a system whose functionality and content is available in multiple devices, many of the unchanged characteristics found across the platforms may indicate the identity of the system and some of its impermeable signs. For instance, the control buttons on the mix page, the buttons for liking the mix and for sharing the mix (Figure 1) are present with their meanings unchanged in all the platforms. This indicates that they probably are some of the essential impermeable signs and part of the system's identity. However, these relations are still not so clear.

Type I Manipulations. Only a few type I (renaming) manipulations were found: the label "Created" on the mobile versions is written "Published" on the desktop web version (Figure 4); there is a slight difference between the desktop and the mobile versions of the mix feed icons (Figure 5); and the label "recent" in the mobile web version is written "new" in the other versions (Figure 6). Although they may not lead the user to a failure, those unnecessary inconsistencies should be avoided. Type II Manipulations. Type II manipulations (reordering) are most common and inevitable when it comes to cross-platform systems working in different devices. This happens because the display size and resolution of each device force designers to reorganize the layout to fit all the interface elements, or even to separate elements into two or more screens to avoid pruning some essential impermeable signs. Hence, it is inevitable to break the interface layout consistency to fit everything.

However, it is possible to find on 8tracks some type II manipulations that resulted in unnecessary inconsistencies. For instance, in the mix description (Figure 4), the labels for the mix like and play counts are displayed in one order in the desktop version and in the reverse order in the mobile versions, even though there are no space constraints. Although the latter example may not constitute a serious communication breakdown, it may negatively affect some cognitive aspects. For instance, a person who is already used to the order of the labels for mix like and play counts in a plat-form will have to learn the order of the elements again in the other platforms. If instead of labels the interface elements were buttons, for example, the same reordering would represent a bigger issue, since a distracted person might press the wrong button because the order (that the person was already used to) has changed.

Another reordering problem found was that the same groups of mixes are arranged in the desktop web in a different order from the mobile web. Since the order of elements may also convey a relation of importance between them, there is an ambiguity on what the designer thought was more important to the user: the desktop web prioritizes the users' mixes (i.e., their own mixes, liked mixes and mix feeds from whom he follows) while the mobile web prioritizes general featured and hot mixes.

Type III Manipulations. After the vertical analysis, the evaluator already has a clear vision of which features and components are present or not in each version, which makes it easy to identify type III (pruning) manipulations. Based on the collated type III manipulations, it became evident that the versions are complementary, i.e., although they share a set of data and functions, some of them provide access to data or functions that are not available in the others. Some of the observed pruning operations from the desktop to the mobile versions may be justified by either reduced platform capabilities or little utility. For instance, the functionality of creating mixes is not available in the mobile versions, maybe because of either platform limitations (slower internet connection when using 3G to upload each track and limited processing power in comparison to a desktop); or little utility, since the user will be using the system mostly on the go and therefore will not be interested or even capable of spending too much time looking at the device screen. Thus, the 8track's designer may have intended that the users should only listen in their mobile devices to what had been produced on their desktop computers.

In the desktop version, since some mix arts may contain inappropriate content to be shown during the service use in a public or work environment, the user may flag some arts as Not Safe For Work (NSFW) and configure them not to be shown. This feature is not available on the mobile versions, probably because Smartphones are highly personal; therefore there is little utility in avoiding displaying any type of inappropriate content, since only the user would see it on his device.

