Dimension Spectrum for Sofic Systems

We study the dimension spectrum of sofic systemwith the potential functions beingmatrix 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.


Introduction
Let (Σ  , ) 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][2][3][4][5]), one considers a special type of potential functions  : Σ  → (R  , R  ) which take values on the set of  ×  matrices.For  ∈ R, define the topological pressure as follows: whenever the limit exists and Σ , denotes the collection of cylinders in Σ  .(Here ‖‖ is the matrix norm; that is, ‖‖ = 1 1× 1 ×1 , where 1 ×1 is the  × 1 column vector with entries being 1's).In [4], if  is positive, that is,  , > 0, and Hölder potential with Σ  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.
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 (Σ  , ).The authors extend this result to the case that  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  = 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  ≥ 2, the thermodynamic properties relate to fractal dynamics of given sofic affine-invariant sets (cf.[7]).
Throughout this paper we assume  : Σ  → S is right resolving.
In this paper, we study that the dimension spectrum with  : S → (R  , R  ) is a matrix-valued potential on S taking values on the set of  ×  matrices.To be precise, let Γ  (+) (S) be the collection of  ×  nonnegative (positive) matrices which are  continuous on S; notation  stands for the  of Hölder continuous and  for continuous, the same for Γ  (+) (Σ  ).For  ∈ R, 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.  S be a sofic shift induced by Σ  and let  ∈ Γ  + (S) be a matrix-valued potential on S which depends on  coordinates.Then (1) 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 S. We emphasize here that our method makes the discussion of the limiting measure on infinite-coordinate systems possible.Let Since  ∈ Γ  + (Σ  ), we have | log   | ≤  − for some 0 <  < 1 ([4, Lemma 2.2]).The following result deals with the dimension spectrum for infinite-coordinate .

Thearom B. 𝐿𝑒𝑡 𝑁 ∈ Γ 𝐻
+ (S) be a matrix-valued potential on S which depends on infinite many coordinates.Then (1) where  is the maximal eigenvalue of .
The block map Φ () plays an important role in this method and one of the advantages of this method is that we can prove that (3) holds for  ∈ Γ  + (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   -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.

Proof of Theorem A
This section gives a proof for Theorem A. We recall some definitions first.Denote by M(Σ  ) the set of probability measures on Σ  and M  (Σ  ) the subset of -invariant measures of M(Σ  ), M(S) and M  (S) are defined similarly.
(2) For  ∈ R, the   -spectrum of  is defined by where  denotes the maximal eigenvalue of .

Advances in Mathematical Physics
Our method is motivated by the idea which is proposed in [5] and the intrinsic property of the sliding block codes Φ ()  and ; we formulate it briefly.
(1) Since  depends on  coordinates, we construct Φ  from Φ () as mentioned above.Then the pullback potential on Σ ,+−1 from Φ  is also defined.We extend the idea of the proof of Lemma 4.3 of [5] to construct an invariant, ergodic probability measure on Σ ,+−1 and extend this measure to some limiting measure which supports the whole Σ  .(2) For all  ∈ S  we define a measure on  by measuring one of its preimages with the measure in Σ  which is constructed in Step 1.Although the measure in Σ  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   -spectrum preserved under the factor  which is induced from the limit of Φ  ; that is,  = lim  → ∞ Φ  .Therefore, the differentiability and the dimension spectrum can be preserved from .
Proof of Theorem .We divide the proof in the following 4 steps.
Step 3. Since  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  ∈ S  , define a sequence Hence there exists a constant  3 > 0 such that Thus there exists a  4 > 0 such that Since S is compact, then let ] ∈ M(S) be the limiting measure of Combining the fact that lim  → ∞ Φ + =  with the above computations it yields ] ≪  and  ≪ ].Up to a small modification of the proof in Theorem 1.1 of [4] we also have that ] is ergodic.The Radon-Nikodym theorem applies to show that there is a constant  > 0 such that ]([]) = (Φ −1  ([])) for ]-a.e. ∈ S  and  ≥ .It follows from that ] and  are both invariant probability measures.We obtain  = 1 and for all Step 4. From the above computation we obtain that if  ∈ S  with  ≥  and  (1) ∈   , then ]([]) = ([ (1) ]) = ([ (1) ]).
With the positivity of  implements there exists a constant  2 > 0 such that for any  ∈ Σ  , ,  ∈ N we have where  *  = ( −1 ).Theorem 1.3 of [4] is applied to show that for all  ∈ R \ {0},  ] is differentiable and if  =    (), where  denotes the maximal eigenvalue of .Finally, the differentiability for   () with  ̸ = 0 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, ] ∈ M  (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  ∈ Γ  + (S).

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.

