Dimension spectrum for sofic systems.

1. Introduction

Let ([[SIGMA].sub.A], T) be a subshift of finite type (SFT) with A being the incidence matrix and T being its shift map. Motivated by the study of the iterated function systems (IFS) and generalized Sierpinski carpets (GSC, cf. [1-5]), one considers a special type of potential functions M: [[SIGMA].sub.A] [right arrow] L([R.sup.d], [R.sup.d]) which take values on the set of d x d matrices. For q [member of] R, define the topological pressure as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

whenever the limit exists and [[SIGMA].sub.A,n] denotes the collection of n-cylinders in [[SIGMA].sub.A]. (Here [parallel]A[parallel] is the matrix norm; that is, [parallel]A[parallel] = [1.sub.1xd]A[1.sub.dx1], where [1.sub.dx1] is the d x 1 column vector with entries being 1's). In [4], if M is positive, that is, [M.sub.i,j] > 0, and Holder potential with [[SIGMA].sub.A] is topologically mixing, the authors prove that the Gibbs measure for M exists uniquely and the system admits the multifractal analysis. More precisely, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

be the level set for the upper Lyapunov exponent. Then the Hausdorff dimension of the level set is obtained as follows.

Theorem 1 (see [4, Theorem 1.3]). Let ([[SIGMA].sub.A], T) be a SFT and let M: [[SIGMA].sub.A] [right arrow] L([R.sup.d], [R.sup.d]) be Holder continuous. Then [P.sub.M](q) is differentiable and for any [alpha] = [P'.sub.M](q), q [not equal to] 0,

[dim.sub.H][E.sub.M]([alpha]) = [1/log m]{-[alpha]d + [P.sub.M](q)}, (3)

where [dim.sub.H] denotes the Hausdorff dimension.

The study of the thermodynamic properties with these potentials relates deeply to the fractal properties of the given IFS or GSC. We emphasize that the formula (3) set up the fine structure in the Hausdorff dimension point of view for ([[SIGMA].sub.A], T). The authors extend this result to the case that M is nonnegative with some additional irreducible conditions' the reader may refer to [3] for the detail. When the underlying space S is a sofic shift and d = 1, that is, the potential function is finitary real valued, there raises a natural equilibrium measure called semigroup measure proposed by Kitchens and Tuncel [6]. When d [greater than or equal to] 2, the thermodynamic properties relate to fractal dynamics of given sofic affine-invariant sets (cf. [7]).

Theorem 1 investigates the dimension spectrum of SFTs; it is natural to ask whether the formula is preserved by passing to their factors (sofic shifts). To be advanced, does the formula hold for those shifts beyond sofic shifts such as the case for the specification property? Recent research revealed that some properties of SFTs are preserved for the cases beyond the specification (cf. [8-10]). This study intends to show that the formula of dimension spectrum of SFTs is passing to their factors, that is, sofic shifts, and their Hausdorff dimension can be expressed explicitly for some class of sofic shifts.

Let S be a sofic shift which is a subshift of the shift space and let S be the shift map on it. The well-known Curtis-Lyndon-Hedlund Theorem ([11, Theorem 6.2.9]) demonstrates that if [phi]: [[SIGMA].sub.A] [right arrow] S is a function, then [phi] is a homomorphism if and only if [phi] is a sliding block code. The sliding block code is induced from block map, that is, a map [[PHI].sup.(L)]: [[SIGMA].sub.A,L] [right arrow] A(S) for some L [greater than or equal to] 1. For all k [greater than or equal to] 1 and if I [member of] [[SIGMA].sub.A,k+L-1], define [[PHI].sub.k]: [[SIGMA].sub.A,k+L-1] [right arrow] [S.sub.k] by [[PHI].sub.k]([I]) := [[PHI].sup.(L)]([[pi].sup.(L)]([T.sup.k-1]([I]))), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the restriction map of a cylinder to [Z.sup.L] lattice. Then [phi]: [[SIGMA].sub.A] [right arrow] S is thus defined as the limit of [[PHI].sub.k]; that is, [phi] = [lim.sub.k[right arrow][infinity]][[PHI].sub.k] and S := {[phi](x): x [member of] [[SIGMA].sub.A]}. (We refer reader to [11,12] for more detail). We call [phi] right resolving if for all [I.sub.1] = [([i.sub.1;l]).sup.L-2.sub.l=0] and any [I.sub.2] = [([i.sub.2;l]).sup.L- 2.sub.l=0] and [I.sub.3] = [([i.sub.3;l]).sup.L-2.sub.l=0] [member of] [[SIGMA].sub.A,L-1] such that [I.sub.1] [direct sum] [I.sub.2] and [I.sub.1] [direct sum] [I.sub.3] [member of] [[SIGMA].sub.A,L] we have

[[PHI].sup.(L)]([[I.sub.1] [direct sum] [I.sub.2]]) [not equal to] [[PHI].sup.(L)]([[I.sub.1] [direct sum] [I.sub.3]]). (4)

where [I.sub.1] [direct sum] [I.sub.2] = [i.sub.1;0], [i.sub.1;1], .... [i.sub.2;L-2] [member of] [[SIGMA].sub.A,L] if and only if [i.sub.1;l+1] = [i.sub.2;l] [for all]l = 0, ..., L - 2. If [I.sub.1], [I.sub.2] [member of] A([[SIGMA].sub.A]), then we define [I.sub.1] [direct sum] [I.sub.2] := [I.sub.1][I.sub.2]. Throughout this paper we assume [phi]: [[SIGMA].sub.A] [right arrow] S is right resolving.

