Research Article | Open Access
Dimension Spectrum for Sofic Systems
We study the dimension spectrum of sofic system with the potential functions being matrix valued. For finite-coordinate dependent positive matrix potential, we set up the entropy spectrum by constructing the quasi-Bernoulli measure and the cut-off method is applied to deal with the infinite-coordinate dependent case. We extend this method to nonnegative matrix and give a series of examples of the sofic-affine set on which we can compute the spectrum concretely.
Let be a subshift of finite type (SFT) with being the incidence matrix and being its shift map. Motivated by the study of the iterated function systems (IFS) and generalized Sierpiński carpets (GSC, cf. [1–5]), one considers a special type of potential functions which take values on the set of matrices. For , define the topological pressure as follows: whenever the limit exists and denotes the collection of -cylinders in . (Here is the matrix norm; that is, , where is the column vector with entries being 1’s). In , if is positive, that is, , and Hölder potential with is topologically mixing, the authors prove that the Gibbs measure for exists uniquely and the system admits the multifractal analysis. More precisely, let 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 be a SFT and let be Hölder continuous. Then is differentiable and for any , , where 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 . The authors extend this result to the case that is nonnegative with some additional irreducible conditions’ the reader may refer to  for the detail. When the underlying space is a sofic shift and , that is, the potential function is finitary real valued, there raises a natural equilibrium measure called semigroup measure proposed by Kitchens and Tuncel . When , the thermodynamic properties relate to fractal dynamics of given sofic affine-invariant sets (cf. ).
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 be a sofic shift which is a subshift of the shift space and let be the shift map on it. The well-known Curtis-Lyndon-Hedlund Theorem ([11, Theorem ]) demonstrates that if is a function, then is a homomorphism if and only if is a sliding block code. The sliding block code is induced from block map, that is, a map for some . For all and if , define by , where is the restriction map of a cylinder to lattice. Then is thus defined as the limit of ; that is, and . (We refer reader to [11, 12] for more detail). We call right resolving if for all and any and such that and we have where if and only if . If , then we define . Throughout this paper we assume is right resolving.
In this paper, we study that the dimension spectrum with is a matrix-valued potential on taking values on the set of matrices. To be precise, let be the collection of nonnegative (positive) matrices which are ★ continuous on ; notation ★ stands for the of Hölder continuous and for continuous, the same for . For , let be defined similarly as in (1) and the level set for the upper Lyapunov exponent for is also defined similarly as (2):
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. be a sofic shift induced by and let be a matrix-valued potential on which depends on coordinates. Then (1)for all is differentiable;(2)if , where is the maximal eigenvalue of .
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 . We emphasize here that our method makes the discussion of the limiting measure on infinite-coordinate systems possible. Let Since , we have for some ([4, Lemma 2.2]). The following result deals with the dimension spectrum for infinite-coordinate .
Thearom B. be a matrix-valued potential on which depends on infinite many coordinates. Then(1)for all is differentiable;(2)if , where is the maximal eigenvalue of .
The block map plays an important role in this method and one of the advantages of this method is that we can prove that (3) holds for by using the matrix theory argument (Perron-Frobenius Theorem ). We will show there are some interesting examples of sofic affine set that we can compute their rigorous formulae for -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 the set of probability measures on and the subset of -invariant measures of , and are defined similarly.
Definition 2. We say that is quasi-Bernoulli if there exists a constant such that
For , the -spectrum of is defined by where denotes the maximal eigenvalue of .
Our method is motivated by the idea which is proposed in  and the intrinsic property of the sliding block codes and ; we formulate it briefly.(1)Since depends on coordinates, we construct from as mentioned above. Then the pullback potential on from is also defined. We extend the idea of the proof of Lemma 4.3 of  to construct an invariant, ergodic probability measure on and extend this measure to some limiting measure which supports the whole .(2)For all we define a measure on by measuring one of its preimages with the measure in which is constructed in Step . Although the measure in satisfies the Markov property and probability properties, the measure on cannot share the same properties. However, the space is still compact and the standard argument allows us to find an invariant and ergodic measure on .(3)Combining steps with 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 -spectrum preserved under the factor which is induced from the limit of ; that is, . Therefore, the differentiability and the dimension spectrum can be preserved from .