Type III manipulations can indicate some interesting communication breakdowns across platforms, because the lack of some features between certain versions may represent an ambiguity for the user. In our study, one example is the feature of adding a track to the favorites. To do so, the user must press the track's star button placed either next to where the current track is being shown or on the tracklist, which shows only the mix tracks already played. The vertical analysis of the mobile web version revealed that, on the tracklist page (Figure 7), there is only a button to buy each track and you can neither see the tracks you have added to your favorites nor add them; therefore you can only add a track to your favorite (or know that it has already been added to your favorite tracks) while it is still playing. When contrasted to the desktop and the mobile native versions, this issue is amplified, because the user in the desktop version can buy the track or add it to the favorites using buttons on the tracklist, and in the mobile native the user can only add but not buy the track in the tracklist (Figure 7). Therefore a person used to these versions may get frustrated when trying to add a track to the favorites on the mobile web version after he has listened to the whole song, only to discover that he cannot do it. Since in 8tracks the user cannot go back to a previous track and the tracklist is shuffled each time the user listens again to a mix from the beginning, he would be forced to go to another version to see the full tracklist so far and then add the track to his favorites. Since pruning the button to buy a track from the tracklist does not represent a problem, but removing the star button (to add a track to the favorites) does, it is possible to infer that the tracklist and the star button together make up an impermeable sign, and therefore should not be separated. Still regarding pruning manipulations, removing the dashboard page on the mobile native version per se would not lead to a communication problem across platforms. However, confusion arises because there is a similar home button that shows the lefthand menu (Figure 3) instead of leading the user to a dashboard page (Type IV manipulationchanging the meaning of a sign), which happens in the other platforms.

Pruning manipulations can also be observed on the 8tracks meta-linguistic signs. Adding a new FAQ to the mobile native version results in interesting observations. First, its benefit to the mobile native version is clear, once it avoids the same communication breakdown as the mobile web version regarding the confusion about which functionalities are available or not. Second, the mobile native's FAQ contains a piece of information about the purpose of the favorite tracks feature (i.e., save tracks to buy them later), which is missing from the meta-linguistic signs in the desktop version. Hence, if a user learns about this feature from the native app, he may understand it better than another user who inferred its behavior by its name on the desktop version.

A similar situation can be observed regarding the feature of sorting search results by hot, new or popular. As mentioned before, a common breakdown in the desktop and mobile native versions is the lack of clarity on the difference between the hot and popular criteria. However, if a person has used this feature on the mobile web version, he can eliminate this doubt, since the labels there are more detailed: "hot this week" and "popular this month". In the mobile web version the breakdown regards not knowing whether a sorting or a filtering operation was applied to the results after selecting one of the criteria. This problem can be either avoided or transformed into another issue, depending on which version was used prior to the mobile web one. If the user has already used the desktop version, he will know the results are sorted by one of those criteria. If the user has had previous experience only with the mobile native, he may mistakenly assume a similar behavior, i.e., that the results were filtered (Figure 3). Finally, contrasting the desktop and mobile versions altogether may lead to another breakdown, since the user may no longer be sure whether each version is behaving differently or similarly, and in case they are indeed doing the same operation, they might not know whether it is sorting or filtering the results.

The latter breakdown is probably due to one of two reasons: an unfortunate type I manipulation, because as stated before you should not change a label if you are changing an impermeable sign, and therefore you mislead the user into thinking that something means something else (sort versus filter); or a type II manipulation that went wrong, i.e., the label in the mobile native app was referring only to the tags but was misplaced next to the sorting options as well.

Type IV Manipulations. Regarding meaning-changing manipulations, one type IV manipulation (only the meaning of a sign is changed) is responsible for a potential issue when the user selects a mix to go to its page: in the desktop and mobile web apps, when entering the mix page, the mix does not start playing right away (the user needs to press a play button), but the same is not true in the mobile native app. The fact that the mix will start playing once selected in the latter version may frustrate a person used to the other meaning of selecting a mix on the web versions of the system, and vice-versa.

Type V Manipulations. On the mobile web version, a button (Figure 6) on the header of the search results page (which triggers the interface to go to the home or dashboard page) can cause a breakdown because it does not resemble a home button and therefore its meaning cannot be easily interpreted or even learned by the user. When contrasting this version with the mobile native and desktop web versions, this button can cause additional breakdowns, because a very similar sign is used in those versions (Figure 6) with different meanings. Thus, we have the occurrence of a type V manipulation (meanings and lexical items are changed).

In the mobile native the meaning of this sign is to display the mix tracklist, while in the desktop web the meaning is to display the search results in a list (instead of in a grid). The same sign did not create a vertical communication problem in the desktop web and mobile native versions; it created a vertical communication problem in the mobile web version; and it also created a horizontal communication breakdown, since the user may get confused about which action is associated to the button on each platform, leading to temporary failures.