In this paper, we study that the dimension spectrum with N: S [right arrow] L([R.sup.d], [R.sup.d]) is a matrix-valued potential on S taking values on the set of d x d matrices. To be precise, let [[GAMMA].sup.*.sub.(+)](S) be the collection of d x d nonnegative (positive) matrices which are * continuous on S; notation * stands for the H of Holder continuous and C for continuous, the same for [[GAMMA].sup.*.sub.(+)]([[SIGMA].sub.A]). For q [member of] R, let [P.sub.N](q) be defined similarly as in (1) and the level set for the upper Lyapunov exponent for N is also defined similarly as (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

The main results of the present paper are the following. We want to mention here that our results were independently investigated by Feng and Huang [13, Theorem 1.4] via different approach. Our method, except for providing another point of view for the mathematical demonstration, can be applied for evaluating the topological pressure rigorously.

Thearom A. Let S be a sofic shift induced by [[SIGMA].sub.A] and let N [member of] [[GAMMA].sup.(C).sub.+](S) be a matrix-valued potential on S which depends on k coordinates. Then

(1) for all q [member of] R\{0}, [P.sub.N](q) is differentiable;

(2) if [alpha] = [P'.sub.N](q),

[dim.sub.H][E.sub.N]([alpha]) = [1/log [lambda]]{-[alpha]q + [P.sub.N](q)}. (6)

where [lambda] is the maximal eigenvalue of A.

Theorem A deals with the finite-coordinate dependent matrix potentials. This method also allows us to set up the dimension spectrum for infinite-coordinate dependent one for S. We emphasize here that our method makes the discussion of the limiting measure on infinite-coordinate systems possible. Let

[[eta].sub.n] := sup{[M.sub.i,j](x)/[M.sub.i,j](y): I [member of] [[SIGMA].sub.A,n], x,y [member of] [I], 1 [less than or equal to] i, j [less than or equal to] d}. (7)

Since M [member of] [[GAMMA].sup.H.sub.+]([[SIGMA].sub.A]), we have [absolute value of log [[eta].sub.n]] [less than or equal to] C[[lambda].sup.-[alpha]n] for some 0 < [alpha] < 1 ([4, Lemma 2.2]). The following result deals with the dimension spectrum for infinite-coordinate N.

Theorem B. Let N [member of] [[GAMMA].sup.H.sub.+](S) be a matrix-valued potential on S which depends on infinite many coordinates. Then

(1) for all q [member of] R\{0}, [P.sub.N](q) is differentiable;

(2) if [alpha] = [P'.sub.N](q),

[dim.sub.H][E.sub.N]([alpha]) = [1/log [lambda]]{-[alpha]q + [P.sub.N](q)}, (8)

where [lambda] is the maximal eigenvalue of A.

The block map [[PHI].sup.(L)] plays an important role in this method and one of the advantages of this method is that we can prove that (3) holds for N [member of] [[GAMMA].sup.H.sub.+](S) by using the matrix theory argument (Perron-Frobenius Theorem [6]). We will show there are some interesting examples of sofic affine set that we can compute their rigorous formulae for [L.sup.q]-spectrum and the pressure functions; then the dimension spectrum is thus derived by simple computation.

The content of the paper is following. In Section 2, we present the proof of Theorem A and the proof of Theorem B is given in Section 3. Section 4 extends Theorems A and B to nonnegative matrix-valued potential functions and investigates some examples.

2. Proof of Theorem A

This section gives a proof for Theorem A. We recall some definitions first. Denote by M([[SIGMA].sub.A]) the set of probability measures on [[SIGMA].sub.A] and [M.sub.T]([[SIGMA].sub.A]) the subset of T-invariant measures of M([[SIGMA].sub.A]), M(S) and [M.sub.S](S) are defined similarly.

Definition 2. (1) We say that [mu] [member of] [M.sub.T]([[SIGMA].sub.A]) is quasi-Bernoulli if there exists a constant C > 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

(2) For q [member of] R, the [L.sup.q]-spectrum of [mu] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [lambda] denotes the maximal eigenvalue of A.

Our method is motivated by the idea which is proposed in [5] and the intrinsic property of the sliding block codes [[PHI].sup.(L)] and 0; we formulate it briefly.