Proof of Theorem . We divide the proof in the following steps.
Step 1. Let be a sliding block code from to . For , define from to by : where denotes the projection map to coordinate on for all . Define a matrix potential on by, if , Then is well defined for all from the fact that depends on -coordinate. Write . Define for and ; we setup a matrix which is indexed by the elements of as follows: where denotes the matrix with entries which are all zeros. For all with , we denote by the indicator matrix of , It is obvious that if is mixing, then is primitive. Therefore, if we assume , there exists a uniform constant such that for all and there exists a path with , , , , and Combining the fact of , Thus and also . Let be an ordered set by the lexigraphic ordering and we rearrange according to this ordering. Since is primitive, Perron-Frobenius Theorem is applied to show that there exist eigenvalues and with corresponding eigenvectors and , respectively, for . We may also assume That is, For all and , let and We define a measure as follows: It follows from (19) that if , That is, It follows from the same computation we also have that This implies that , It can be easily checked that and then we define The Kolmogorov consistence theorem is applied to show that there exists a measure on such that
Step 2. In this step, we will define a measure on . Since is onto, for all , we define an ordered set as and set a measure on : We note here that is not invariant. Let for all and assume , Hence Set where denotes the -coordinate of vector of for or . Since is right resolving, for all there is at least one and at most preimages of such that . Therefore, if and , it follows from (33) that By the positivity of , , and is finite, we can conclude that there exist two constants and such that and ; then This means that for all there exists a constant such that And thus for some .
Step 3. Since is not invariant, we follow the proof of  to construct an invariant and ergodic measure satisfying the property of (21) in this step. For all , define a sequence . It follows from (37) that if , Hence there exists a constant such that Thus there exists a such that Since is compact, then let be the limiting measure of Combining the fact that with the above computations it yields and . Up to a small modification of the proof in Theorem 1.1 of  we also have that is ergodic. The Radon-Nikodym theorem applies to show that there is a constant such that for -a.e. and . It follows from that and are both invariant probability measures. We obtain and for all
Step 4. From the above computation we obtain that if with and , then . Moreover, there exists such that With the positivity of implements there exists a constant such that for any we have This demonstrates is quasi-Bernoulli and so are and . Hence is a quasi-Bernoulli measure. According to the fact that right-resolving factor cannot increase the topological entropy, we can assert that where . Theorem 1.3 of  is applied to show that for all is differentiable and if , where denotes the maximal eigenvalue of . Finally, the differentiability for with comes from the fact since is right resolving and is the pullback potential of . This completes the proof.
Remark 3. We remark that in the proof of Theorem A, 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 .
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 . The first statement is an immediate consequence of Theorem A since depends on -coordinate. It is still remaining to prove the second statement.
For with , let and let be its maximal eigenvalue as in Theorem A. Since is primitive and is positive, is also primitive for all . We claim that for . Indeed, let and be indexed by and , respectively. For all , where and . Therefore, for , for some . Hence, . Taking we have Using the same argument, we also have On the other hand, for with being fixed, for some . Similarly we have and there exists such that The fact that asserts and there exists such that for we have This demonstrates as for some . Define . implies It can also be checked that satisfies the quasi-Bernoulli property and for all , Using the same proof of Theorem A, we have Combining Theorems 1 with , we conclude that is thus differentiable and the desired equality (8) follows. This completes the proof.
This section illustrates several examples that help for the understanding of our results.
4.1. Computation of Dimension Spectrum
Suppose is an irreducible subshift of finite type and is a factor. Chazottes and Ugalde  indicate that if a matrix-valued push-forward potential function is row allowable and is positive on periodic points, then there exists a unique Gibbs measure on . Here is called row allowable if there is no zero row in . Before extending our results to nonnegative matrix-valued potential functions, we give the definition of a column allowable matrix first.
