A noncommutative Ruelle's Theorem for a normalized potential taking values on positivity-improving operators

TL;DR

This paper extends Ruelle's thermodynamic formalism to a noncommutative setting involving positivity-improving operators on finite-dimensional C*-algebras, establishing a version of Ruelle's theorem and defining a new entropy concept.

Contribution

It introduces a noncommutative analogue of Ruelle's operator for positivity-improving operators, proving a version of Ruelle's theorem and defining a generalized entropy for Gibbs eigenstates.

Findings

Established a noncommutative Ruelle's theorem under positivity-improving assumptions.

Defined a new entropy concept for noncommutative Gibbs eigenstates.

Provided examples including quantum channels and matrix algebras.

Abstract

Let \(\mathcal{A}\) be a finite-dimensional real (or complex) C*-algebra, \(\Omega_{A}\) an aperiodic subshift of finite type, and \(\mathcal{C}(\Omega_{A}; \mathcal{A})\) the set of continuous functions from \(\Omega_{A}\) to \(\mathcal{A}\). The shift $\sigma$ provides dynamics. Given a real Lipschitz potential $\varphi\in \mathcal{C}(\Omega_{A}; \mathfrak{L}(\mathcal{A}))$, where $\mathfrak{L}(\mathcal{A})$ is the set of linear operators acting on a real $\mathcal{A}$, we introduce a noncommutative analogue of Ruelle's operator, which acts on \(\mathcal{C}(\Omega_{A}; \mathcal{A})\). Assuming the positivity-improving hypothesis, we prove a version of Ruelle's Theorem whenever the operator is normalized. An eigenstate (a linear functional) invariant for the action of the noncommutative Ruelle's operator will play the role of the Gibbs probability of Thermodynamic Formalism; to be called a Gibbs eigenstate. We introduce the concept of entropy for a Gibbs eigenstate (obtained from a certain family of potentials $\varphi$) - generalizing the classical one. In our setting, there is currently no direct relationship with cocycles and Lyapunov exponents. We present examples illustrating the novelty of the cases that can be considered, ranging from topics related to quantum channels to Pauli matrices. Interesting cases: $\mathcal{A}=M_{N \times N}(\mathbb{R})$ and $\mathcal{A}= \mathbb{R}^{N}$.

Equations

$$\eta :\mathcal{C}({\left\{1,2\right\}}^{\mathbb{N}};M_{2\times 2}(\mathbb{R}))\to \mathbb{R}.$$

$$g\mapsto \eta (g):=\int \widehat{\mathrm{tr}}(g(x))d\mu (x)\text{,}$$

$${\Omega }_A:=\left\{(x_n{)}_{n\in \mathbb{N}}|x_n\in \Sigma ,A(x_n,x_{n+1})=1,\forall n\in \mathbb{N}\right\}\text{.}$$

$${\left|g\right|}_{\infty }:=\underset{x\in {\Omega }_A}{\sup }{\left|g\right|}_X\text{,}$$

$$d_{\theta }(x,y)$$

$$N(x,y)$$

$${\left|g\right|}_{\theta }:=\underset{x\ne y\in {\Omega }_A}{\sup }\frac{{\left|g(x)-g(y)\right|}_X}{d_{\theta }(x,y)}\text{,}$$

$${\Vert g\Vert }_{\theta }:={\left|g\right|}_{\infty }+{\left|g\right|}_{\theta }\text{.}$$

$$\sigma (x)=\sigma ((x_n{)}_{n\in \mathbb{N}}):=(x_{n+1}{)}_{n\in \mathbb{N}}\text{.}$$

$$\left(L_{\varphi}g\right)(x):=\sum_{y\in f^{-1}(x)}\varphi_{y}\big{(}g(y)\big{)}\text{.}$$

$$L_{\phi }({\mathbb{I}}_{\mathcal{C}})={\mathbb{I}}_{\mathcal{C}}\text{,}$$

$$\sum _{\begin{array}{c}i\in \Sigma \\ A(i,x_1)=1\end{array}}{\phi }_{ix}{\mathbb{I}}_{\mathcal{A}}={\mathbb{I}}_{\mathcal{A}}\text{,}\forall x\in {\Omega }_A\text{.}$$

$${\phi }_{(1,y)}=\left(\widehat{\mathrm{tr}}(\cdot )\right)A_1\text{and}{\phi }_{(2,y)}=\left(\widehat{\mathrm{tr}}(\cdot )\right)A_2,$$