(1) Since N depends on k coordinates, we construct [[PHI].sub.k] from [[PHI].sup.(L)] as mentioned above. Then the pullback potential on [[SIGMA].sub.A,k+L-1] from is also defined. We extend the idea of the proof of Lemma 4.3 of [5] to construct an invariant, ergodic probability measure on [[SIGMA].sub.A,k+L-1] and extend this measure to some limiting measure which supports the whole [[SIGMA].sub.A].

(2) For all J [member of] [S.sub.n] we define a measure on J by measuring one of its preimages with the measure in [[SIGMA].sub.A] which is constructed in Step 1. Although the measure in [[SIGMA].sub.A] satisfies the Markov property and probability properties, the measure on S cannot share the same properties. However, the space M(S) is still compact and the standard argument allows us to find an invariant and ergodic measure on S.

(3) Combining steps 1 with 2 we are able to show that the limiting measure is Gibbs-like and satisfies the quasi-Bernoulli property (we emphasize here that this measure is not necessary a Gibbs measure) and the [L.sup.q]-spectrum preserved under the factor [phi] which is induced from the limit of [[PHI].sub.k]; that is, [phi] = [lim.sub.k[right arrow][infinity]][[PHI].sub.k]. Therefore, the differentiability and the dimension spectrum can be preserved from [phi].

Proof of Theorem A. We divide the proof in the following 4 steps.

Step 1. Let [[PHI].sup.(L)]: [[SIGMA].sub.A,L] [right arrow] A(S) be a sliding block code from [[SIGMA].sub.A,L] to A(S). For k [greater than or equal to] 1, define [[PHI].sub.k] = [[PHI].sup.(k+L-1)] from [[SIGMA].sub.A,k+L-1] to [S.sub.k] by [[PHI].sub.k]([I]) = [J] = [[j.sub.0], ..., [j.sub.k-1]]:

[j.sub.l] = [[PHI].sup.(L)]([[pi].sub.[0,L-1]]([T.sup.l-1]([I]))) [for all]l = 1, ..., k, (11)

where [[pi].sub.[0,L-1]]([I]) = [i.sub.0][i.sub.1] ... [i.sub.L-1] denotes the projection map to coordinate [0, L - 1] on [Z.sup.1] for all I [member of] [[SIGMA].sub.A]. Define a matrix potential M on [[SIGMA].sub.A,k+L-1] by, if I [member of] [[SIGMA].sub.A,k+L-1],

M([I]) = N([[PHI].sub.k]([I])). (12)

Then M is well defined for all I [member of] [[SIGMA].sub.A,k+L-1] from the fact that N depends on k-coordinate. Write [[SIGMA].sup.(k)] = [[SIGMA].sub.A,k+L-1]. Define [q.sup.(k)] = #[[SIGMA].sub.A,k+L-1] for k [greater than or equal to] 1 and [q.sup.(0)] = #A([[SIGMA].sub.A]); we setup a matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is indexed by the elements of [[SIGMA].sup.(k-1)] as follows: [for all][I.sub.1],[I.sub.2] [member of] [[SIGMA].sup.(k-1)]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where [0.sub.dxd] denotes the d x d matrix with entries which are all zeros. For all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we denote by I(A) the indicator matrix of A; that is, I(A) [member of] [M.sub.nxn](R),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

It is obvious that if [[SIGMA].sub.A] is mixing, then I(A) is primitive. Therefore, if we assume H = [H.sup.(k)], there exists a uniform constant m > 0 such that for all [I.sub.1] and [I.sub.2] [member of] [[SIGMA].sup.(k-1)] there exists a path [([I'.sub.l]).sup.R.sub.l=1] with R [less than or equal to] m, [I'.sub.l] [member of] [[SIGMA].sup.(k-1)], [I'.sub.l] = [I.sub.1], [I'.sub.R] = [I.sub.2], and

I' := [I'.sub.1] [direct sum] ... [direct sum] [I'.sub.R] [member of] [[SIGMA].sup.(k+R-2)]. (15)

Combining the fact of N [member of] [[GAMMA].sup.C.sub.+](S),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

and also [H.sup.m] > 0. Let [A.sup.(k)] = {[I.sup.(l)] [member of] [[SIGMA].sup.(k-1)]: l = 1, ..., [q.sup.(k-1)]} be an ordered set by the lexigraphic ordering and we rearrange H according to this ordering. Since H is primitive, Perron-Frobenius Theorem is applied to show that there exist eigenvalues [[rho].sub.L] and [[rho].sub.R] > 0 with corresponding eigenvectors L and R > 0, respectively, for H. We may also assume

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

For all I [member of] [[SIGMA].sup.(k+j)] and j [greater than or equal to] 0, let [[pi].sup.(k)] = [[pi].sub.[0,k+L-2]] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

We define a measure as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

It follows from (19) that if I [member of] [[SIGMA].sup.(k+j-1)],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

It follows from the same computation we also have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

This implies that [for all]n [greater than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

It can be easily checked that [[rho].sub.L] = [[rho].sub.R] and then we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

