Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.


Introduction and results
In this paper, we study the thermodynamic formalism for matrix products. We will characterize the structure of equilibrium states of pressure functions, and also examine the Gibbs properties of such states. This work was first carried out in [11] in the case that the involved matrices are non-negative and satisfy a kind of irreducibility. Some applications were given in the multifractal analysis of the top Lyapunov exponents of matrix products [11,6,8] (see also [10]). In this paper, we will consider arbitrary complex matrices.
Let (Σ, σ) be the one-sided full shift over the alphabet {1, . . . , ℓ} (cf. [2]) and let where [J] denotes the n-th cylinder {x = (x i ) ∞ i=1 ∈ Σ : x 1 · · · x n = J} in Σ. The term M * (µ) is called the Lyapunov exponent of {M i } ℓ i=1 with respect to µ. It also takes values in the set R ∪ {−∞}. The following variational principle for P was proved in [3] in a more general sub-additive setting: where h(µ) denotes the measure-theoretic entropy of µ with respect to σ (cf. [19]). We remark that (1.3) was proved earlier in [7,15] when the matrices are non-negative or invertible, respectively. For given q > 0, let Each element µ in I q is called a q-equilibrium state of P . Since both M * (·) and h(·) are upper semi-continuous on M σ (Σ), I q is a non-empty closed convex subset of M σ (Σ). In particular, I q contains ergodic elements (each extreme point of I q is an ergodic measure).
Our main purpose is to characterize the structure of I q . This question was partially raised from [16]. A complete characterization is given in Theorem 1.7. In the following, we shall present the setting and results. Proofs of the results are postponed until §2.
The above definition is adopted from [1, p. 48]. If {M i } ℓ i=1 is irreducible over F d , then there exist D > 0 and k ∈ N such that for any words I, J ∈ Σ * = ∞ n=1 {1, . . . , ℓ} n , there exists a word K in k n=1 {1, . . . , ℓ} n such that For a proof, see [8,Proposition 2.8]. This property is crucial in the proof of the following proposition.
is irreducible over F d , then for each q > 0, P has a unique q-equilibrium state µ q . Furthermore, µ q has the following Gibbs property: for all n ∈ N and J ∈ Σ n . Moreover, P is differentiable over (0, ∞) and P ′ (q) = M * (µ q ) for q > 0. Remark 1.3. Proposition 1.2 is an analogue of Bowen's theory about the equilibrium state of Hölder continuous additive potentials (cf. [2]). See [18,19] for backgrounds and more details about the classical thermodynamic formalism of additive potentials. Proposition 1.2 was first proved in [11] for non-negative matrices under a different irreducibility assumption (that is, there exists r ∈ N so that r i=1 (M 1 + · · · + M ℓ ) r is a strictly positive matrix). An extension was recently given in [9,Theorem 5.5] to certain sub-additive potentials.
Let us next consider the non-irreducibility case. Denote the n × m zero matrix by 0 n×m .
. . , t}, satisfy the following two properties: Considering the partition (1.7) in the above proposition, we set . . , ℓ} and j ∈ {1, . . . , t}. Hence M i 1 · · · M in = 0 d×d for all n > t by (1.8). To see the converse, assume contrarily that {A In the following, we always assume that The following is the main result of our paper. Theorem 1.7. In the above general setting, it holds that (ii) P is a real-valued convex function on (0, ∞), and P (q) = max{P j (q) : j ∈ Λ} for all q > 0. (iii) if q > 0 and µ j,q , j ∈ Λ, is the unique q-equilibrium state for P j , then here δ x denotes the Dirac measure at x), and µ = pµ 1 + (1 − p)µ 2 for some 0 < p < 1. It is easy to check that Remark 1.9. The pressure function for products of matrices has been studied in the literature under some stronger conditions. Let satisfies the strong irreducibility and contraction conditions (cf. [1,13]). Guivarc'h and Le Page showed in [13,Theorem 8.8] that the pressure function P of {M i } ℓ i=1 corresponds to the logarithm of the spectral radius of certain Ruelle transfer operator and moreover, P is real analytic on (0, ∞), and it can be extended to an analytic function on {z ∈ C : ℜz > 0}. This strengthens an early result of Le Page [17].