$$A_1+A_2=I.$$

$$g\mapsto [y\to f(y)=L_{\phi }(g)(y)]=[y\to \frac{1}{2}(\widehat{\mathrm{tr}}g(1,y))A_1+\frac{1}{2}(\widehat{\mathrm{tr}}g(2,y))A_2]\text{,}$$

$$g\mapsto L_{\phi }(g)(y)=\frac{1}{2}(\widehat{\mathrm{tr}}g(1,y))A_1+\frac{1}{2}(\widehat{\mathrm{tr}}g(2,y))A_2\text{.}$$

$$\phi (x)=\left[\begin{array}{cc}1/4& 1/4\\ 1/4& 1/4\end{array}\right]$$

$$g\to L_{\phi }(g)(y)=\left[\begin{array}{cc}1/4& 1/4\\ 1/4& 1/4\end{array}\right](g_1(1x)+g_2(1x)+g_1(2x)+g_2(2x)).$$

$${\phi }_x(g(x))=\left(\widehat{\mathrm{tr}}g(x)\right)P(x),$$

$$\sum _{\begin{array}{c}i\in \Sigma \\ A(i,x_1)=1\end{array}}P(ix)=\mathrm{Id}\text{,}$$

$$\eta (g):=\int \widehat{\mathrm{tr}}g(x)d{\mu }_{\widehat{\mathrm{tr}}P}(x).$$

$$\eta (L_{\phi }g)$$

$$=\int \widehat{\mathrm{tr}}\left(\sum _{\begin{array}{c}i\in \Sigma \\ A(i,y_1)=1\end{array}}\left(\widehat{\mathrm{tr}}g(iy)\right)P(iy)\right)d{\mu }_{\widehat{\mathrm{tr}}P}(y)$$

$$=\int \sum _{\begin{array}{c}i\in \Sigma \\ A(i,y_1)=1\end{array}}\left(\widehat{\mathrm{tr}}g(iy)\right)\left(\widehat{\mathrm{tr}}P(iy)\right)d{\mu }_{\widehat{\mathrm{tr}}P}(y)$$

$$\displaystyle=\int\Big{(}L_{\widehat{\operatorname{tr}}\,P}\widehat{\operatorname{tr}}\,g\Big{)}(y)\operatorname{d}\!\mu_{\widehat{\operatorname{tr}}\,P}(y)\text{.}$$

$$=\int \widehat{\mathrm{tr}}g(x)d{\mu }_{\widehat{\mathrm{tr}}P}(x)=\eta (g)$$

$$g:\{1,2{\}}^{\mathbb{N}}\to M_{2\times 2}(\mathbb{R}).$$

$$P=\left[\begin{array}{cc}{p}_{11}& p_{12}\\ p_{21}& p_{22}\end{array}\right]\text{, where }{p}_{ij}>0\text{.}$$

$$P_1=\left[\begin{array}{cc}\sqrt{p_{11}}& 0\\ 0& \sqrt{p_{21}}\end{array}\right]\text{,}{P}_2=\left[\begin{array}{cc}\sqrt{p_{12}}& 0\\ 0& \sqrt{p_{22}}\end{array}\right]\text{,}$$

$$K=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\otimes P_1+\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\otimes P_2\text{.}$$

$$x\in \Omega \to a(x)=\left[\begin{array}{cc}{a}_{11}(x)& a_{12}(x)\\ a_{21}(x)& a_{22}(x)\end{array}\right]\in M_{2\times 2}(\mathbb{R}),$$

$$\epsilon (a\otimes \mathrm{Id}):={\mathrm{Tr}}_1\left(K^{\ast }(a\otimes \mathrm{Id})K\right)$$

Full text

AMS 2010 Classification Number - 37D35; 94A40

1 Introduction

Ruelle’s Theorem plays an important role in the celebrated thermodynamic formalism approach to classical statistical mechanics [45, 21, 33, 34, 41, 54]. It provides us with a very useful global description of the equilibrium states. These states are traditionally represented by probability measures on the configuration space. We denote by $\sigma$ the shift and by $\Omega_{A}\subset\{1,2,...,k\}^{\mathbb{N}}$ a subshift of finite type (see Definition 1). Such stationary probabilities for the shift describe the long-term statistical behavior of the system under a given interaction potential $B:\Omega_{A}\to\mathbb{R}$, that is, the likelihood of observing particular configurations in the thermodynamic regime. The potential $B$ here corresponds to the negative of the Hamiltonian of Statistical Mechanics.

Since we are primarily interested in obtaining results involving real quantities, such as entropy, etc., we chose to describe the model using a real $C^{*}$-algebra, but the formulation using a complex $C^{*}$-algebra would also be compatible with our reasoning.