The Kolmogorov consistence theorem is applied to show that there exists a measure [mu] on [[SIGMA].sub.A] such that

[mu]([I]) = [eta]([I]) [for all]I [member of] [[union].sub.n [greater than or equal to] 0][[SIGMA].sup.(k+n)]. (27)

Step 2. In this step, we will define a measure on [M.sub.S](S). Since

[[PHI].sub.k+j]([[SIGMA].sup.(k+j)]) = [S.sub.k+j] (28)

is onto, for all J [member of] [S.sub.k+j], we define an ordered set as

[P.sub.J] := {[I.sup.(l)] [member of] [[SIGMA].sup.(k+j)]: [[PHI].sub.k+j]([[I.sup.(l)]]) = J} (29)

and set a measure on S:

[[xi].sub.1]([J]) = [[eta].sub.L]([[I.sup.(1)]]), [for all]J [member of] [S.sub.k+j], [I.sup.(1)] [member of] [P.sub.J]. (30)

We note here that [[xi].sub.1] is not invariant. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

for all J [member of] [S.sub.k+j] and assume [I.sub.1],[I.sub.2] [member of] [P.sub.J],

[[PHI].sub.k]([[phi].sup.(k)]([T.sup.l]([[I.sub.1]]))) = [[PHI].sub.k]([[phi].sup.(k)]([T.sup.l]([[I.sub.2]])))

[for all]l = 0, ..., j - 1. (32)

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where [B.sub.i]([I]) denotes the i-coordinate of vector of B([I]) for B = L or R. Since [phi] = [lim.sub.k[right arrow][infinity]][[PHI].sub.k] is right resolving, for all J [member of] [S.sub.n] there is at least one and at most K preimages of I [member of] [[SIGMA].sup.(n)] such that [[PHI].sub.n]([I]) = J. Therefore, if j [greater than or equal to] 0 and J [member of] [S.sub.k+j], it follows from (33) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

By the positivity of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [q.sup.(k-1)] is finite, we can conclude that there exist two constants P and Q > 0 such that 1 [less than or equal to] [U.sub.L]/[V.sub.L] [less than or equal to] P and 1 [less than or equal to] [U.sub.R]/[V.sub.R] [less than or equal to] Q; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

This means that for all J [member of] [S.sub.k] there exists a constant [C.sub.1] > 0 such that

[C.sup.-1.sub.1][[xi].sub.1]([J]) [less than or equal to] [[eta].sub.L]([[PHI].sup.-1.sub.k]([J])) [less than or equal to] [C.sub.1][[xi].sub.1]([J]). (37)

And thus

[C.sup.-1.sub.2][xi]([J]) [less than or equal to] [eta]([[PHI].sup.-1.sub.k]([J])) [less than or equal to] [C.sub.2][xi]([J]), (38)

for some [C.sub.2] > 0.

Step 3. Since [xi] is not invariant, we follow the proof of [4] to construct an invariant and ergodic measure satisfying the property of (21) in this step. For all J [member of] [S.sub.k], define a sequence [{[[SIGMA].sup.n-1.sub.l=0][xi] [omicron] [S.sup.-l]([J])}.sub.n [greater than or equal to] 1]. It follows from (37) that if l [greater than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Hence there exists a constant [C.sub.3] > 0 such that

[C.sup.-1.sub.3][[eta].sub.L]([[PHI].sup.-1.sub.k][J]) [less than or equal to] [[xi].sub.1] [omicron] [S.sup.-l]([J]) [less than or equal to] [C.sub.3][[eta].sub.L]([[PHI].sup.-1.sub.k][J]). (40)

Thus there exists a [C.sub.4] > 0 such that

[C.sup.-1.sub.4][eta]([[PHI].sup.-1.sub.k]([J])) [less than or equal to] [xi] [omicron] [S.sup.-l]([J]) [less than or equal to] [C.sub.4][eta]([[PHI].sup.-1.sub.k]([J])). (41)

Since S is compact, then let v [member of] M(S) be the limiting measure of

[{[n-1.summation over (l=0)][xi] [omicron] [S.sup.-l]([J])}.sub.n [greater than or equal to] 1]. (42)

Combining the fact that [lim.sub.n[right arrow][infinity]][[PHI].sub.k+n] = [phi] with the above computations it yields v [much less than] [mu] and [mu] [much less than] v. Up to a small modification of the proof in Theorem 1.1 of [4] we also have that v is ergodic. The Radon-Nikodym theorem applies to show that there is a constant C > 0 such that v([J]) = C[mu]([[PHI].sup.-1.sub.l]([J])) for v-a.e. J [member of] [S.sub.l] and l [greater than or equal to] k. It follows from that v and [mu] are both invariant probability measures. We obtain C = 1 and for all J [member of] S

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

Step 4. From the above computation we obtain that if J [member of] [S.sub.l] with l [greater than or equal to] k and [I.sup.(1)] [member of] [P.sub.J], then v([J]) = [mu]([[I.sup.(1)]]) = [eta]([[I.sup.(1)]]). Moreover, there exists [Q.sub.1] > 0 such that