Another type V manipulation is responsible not for a new communication issue, but for amplifying a breakdown identified during the mobile version's vertical analysis: the tags on the mix description (Figure 4) resemble buttons, but they are not actually clickable. Since in the desktop web and mobile native versions the same tag buttons (although different in color) are clickable (triggering a search by a tag when clicked), this manipulation may lead the user to frustration, since he is likely try to click on the tags on the mobile web version if he is already used with the other system versions.

Conclusion and Future Work

This paper described a first effort to apply Semiotic Engineering and its methods to the evaluation of the quality of cross-platform systems, which we defined as crosscommunicability. We presented an analytical study using a new approach to SIM to assess how well the designer-to-user meta-communication message gets across to users through each system version separately and also through the cross-platform system as a whole, taking into account the possible ambiguities and communicative breakdowns when traversing between versions. The proposed evaluation of the cross-communicability of a system consists of two analyses: a vertical analysis, which is mostly an application of the SIM to each system version; and a horizontal analysis, in which the signs and meta-messages from the previous analyses are contrasted and the differences are classified according to the type of sign manipulations that would be necessary to transform one version into another. This classification is based on a semiotic framing originally proposed for EUD.

We could identify through our analytical study some interesting facts when contrasting the meta-messages of the different system versions: ambiguities and new communication breakdowns may arise in one or more system versions; individual breakdowns may be transmitted from one version to another; individual breakdowns in one version may be amplified by individual characteristics or issues of other versions; and also some individual breakdowns in one version may be overcome due to signs present on the other versions.

Thus, we believe that by looking at the results of a SIM evaluation informed by our extended approach, the designer of cross-platform systems may become more aware of several communicative breakdowns regarding composition, continuity (i.e., how tasks state and actions are migrated across platforms) and consistency aspects, which can help them to better design or redesign each system version. The proposed sign organization by manipulations may also help to improve existing cross-platform development processes, as in the graceful degradation [START_REF] Florins | Graceful degradation of user interfaces as a design method for multiplatform systems[END_REF] method, for instance. Results from a horizontal analysis could serve as a basis to refine the set of transformational rules applied during the graceful degradation approach, as well as to validate the priority ordering between the rules in the method.

As future work, we plan to triangulate the results obtained with endogenous and exogenous sources. We also plan to conduct a study comparing the extended SIM with more traditional inspection approaches, such as heuristic evaluation and cognitive walkthrough. Also we intend to investigate more closely the relation between the system's identity and the quality of a cross-platform system, as well as how we can properly define which signs are impermeable and essential to one or more platforms, and to the system as a whole. We also plan to formulate a set of heuristics as a result of consistently applying SIM to multiple cross-platform applications. And finally, we plan to extend CEM to cross-platform systems as well.

Fig. 1 .

 1 Fig. 1. Part of the desktop web version's mix page.

Fig. 2 .

 2 Fig. 2. Example of the mobile web version of a search where the user combined multiple tags as the search criteria.

Fig. 4 .

 4 Fig. 4. From top to bottom, the mix description appearance on the desktop web, mobile web and mobile native versions, respectively. The highlighted tags are not clickable only in the mobile web version.

Fig. 5 .

 5 Fig. 5. Example of type I manipulation and unnecessary inconsistency: difference in the appearance of the mix feed icon used in the mobile versions (on the left) and in the desktop web version (on the right).

Fig. 6 .

 6 Fig. 6. From top to bottom, part of the search results page on the desktop and mobile web versions and part of a mix page on the mobile native version, respectively. The highlighted elements reveal type I (rectangle) and type V manipulations (circle).

Fig. 7 .

 7 Fig. 7. From top to bottom, a part of the tracklist page on the desktop web, mobile web and mobile native versions, respectively. The highlighted signs are responsible for a horizontal communication issue.

In this paper designers should be interpreted as whoever speaks for the design team of a given application.