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[] for some ,   ,   > 0. Similarly we have and there exists  1 > 0 such that The fact that lim  → ∞   = 1 asserts and there exists  2 > 0,  ∈ N such that for  ≥  we have This demonstrates  () → μ as  → ∞ for some μ It can also be checked that μ satisfies the quasi-Bernoulli property and for all  ∈ R \ {0}, Advances in Mathematical Physics 7 Using the same proof of Theorem A, we have Combining Theorems 1 with A, we conclude that   () is thus differentiable and the desired equality (8) follows.This completes the proof.

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

Computation of Dimension Spectrum.
Suppose  is an irreducible subshift of finite type and  :  →  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 .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  ∈ M × (R) column allowable if for all 1 ≤  ≤ , we have ∑  =1   ≥ 1.We also denote by N  the collection of column allowable matrices of size  × .
It can be easily verified that N  forms a semigroup under matrix product.(59) This completes the proof.
Proof.We give the proof for the case that all elements in Λ are equal length and the case for different length is in the same fashion.It follows from the proof in Theorem A that  () can be constructed which is indexed by the Σ () .Since for any  ∈ Λ, || =  +  − 1 (we assume that Λ is equal length and the definition Λ ⊂ ⋃ ≥ Σ () allows us to define all elements which have equal length of  +  − 1) we also assume that Λ consist of only one element; say  * .Without loss of generality, assume  * ∈ Σ (−1) .It suffices to show that  () is primitive.Indeed, for any  1 and  2 ∈ Σ (−1) , since  ∈ N  , Lemma 5 is thus applied to show that This means that  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   -spectrum plays an important role for the computing of dimension spectrum.We emphasize that for a measure  ∈ M  (Σ  ), it is not easy to compute the rigorous formula for   .If the measure  is given as in Theorem A, the following theorem provides a class of matrix-valued potentials for which we can compute its   -spectrum explicitly.Let  ∈ Γ  (S) ∩ N  depend on -coordinate and  :=  () as defined in Theorem A; we define a matrix R() ∈ M  (−1) × (−1) from  (recall that  () = #Σ ,+−1 ) as follows: () ∈ R denotes the maximal eigenvalue of  ∈ M × (R).
Proposition 7.Under the same assumptions of Theorem 6, assume that satisfies that Assume that  () and ] ∈ M  (S) are as defined in Theorem A. Then where Θ() is the maximal root of the characteristic polynomial of R() ∈ M  (−1) × (−1) . Proof.
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 Σ  be the golden mean shift with and the right-resolving sliding block code with  = 2: Define a matrix potential on S 1 , that is,  = 1, as in Proposition 7 by (71) On the other hand, one can easily compute that and Proposition 7 applies to show that where  = 1 + √ 5 2 . (73) 4.2.Computation of Pressure.Let (Σ  , ) be a subshift of finite type and   () be its pressure for  ∈ R. If   () is differentiable, Theorem 1.3 of [4] demonstrates that the dimension spectrum can be computed via the formula of   ().However, the computation of the explicit formula for   () is not easy.If (S, ) a sofic system, we provide a wide class of matrix potential on S 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 [4].
Theorem 9. Let  ∈ Γ  + (S).We have for any  =    () with  ̸ = 0 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   () and dim    ().Theorem 10.If  ∈ Γ  (S) depends on -coordinate, then it satisfies the following properties.
(1) Let  =  () be the matrix constructed in Theorem A which is primitive.
Proof.Define F() ∈ M  (−1) × (−1) (R) by where  ∈ M ×1 (R) is a column vector with  = 1.Since  is primitive, then the left and right eigenvectors are positive; that is, ,  > 0. 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. (1)
In Theorem A, we always assume that if one is regarded as Σ  = (, ) where  = { :  ∈ Σ (−1) } and edges, Then there is only one level from  1 to  2 ∈ Σ () ; that is, the number of levels of ( 1 ,  2 ) for any  1 and  2 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  1 to  2 , we only need to modify  ()  1 , 2 by where ( 1 ⊕  2 ) denotes the number of levels from  1 to  2 .Since  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  such that for any  ∈ Σ (3) One can also easily check that for those classes of Theorem 10, if ] ∈ M  (S) is the measure in Theorem A, Then the   -spectrum is where  is the maximal eigenvalue of  () and Θ() 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.

Figure 1 :
Figure 1: The substitution rules associated with   .

𝐾Figure 2 :
Figure 2: The first and second figures are fourth and seventh steps of iteration, respectively.
∈Σ () can be diagonalized simultaneously.That is, there exists a unique  ∈ M × (R) such that ([]) −1 := ([]) is a diagonal matrix for all  ∈ Σ().Since  is primitive, there exist  and  > 0 such that (19) holds.We first compute the   -spectrum   , where  ∈ M  (Σ  ) is defined in the proof of Theorem A with the property that there exists a constant   > 0 such that for each  ∈ Σ , , be constructed as in Step 1 of the proof of Theorem A. Since elements of  are mutually commuted, then the set of Advances in Mathematical Physics matrices {([])}