[Q.sup.-1.sub.1][[eta].sub.L]([[I.sup.(1)]]) [less than or equal to] [eta]([[I.sup.(1)]]) [less than or equal to] [Q.sub.1][[eta].sub.L]([[I.sup.(1)]]). (44)

With the positivity of M implements there exists a constant [Q.sub.2] > 0 such that for any x [member of] [[SIGMA].sub.A] n, l [member of] N we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

This demonstrates [[eta].sub.L] is quasi-Bernoulli and so are [eta] and [mu]. Hence v [member of] [M.sub.S](S) is a quasi-Bernoulli measure. According to the fact that right-resolving factor [phi] cannot increase the topological entropy, we can assert that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

where [[phi].sub.*][mu] = [mu]([[phi].sup.-1]). Theorem 1.3 of [4] is applied to show that for all q [member of] R\{0}, [[tau].sub.v] is differentiable and if [alpha] = [P'.sub.N](q),

[dim.sub.H][E.sub.N]([alpha]) = [1/log [lambda]](-[alpha]q + [P.sub.N](q)), (47)

where [lambda] denotes the maximal eigenvalue of A. Finally, the differentiability for [P.sub.N](q) with q [not equal to] 0 comes from the fact [P.sub.N](q) = [P.sub.M](q) since [phi] is right resolving and M is the pullback potential of N. This completes the proof. []

Remark 3. We remark that in the proof of Theorem A, v [member of] [M.sub.S](S) is not a Gibbs measure, and in the following, we will show that this method allow us to approximate the potential depending on infinite coordinate for N [member of] [[GAMMA].sup.H.sub.+](S).

3. Proof of Theorem B

In this section, we extend our result to the matrix-valued potentials that are infinite-coordinate dependent.

Proof of Theorem B. The first statement is an immediate consequence of Theorem A since [N.sup.(k)] depends on k-coordinate. It is still remaining to prove the second statement.

For k [greater than or equal to] 1, I [member of] [[SIGMA].sup.(k)] with I [member of] [P.sub.J], let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and let [[rho].sup.(k)] its maximal eigenvalue as in Theorem A. Since I([H.sup.(k)]) is primitive and [M.sup.(k)] is positive, [H.sup.(k)] is also primitive for all k [member of] N. We claim that [[eta].sup.-1.sub.k][[rho].sup.(k)] [less than or equal to] [[rho].sup.(k+1)] [less than or equal to] [[eta].sub.k][[rho].sup.(k)] for k [greater than or equal to] 1. Indeed, let [H.sup.(k)] and [H.sup.(k+1)] be indexed by [[SIGMA].sup.(k-1)] and [[SIGMA].sup.(k)], respectively. For all I [member of] [[SIGMA].sup.(k+1)], [J] = [[j.sub.0] ... [j.sub.k]] := [[PHI].sub.k+1]([I]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where [J.sup.*] := [j.sub.0] ... [j.sub.k-1] and [I.sup.*] := [i.sub.0] ... [i.sub.k+L-1]. Therefore, for m [greater than or equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (49)

for some C > 1. Hence, [[parallel][H.sup.(k+1)m][parallel].sup.1/m] [less than or equal to] [C.sup.1/m]([[eta].sub.k])[[parallel][H.sup.(k)m][parallel].sup.1/m]. Taking m [right arrow] [infinity] we have

[[rho].sup.(k+1)] [less than or equal to] [[eta].sub.k][[rho].sup.(k)], for k [greater than or equal to] 1. (50)

Using the same argument, we also have

[[rho].sup.(k+1)] [greater than or equal to] [[eta].sup.-1.sub.k][[rho].sup.(k)], for k [greater than or equal to] 1. (51)

On the other hand, for I [member of] [[SIGMA].sup.(k+j)] with j [greater than or equal to] 0 being fixed,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

for some D, D', D" > 0. Similarly we have

[[eta].sup.(k+1).sub.L]([I]) [greater than or equal to] [(D"[[eta].sup.2j.sub.k]).sup.-1][[eta].sup.(k).sub.L]([I]) (53)

and there exists [D.sub.1] > 0 such that

[([D.sub.1][[eta].sup.2j.sub.k]).sup.-1][[mu].sup.(k)]([I]) [less than or equal to] [[mu].sup.(k+1)]([I]) [less than or equal to] [D.sub.1][[eta].sup.2j.sub.k][[mu].sup.(k)]([I]). (54)

The fact that [lim.sub.k[right arrow][infinity]][[eta].sub.k] = 1 asserts and there exists [D.sub.2] > 0, n [member of] N such that for k [greater than or equal to] n we have

[D.sup.-1.sub.2][[mu].sup.(k)]([I]) [less than or equal to] [[mu].sup.(k+1)]([I]) [less than or equal to] [D.sub.2][[mu].sup.(k)]([I]). (55)

This demonstrates [[mu].sup.(k)] [right arrow] [??] as k [right arrow] [infinity] for some [??]. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [v.sup.(k)] = [[phi].sub.*][[mu].sup.(k)] implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

It can also be checked that [??] satisfies the quasi-Bernoulli property and for all q [member of] R\{0},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

Using the same proof of Theorem A, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (58)

Combining Theorems 1 with A, we conclude that [P.sub.N](q) is thus differentiable and the desired equality (8) follows. This completes the proof. []

4. Examples

This section illustrates several examples that help for the understanding of our results.