Definition 4. We call column allowable if for all , we have . We also denote by the collection of column allowable matrices of size .
It can be easily verified that forms a semigroup under matrix product.
Lemma 5. If and , then .
Proof. Indeed, for all , This completes the proof.
For nonnegative matrix-valued potential , we have the following result.
Theorem 6. Let depend on -coordinate and there exists a finite set such that for all and there exists such that and for all , and then (6) holds.
Proof. We give the proof for the case that all elements in are equal length and the case for different length is in the same fashion. It follows from the proof in Theorem that can be constructed which is indexed by the . Since for any (we assume that is equal length and the definition allows us to define all elements which have equal length of ) we also assume that consist of only one element; say . Without loss of generality, assume . It suffices to show that is primitive. Indeed, for any and , since , Lemma 5 is thus applied to show that This means that . The other case can be done similarly. Therefore, the same proof as in Theorem leads to (6) and the proof is completed.
In the proof of Theorem , the -spectrum plays an important role for the computing of dimension spectrum. We emphasize that for a measure , it is not easy to compute the rigorous formula for . If the measure is given as in Theorem , the following theorem provides a class of matrix-valued potentials for which we can compute its -spectrum explicitly. Let depend on -coordinate and as defined in Theorem ; we define a matrix from (recall that ) as follows: denotes the maximal eigenvalue of .
Proposition 7. Under the same assumptions of Theorem 6, assume that satisfies that Assume that and are as defined in Theorem . Then where is the maximal root of the characteristic polynomial of .
Proof. Let be constructed as in Step 1 of the proof of Theorem . Since elements of are mutually commuted, then the set of matrices can be diagonalized simultaneously. That is, there exists a unique such that is a diagonal matrix for all . Since is primitive, there exist and such that (19) holds. We first compute the -spectrum , where is defined in the proof of Theorem A with the property that there exists a constant such that for each , This induces We note here that the second equality comes from the positivity of , and is invertible. Since , the proof is completed.
Here we give a concrete example for the dimension spectrum of sofic system.
Example 8. Let be the golden mean shift with and the right-resolving sliding block code with : Define a matrix potential on , as in Proposition 7 by Then and . A little modification of the proof of Theorem indicates that is differentiable. Suppose with ; Theorem 6 is applied to show that On the other hand, one can easily compute that and Proposition 7 applies to show that
4.2. Computation of Pressure
Let be a subshift of finite type and be its pressure for . If is differentiable, Theorem 1.3 of  demonstrates that the dimension spectrum can be computed via the formula of . However, the computation of the explicit formula for is not easy. If a sofic system, we provide a wide class of matrix potential on for which we can compute its rigorously which leads to the dimension spectrum of . We first give a theorem which is analogous to Theorem 1.3 of .
Theorem 9. Let . We have for any with
Proof. Up to a minor modification, the proof is identical to the proof of Theorem 1.3 of  and we omit it here.
We prove the following class for which we can compute its and .
Theorem 10. If depends on -coordinate, then it satisfies the following properties. (1)Let be the matrix constructed in Theorem which is primitive.(2)Let be induced from as in the proof of Theorem . If there exists a sequence of real numbers and is a row vector such that for any we have Then where denotes the maximal eigenvalue of defined in (77) and thus it is differentiable. Furthermore, (74) can be computed explicitly.
Proof. Define by where is a column vector with . Since is primitive, then the left and right eigenvectors are positive; that is, . Combining (75) with Perron-Frobenius Theorem we have The second equality exists because is finite-coordinate dependent and 4th equality comes from the right-resolving property of . This completes the proof.
Remark 11. In Theorem , we always assume that if one is regarded as where and edges,
Then there is only one level from to ; that is, the number of levels of for any and is equal to one, and thus can be constructed with the entry which is a single smaller matrix. However, if there is more than one level from to , we only need to modify by
where denotes the number of levels from to . Since is right resolving, Theorem still follows.
In the assumption (75) of Theorem 10, one can easily check that the result remains if there exists a column vector such that for any