Unlike the classical approach, commonly used in statistical mechanics, which analyzes finite subsystems and then takes a thermodynamic limit, Ruelle’s Theorem identifies equilibrium states directly; Kolmogorov-Sinai entropy for the shift plays an important role. It characterizes them as the product of an eigenfunction and an eigenmeasure of a positive linear operator on continuous functions, known as the transfer (or Ruelle) operator. This Theorem also helps to establish an associated variational principle, according to which the topological pressure is given by the Legendre–Fenchel transform of the Kolmogorov-Sinai entropy (which is an affine function). It is well known that the study of equilibrium states for Hölder potentials defined on $\{1,2,...,k\}^{\mathbb{N}}$ helps to analyze equilibrium states for Hölder potentials defined on $\{1,2,...,k\}^{\mathbb{Z}}$ (see Section 1 in [41]).

Ruelle’s Theorem for a potential $B:\Omega_{A}\to\mathbb{R}$ is a dynamical counterpart of the Perron-Frobenius Theorem, which, as noted in [24], may be regarded as concerning the spectral properties of positive linear operators on finite-dimensional commutative C*-algebras.

To clarify this interpretation, recall, from Gelfand duality [14, Theorem 2.1.11], that the finite-dimensional commutative C*-algebras are precisely those of the form $\mathbb{R}^{n}$. For a linear self-mapping of $\mathbb{R}^{n}$, being positive is equivalent to its matrix having positive entries. Note that we are referring here to the notion of positivity in the sense of preserving a positive cone, not to the notion of (semi-)definite positivity of a linear operator on a Hilbert space. This is particularly important regarding the linear self-mappings of $\mathbb{R}^{n}$, since this set can be seen both as a finite-dimensional commutative C*-algebra and as a Hilbert space. Note also that both the matrix which represents a linear self-mapping of $\mathbb{R}^{n}$ and the positive cone $\mathbb{R}^{+n}$ depend on the choice of a basis.

Extending this interpretation, we may regard Ruelle’s Theorem as replacing finite-dimensional commutative real C*-algebras by infinite-dimensional real commutative C*-algebras of the form $\mathcal{C}(\Omega_{A};\mathbb{R})$, and positive matrices by transfer operators. This observation is the starting point for the version of Ruelle’s Theorem we are going to prove here.

It is also relevant to note that there are many other successful versions of the Perron-Frobenius theory that have been proved for different settings; see, for instance [36, 30, 23, 50, 40, 18, 56, 26, 12, 1, 37].

It is natural to look for noncommutative analogues of Ruelle’s Theorem, which can be eventually used in applications in quantum statistical mechanics (see, for instance [24], [9], [47], [12], [13], [35] or [31]). Perhaps the most successful example was Matsui’s quantum channel formalism, inaugurated in [36]. There, it is shown that an analogue of Ruelle’s Theorem holds for a wide class of transfer operators on UHF111Uniformly HyperFinite (see [49, 22, 44]). For example, the CAR algebra. algebras, which are a family of noncommutative C*-algebras associated with infinite chains of quantum spin particles; our setting is somewhat different.

In the present paper, we prove a version of Ruelle’s Theorem for a class of finite-dimensional eventually noncommutative C*-algebras different from those of [36]. More precisely: for the class of C*-algebras of continuous functions of aperiodic subshifts of finite type222Sometimes called topological Markov shifts. into finite-dimensional real $C^{*}$-algebras. In order to do so, we borrow several definitions and techniques from [41, 54, 21, 34, 33, 53].

In the classical Ruelle theorem for a positive Lipschitz potential $B:\Omega_{A}\to(0,\infty)$, the interaction potential enters the transfer operator through a strictly positive scalar function in exponentiated form. This strict positivity of $B$ plays a fundamental role: it ensures that the transfer operator eventually maps non-zero positive functions to strictly positive ones, thereby enabling uniqueness and convergence results for eigenfunctions and eigenmeasures. This is an important issue to consider when attempting to obtain noncommutative generalizations of classical theory.

In our noncommutative setting, we sought a natural generalization of this structure. While it is tempting to consider positive or even completely positive operator-valued potentials — the latter being a widely accepted generalization of scalar positivity [19, 20, 46, 48, 5, 6] — these notions alone are insufficient for our purposes. In particular, they do not guarantee that the transfer operator satisfies the conditions needed in [29, Theorem 4.1, hypothesis (a)], which require a form of strict positivity.