4.1. Computation of Dimension Spectrum. Suppose X is an irreducible subshift of finite type and [pi]: X [right arrow] Y is a factor. Chazottes and Ugalde [14] indicate that if a matrix-valued push-forward potential function N is row allowable and is positive on periodic points, then there exists a unique Gibbs measure on Y. Here N is called row allowable if there is no zero row in N. Before extending our results to nonnegative matrix-valued potential functions, we give the definition of a column allowable matrix first.

Definition 4. We call A [member of] [M.sub.nxn](R) column allowable if for all 1 [less than or equal to] j [less than or equal to] n, we have [[SIGMA].sup.n.sub.i=1][A.sub.ij] [greater than or equal to] 1. We also denote by [N.sub.n] the collection of column allowable matrices of size nxn.

It can be easily verified that [N.sub.n] forms a semigroup under matrix product.

Lemma 5. If A and B [member of] [N.sub.n], then AB [member of] [N.sub.n].

Proof. Indeed, for all j = 1, ..., n,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

This completes the proof. []

For nonnegative matrix-valued potential N, we have the following result.

Theorem 6. Let N [member of] [[GAMMA].sup.H](S) [intersection] [N.sub.d] depend on k-coordinate and there exists a finite set [LAMBDA] [subset] [[union].sub.n [greater than or equal to] k][[SIGMA].sup.(n)] =: [[SIGMA].sup.*] such that for all [I.sub.1] and [I.sub.2] [member of] [[SIGMA].sup.*] there exists I [member of] [LAMBDA] such that [I.sub.1] [direct sum] I [direct sum] [I.sub.2] [member of] [[SIGMA].sup.*] and M([I]) = N([[PHI].sub.k]([I])) > 0 for all I [member of] [LAMBDA], and then (6) holds.

Proof. We give the proof for the case that all elements in [LAMBDA] are equal length and the case for different length is in the same fashion. It follows from the proof in Theorem A that [H.sup.(k)] can be constructed which is indexed by the [[SIGMA].sup.(k)]. Since for any [omega] [member of] [LAMBDA], [absolute value of [omega]] = k + L - 1 (we assume that [LAMBDA] is equal length and the definition [LAMBDA] [subset] [[union].sub.n [greater than or equal to] k][[SIGMA].sup.(n)] allows us to define all elements which have equal length of k + L - 1) we also assume that [LAMBDA] consist of only one element; say [I.sup.*]. Without loss of generality, assume [I.sup.*] [member of] [[SIGMA].sup.(k-1)]. It suffices to show that [H.sup.(k)] is primitive. Indeed, for any [I.sub.1] and [I.sub.2] [member of] [[SIGMA].sup.(k-1)], since N [member of] [N.sub.d], Lemma 5 is thus applied to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

This means that [H.sup.2] > 0. The other case can be done similarly. Therefore, the same proof as in Theorem A leads to (6) and the proof is completed. []

In the proof of Theorem A, the [L.sup.q]-spectrum plays an important role for the computing of dimension spectrum. We emphasize that for a measure [mu] [member of] [M.sub.T]([[SIGMA].sub.A]), it is not easy to compute the rigorous formula for [[tau].sub.[mu]]. If the measure [mu] is given as in Theorem A, the following theorem provides a class of matrix-valued potentials for which we can compute its [L.sup.q]-spectrum explicitly. Let N [member of] [[GAMMA].sup.H](S) [intersection] [N.sub.d] depend on k-coordinate and H := [H.sup.(k)] as defined in Theorem A; we define a matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from H (recall that [q.sup.(k)] = #[[SIGMA].sub.A,k+L-1]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

[rho](A) [member of] R denotes the maximal eigenvalue of A [member of] [M.sub.dxd](R).