Proofs of the results
This section is dedicated to the proof of Theorem 1.7. For the convenience of the reader we shall also present complete proofs for Propositions 1.2 and 1.4.
Proof of Proposition 1.2. Let q > 0. Define a sequence of probability measures (ν n,q ) n≥1 on Σ so that ν n,q ([I]) = M I q J∈Σn M J q for all I ∈ Σ n . Let ν q be a limit point of the sequence (ν n,q ) n≥1 in the weak topology. Furthermore, let µ q be a limit point of the sequence in the weak topology. Using (1.5) and a proof essentially identical to that of [11,Theorem 3.2], we see that µ q ∈ M σ (Σ) is ergodic and has the Gibbs property (1.6). Thus Recalling (1.3), this implies µ q ∈ I q .
We remark that although [7, Theorem 1.2] only deals with non-negative matrices, the proof given there works for arbitrary matrices. Alternatively, to show that P ′ (q) = M * (µ q ), we may apply (1.6) and the ergodicity of µ q , and follow [14, proof of Theorem 2.1] (see also [16,Theorem 4.4]).
Proof of Proposition 1.4. We prove the proposition by induction on d. Clearly the proposition is true when d = 1. Assuming there exists an integer p so that the proposition is true for all d ≤ p, we show below that it remains true for d = p + 1. Let L(n, m) be the collection of all n × m matrices with entries in F.
we simply take t = 1 and have nothing else to prove. We may thus assume that Now by the induction assumption, there exist invertible matrices T 2 ∈ L(v, v) and have the desired partitioned form for all i ∈ {1, . . . , ℓ}. It follows that 3 D i T 3 has the desired partitioned form for all i ∈ {1, . . . , ℓ}.
Before proving Theorem 1.7, we shall first prove the following auxiliary result. Let (X, F , µ) be a probability space and T : X → X an ergodic measure-preserving transformation. Let {f n } ∞ n=1 be a sequence of non-negative Borel measurable functions on X such that sup x∈X f 1 (x) < ∞ and for all m, n ∈ N and x ∈ X. If ǫ > 0 and α = lim n→∞ (1/n) log f n dµ, then the following claims hold: (i) If α = −∞, then for µ-almost every x ∈ X, there exists a positive integer n 0 (x) such that for all n ≥ n 0 (x) and m ∈ N. (ii) If α = −∞, then for any N > 0 and µ-almost every x ∈ X, there exists a positive integer n 0 (x) such that for all n ≥ n 0 (x) and m ∈ N.
Proof. We only prove (i). The proof of (ii) is similar.

(2.4)
To see the opposite inequality, take k large enough such that |β − α| < δ, where Since {f n (x)} ∞ n=1 is sub-multiplicative, by [3, Lemma 2.2], we have for any n ≥ 2k, for all x ∈ X, where C = max{1, sup x∈Σ f 1 (x)}. It follows that for n ≥ 2k and m ∈ N, Applying the Birkhoff ergodic theorem to the function 1 k log f k , and combining it with the above inequality, we see that for µ-almost every x ∈ X, there exists an integer for all n ≥ñ 0 (x) and m ∈ N. This together with (2.4) yields (2.2).
As a direct corollary of Proposition 2.1, we have the following. Proof of Theorem 1.7. We only need to prove part (i), since parts (ii) and (iii) follow immediately from (i), the variational principle (1.3), and Proposition 1.2.
By Furstenberg-Kesten's theorem [12] on random matrices, or Kingman's subadditive ergodic theorem (see e.g. [19]), we have for µ-almost every n } ∞ n=1 of non-negative functions on Σ by setting For the rest of the proof, we take a point x = (x i ) ∞ i=1 ∈ Σ such that both (2.5) and (2.6) hold for x.

Extensions and remarks
For an invertible matrix M ∈ R d×d , following [4], we define the singular value function of M as where 0 ≤ q < d, k is the integral part of q, and α i (M) is the i-th largest singular value of M. For q > d, we put φ q (M) = | det(M)| q/d . It is known (see [4, Lemma 2.1]) that φ q is sub-multiplicative in the sense that for any two invertible matrices M 1 , M 2 ∈ R d×d . For a given family of invertible matrices {M i } ℓ i=1 ⊂ R d×d , similar to (1.1), we define Then by [15,Theorem 2.6], or more generally by [3, Theorem 1.1], we have the following variational principle P φ (q) = max{φ q * (µ) + h(µ) : µ ∈ M σ (Σ)}. Similarly we can study the structure of the equilibrium states of P φ (q). It is easy to see that Theorem 1.7 remains true for P φ (q) when 0 ≤ q ≤ 1 or q ≥ d − 1.
Observe also that it is true when q is an integer: if M ∧q is the q-th exterior product of M ∈ R d×d (i.e. the d q × d q matrix whose entries are the q × q minors of M), then α 1 (M ∧q ) = α 1 (M) · · · α q (M) = φ q (M).
This gives a partial answer to [16,Question 6.3].
We remark that some assumption was given in [5] so that an analogue of (1.5) (where · is replaced by φ q (·)) holds; and for such case, an analogue of Proposition 1.2 holds for P φ (cf. [9,Theorem 5.5]).