We therefore impose the alternative condition that the potential takes values in positivity-improving linear mappings: that is, mappings which send every non-zero positive element to a strictly positive one. Notice that this condition is stronger than positivity, but does not imply nor is implied by complete positivity. However, it captures the same essential structure present in the classical case, and enables us to reproduce the key features of the Ruelle Theorem in this more general, noncommutative context. Quantum channels provide examples of linear transformations acting on the set of matrices and preserving the cone of positive matrices (see [39]). Explicit examples of positivity-improving linear mappings acting on the space of $n$ by $n$ matrices appear in Section 8 in [12]. We will consider here functions defined on the symbolic space $\Omega_{A}$ taking values in positivity-improving linear mappings.

To fully understand the significance of the results obtained here, it is necessary to describe first some aspects of the classical Ruelle Theorem as described in [41]. Consider a Hölder (or Lipschitz) positive potential $J:\Omega_{A}\to(0,\infty)\subset\mathbb{R}$, and the associated Ruelle operator $\mathcal{L}_{J}$, which acts on continuous functions $f:\Omega_{A}\to\mathbb{R}$ via $\mathcal{L}_{J}(f)(x)=\sum_{\sigma(y)=x}J(y)f(y).$ The operator $\mathcal{L}_{J}$ preserves the set of continuous positive functions. We say that $J$ is normalized if $\mathcal{L}_{J}(1)=1$; in this case, let $\mathcal{L}_{J}^{*}$ be the dual operator, which acts on the set of probabilities. A probability $\mu_{J}$ that satisfies the fixed-point property for $\mathcal{L}_{J}^{*}$ is called a Gibbs probability (sometimes called Gibbs state) for the normalized potential $B$. The probability $\mu_{J}$ is an eigenprobability for the operator $\mathcal{L}_{J}^{*}$ associated with the eigenvalue $1$. The set of probabilities in this class is invariant for the shift action and is one of the main focuses of research in thermodynamic formalism. They describe the possible equilibrium states for Hölder (or Lipschitz) potentials defined on $\{1,2,...,k\}^{\mathbb{N}}$. The maximal entropy measure $\mu$ is the fixed point for the action of the operator $\mathcal{L}_{\frac{1}{k}}^{*}$.

We will denote by $\mathfrak{L}(\mathcal{A})$ the set of linear operator acting on the finite dimensional C*-algebra $\mathcal{A}$. Given a Lipschitz potential $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$, we will introduce a noncommutative analogue of the Ruelle operator, which will be denoted by $L_{\varphi}$; we will demonstrate, in a noncommutative setting, the existence of the analog of Hölder Gibbs states, and these will be denoted generically by linear operators $\eta$, which will be called eigenstates. In some examples (see sections 3 and 7), we will explicitly exhibit such an eigenstate operator $\eta$; in some of them, a classical Lipschitz equilibrium probability (here called Gibbs probability) will help to express the precise form of this eigenstate $\eta$. We will assume that the noncommutative Ruelle operator is normalized (see Definition 27). An important issue, exemplified in Example 46, is the fact that scalar probability measures are insufficient to describe equilibrium in the noncommutative setting. The variety of examples one can get ranges from topics related to quantum channels to Pauli matrices (see [39], [10], [12] or [13] for definitions). In Example 33 the potential $\varphi:\{1,2\}^{\mathbb{N}}\to\mathcal{L}(M_{2\times 2}(\mathbb{R}))$ is given by a fixed quantum channel acting on the family of continuous functions $g:\{1,2\}^{\mathbb{N}}\to M_{2\times 2}(\mathbb{R})$. In a particular case one can take $\mathcal{A}=\mathbb{R}^{N}$ and $\varphi$ of the form $\varphi:\Omega_{A}\to M_{N\times N}(\mathbb{R})$ (see Example 30).

Given the normalized noncommutative Ruelle operator $L_{\varphi}$, the associated eigenstate $\eta$ will be the one such that is invariant for the action of this Ruelle operator (as for instance in (7)). One can show that such $\eta$ also satisfy a form of shift-invariance (see (38) in Section 5). It would be natural to call such a state of Gibbs eigenstate.