Proposition 7. Under the same assumptions of Theorem 6, assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

satisfies that

N([[J.sub.1]])N([[J.sub.2]]) = N([[J.sub.2]])N([[J.sub.1]]) [for all][J.sub.1] [not equal to] [J.sub.2] [member of] [S.sub.k]. (63)

Assume that [H.sup.(k)] and v [member of] [M.sub.S](S) are as defined in Theorem A. Then

[[tau].sub.v](q) = [1/log [lambda]](-q log [rho] + log [THETA](q)), (64)

where [THETA](q) is the maximal root of the characteristic polynomial of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

be constructed as in Step 1 of the proof of Theorem A. Since elements of N are mutually commuted, then the set of matrices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be diagonalized simultaneously. That is, there exists a unique P [member of] [M.sub.dxd](R) such that PM([I])[P.sup.-1] := D([I]) is a diagonal matrix for all I [member of] [[SIGMA].sup.(k)]. Since H is primitive, there exist L and R > 0 such that (19) holds. We first compute the [L.sup.q]-spectrum [[tau].sub.[mu]], where [mu] [member of] [M.sub.T]([[SIGMA].sub.A]) is defined in the proof of Theorem A with the property that there exists a constant C' > 0 such that for each I [member of] [[SIGMA].sub.A,n],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (66)

This induces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (67)

We note here that the second equality comes from the positivity of L, R and P is invertible. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the proof is completed. []

Here we give a concrete example for the dimension spectrum of sofic system.

Example 8. Let [[SIGMA].sub.A] be the golden mean shift with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68)

and the right-resolving sliding block code with L = 2:

[[PHI].sup.(2)]([00]) = a, [[PHI].sup.(2)]([01]) = b, [[PHI].sup.(2)]([10]) = b. (69)

Define a matrix potential on [S.sub.1], that is, k = 1, as in Proposition 7 by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (70)

Then [LAMBDA] = {[00]} and M([00]) = [N.sub.a] > 0. A little modification of the proof of Theorem A indicates that [P.sub.N](q) is differentiable. Suppose [alpha] = [P'.sub.N](q) with q [not equal to] 0; Theorem 6 is applied to show that

[dim.sub.H][E.sub.N]([alpha]) = -[alpha]q + [P.sub.N](q). (71)

On the other hand, one can easily compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

and Proposition 7 applies to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (73)

4.2. Computation of Pressure. Let ([[SIGMA].sub.A], T) be a subshift of finite type and [P.sub.M](q) be its pressure for q [member of] R. If [P.sub.M](q) is differentiable, Theorem 1.3 of [4] demonstrates that the dimension spectrum can be computed via the formula of [P.sub.M](q). However, the computation of the explicit formula for [P.sub.M](q) is not easy. If (S, S) a sofic system, we provide a wide class of matrix potential on S for which we can compute its [P.sub.N](q) rigorously which leads to the dimension spectrum of [E.sub.N]([alpha]). We first give a theorem which is analogous to Theorem 1.3 of [4].

Theorem 9. Let N [member of] [[GAMMA].sup.H.sub.+](S). We have for any [alpha] = [P'.sub.N](q) with q [not equal to] 0

[dim.sub.H][E.sub.N]([alpha]) = [1/log [lambda]](-[alpha]q + [P.sub.N](q)). (74)

Proof. Up to a minor modification, the proof is identical to the proof of Theorem 1.3 of [4] and we omit it here. []

We prove the following class for which we can compute its [P.sub.N](q) and[ dim.sub.H][E.sub.N]([alpha]).

Theorem 10. If N [member of] [[GAMMA].sup.C](S) depends on k-coordinate, then it satisfies the following properties.

(1) Let H = [H.sup.(k)] be the matrix constructed in Theorem A which is primitive.

(2) Let M [member of] [[GAMMA].sup.C]([[SIGMA].sup.(k).sub.A]) be induced from N as in the proof of Theorem A. If there exists a sequence of real numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and K [member of] [M.sub.1xd](R) is a row vector such that for any I [member of] [[SIGMA].sup.(k).sub.A] we have

KM([I]) = [[chi].sub.I]K. (75)

Then

[P.sub.N](q) = log [THETA](q), (76)

where [THETA](q) denotes the maximal eigenvalue of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined in (77) and thus it is differentiable. Furthermore, (74) can be computed explicitly.

Proof. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (77)

where Y [member of] [M.sub.dx1](R) is a column vector with KY = 1. Since H is primitive, then the left and right eigenvectors are positive; that is, L,R > 0. Combining (75) with Perron-Frobenius Theorem we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (78)

The second equality exists because N is finite-coordinate dependent and 4th equality comes from the right-resolving property of [phi]. This completes the proof. []

Remark 11. (1) In Theorem A, we always assume that if one is regarded as [[SIGMA].sub.A] = (G, E) where G = {I: I [member of] [[SIGMA].sup.(k-1)]} and edges,

E = {([I.sub.1],[I.sub.2]): [I.sub.1] [direct sum] [I.sub.2] [member of] [[SIGMA].sup.(k)]}. (79)

Then there is only one level from [I.sub.1] to [I.sub.2] [member of] [[SIGMA].sup.(k)]; that is, the number of levels of ([I.sub.1],[I.sub.2]) for any [I.sub.1] and [I.sub.2] is equal to one, and thus [H.sup.(k)] can be constructed with the entry which is a single smaller matrix. However, if there is more than one level from [I.sub.1] to [I.sub.2], we only need to modify [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (80)

where l([I.sub.1] [direct sum] [I.sub.2]) denotes the number of levels from [I.sub.1] to [I.sub.2]. Since f is right resolving, Theorem A still follows.

(2) In the assumption (75) of Theorem 10, one can easily check that the result remains if there exists a column vector K such that for any I [member of] [[SIGMA].sup.(k).sub.A]

M([I])K = [[chi].sub.I]K. (81)

(3) One can also easily check that for those classes of Theorem 10, if v [member of] [M.sub.S](S) is the measure in Theorem A, Then the [L.sup.q]-spectrum is

[[tau].sub.v](q) = [1/log [lambda]](-q log [rho] + log [THETA](q)), (82)

where [rho] is the maximal eigenvalue of [H.sup.(k)] and [THETA](q) is the maximal eigenvalue of (77).

In the following example, the computation of pressure helps for the computation of dimension spectrum of sofic affine-invariant set.

Example 12 (sofic affine-invariant set. See [15, Example 1]). Consider [T.sup.2] = [R.sup.2]/[Z.sup.2] which is invariant under

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B)

Let D = {0, ..., 4} x {0, ..., 2} and for any [{[d.sub.k]}.sup.[infinity].sub.k=1] [member of] [D.sup.N] the base T representation is as follows (reader may refer to [7] for details):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (84)

Let A be a matrix index by D which is incidence matrix and [K.sub.T](A) is also defined as the image of [R.sub.T]; that is,

[K.sub.T](A) = {[R.sub.T]{[d.sub.k]}: A([d.sub.k], [d.sub.k+1]) = 1 for k [greater than or equal to] 1}. (85)

Let S be a sofic system that is induced by projecting [[SIGMA].sub.A] on y-direction, see Figure 1, and let N = [(N([s])).sup.2.sub.s=0] be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (86)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (87)

[K.sup.(4).sub.T](S) and [K.sup.(7).sub.T](S) are represented in Figure 2. Define [[PHI].sup.(2)] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (88)

Then one can easily check that if K = (1,1), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be defined as

[[chi].sub.0] = [[chi].sub.1] = [[chi].sub.2] = 2 (89)

[H.sup.(1)] can be defined as follows ((1) of Remark 11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (90)

For any q [member of] R, define F(q) from H by taking K = (1,1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (91)

and then

[P.sub.N](q) = log [THETA](q) = log 6 + q log 2. (92)

Theorem 6 indicates that [alpha] = log 2 and

[dim.sub.H][E.sub.N](log 2) = [1/log 6](-q log 2 + (log 6 + q log 2)) = 1 (93)

which is constant multifractal.

If N is symmetric, we also have the following estimate.

Corollary 13. Let N [member of] [[GAMMA].sup.C](S) depend on k-coordinate and let H = [H.sup.(k)] be the matrix constructed in Theorem A which is primitive. Let M [member of] [[GAMMA].sup.C]([[SIGMA].sup.(k).sub.A]) be induced from N, if for any I [member of] [[SIGMA].sup.(k).sub.A], there exist [a.sub.I], [b.sub.I] [member of] R such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (94)

Then

[P.sub.N](q) = log [THETA](q), (95)

where [THETA](q) denotes the maximal eigenvalue of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (96)

Proof. Since for any I [member of] [[SIGMA].sup.(k).sub.A],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (97)

Theorem 10 is applied to show that [P.sub.N](q) = log [THETA]](q), where [THETA](q) is the maximal eigenvalue of (96). The proof is completed. []

Example 14 (continued). Under the same substitution rule of Example 12, if the potentials on A(S) are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (98)

one can easily check that H(1) is primitive and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (99)

Then

[P.sub.N](q) = log 2 + log (1 + [3.sup.q] + [4.sup.q]). (100)

http://dx.doi.org/10.1155/2014/624523

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Ban is partially supported by the National Science Council, ROC (Contract no. NSC 102-2628-M-259-001-MY3). Chang is grateful for the partial support of the National Science Council, ROC (Contract no. NSC 102-2115-M-035-004-).

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[8] V. Climenhaga and D. J. Thompson, "Intrinsic ergodicity beyond specification: [beta]-shifts, S-gap shifts, and their factors," 2011, http://arxiv.org/abs/1011.2780.

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Jung-Chao Ban, (1) Chih-Hung Chang, (2) Ting-Ju Chen, (1) and Mei-Shao Lin (3)

(1) Department of Applied Mathematics, National DongHwa University, Hualien 970003, Taiwan

(2) Department of Applied Mathematics, FengChia University, Taichung 40724, Taiwan

(3) Department of Mathematics, National Central University, Chungli 32054, Taiwan

Correspondence should be addressed to Chih-Hung Chang; [email protected]

Received 21 August 2013; Accepted 3 February 2014; Published 10 March 2014

Academic Editor: Christian Maes

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Title Annotation:	Research Article
Author:	Ban, Jung-Chao; Chang, Chih-Hung; Chen, Ting-Ju; Lin, Mei-Shao
Publication:	Advances in Mathematical Physics
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Date:	Jan 1, 2014
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