We believe that at this point an explicit example will help to understand the generality of what is being proposed here (we will elaborate on more details in Example 46). This is perhaps the simplest of all possible examples. Take $\mathcal{A}=M_{2\times 2}(\mathbb{R})$ and denote a general matrix-valued function by $g\in\mathcal{C}(\Omega_{A};M_{2\times 2}(\mathbb{R}))$. Given a normalized Lipschitz potential $\varphi$ taking values in linear operators $M_{2\times 2}(\mathbb{R})\to M_{2\times 2}(\mathbb{R})$, we can introduce a normalized noncommutative Ruelle operator $L_{\varphi}$, which will act on functions $g$ of the above form. A state $\eta$ acting of functions $g$ is a linear functional:

[TABLE]

For the case of the noncommutative Ruelle operator $L_{\varphi}$ described in Example 46, the associated eigenstate $\eta$ is defined by:

[TABLE]

where $\mu$ denotes the maximal entropy measure for the action of the full shift on on $\{1,2\}^{\mathbb{N}}$, and $\widehat{\operatorname{tr}}:=\frac{1}{2}\operatorname{tr}$ is the normalized trace. We may call such $\eta$ of Gibbs eigenstate.

In the classical setting, one way to obtain an approximation of the Gibbs state $\mu_{J}$ for the potential $J$, would be to take the limit in $n$ of the iterations of the dual operator $\mathcal{L}_{J}^{*}$. More precisely, given any probability $\rho$, then $\mu_{J}=\lim_{n\to\infty}(\mathcal{L}_{J}^{*})^{n}(\rho)$. In fact, one can show that the action of $\mathcal{L}_{J}^{*}$ is a contraction for the $1$-Wasserstein distance (see [28]). Our main theorem will ensure that one can obtain the eigenstate $\eta$ as a limit in a similar procedure; we will work out some examples.

In Sections 3 and 7, we will present examples that illustrate the novelty of our results. In Section 4, we will present the proof of our main theorem, which is a noncommutative version of the Ruelle Theorem. In Section 5 we will introduce a form of entropy for eigenstates associated with a certain family of normalized potentials, the so-called trace-type potentials (see Definitions 38, 42 and Theorem 43). In Section 6, at the end of the paper, we will comment on similarities and dissimilarities with some previous works. We claim that, to the best of our knowledge, our results are not included in the statements of other works in the literature.

The analysis of the case when $\mathcal{A}$ has infinite dimension introduces several technical difficulties and will be left for future work.

We note that our results are of a different nature from those of the linear cocycle and Lyapunov exponents theory, and that they are described in a vast literature; it is worth listing here some important examples: [2], [11], [7], [8], [25], [52] and [38]. f We do not rule out the possibility that in some future work, Lyapunov exponents and Oseledet’s theorem may be relevant for obtaining results to the present setting.

The concept of entropy described here, associated with the use of a normalized generalized Ruelle operator, is quite distinct from those considered in other texts regarding C*-algebras, such as the following: [3], [4], [43], [17], [35] and [27].

2 Preliminaries and definitions

Let $\Sigma=\left\{1,2,\cdots,k\right\}$ and $A:\Sigma\times\Sigma\to\left\{0,1\right\}$ be a matrix of zeroes and ones. We call elements $i,j\in\Sigma$ symbols and $A$ the transition matrix. When $A(i,j)=1$ we say that $(i,j)$ is allowed and when $A(i,j)=0$ we say that $(i,j)$ is forbidden. In what follows, $A$ is always taken to be aperiodic.

Definition 1**.**

The subshift of finite type $\Omega_{A}$ is given by:

[TABLE]

It is the set of all unilaterally infinite sequences on $k$ symbols with transitions allowed by $A$.

Definition 2**.**

We consider $\Omega_{A}$ to be equipped with the product topology333Sometimes called Tychonoff product topology. and denote the set of continuous functions from $\Omega_{A}$ to a topological space $X$ by $\mathcal{C}(\Omega_{A};X)$.

Definition 3**.**

Let $X$ be a Banach space. We denote its norm by $\left|\cdot\right|_{X}$. We also consider $\mathcal{C}(\Omega_{A};X)$ to be equipped with the uniform norm, and denote it by:

[TABLE]

which makes it a Banach space. Therefore, we mildly abuse notation, since $\left|\cdot\right|_{\mathcal{C}(\Omega_{A};X)}:=\left|\cdot\right|_{\infty}$. In each case, $X$ will always be clear from context whenever we employ the notation $\left|\cdot\right|_{\infty}$.

Definition 4**.**

Let $X$ be a Banach space. We denote its topological dual space by $X^{\prime}$. We also consider $X^{\prime}$ to be equipped with the weak norm, which makes it a Banach space.*

Definition 5**.**

Let $X,Y$ be two Banach spaces, $T:X\to Y$ a bounded linear mapping. We denote the transpose of $T$ by $T^{\prime}:Y^{\prime}\to X^{\prime}$. It is also a bounded linear mapping.

Definition 6**.**

Let $0<\theta<1$ be an arbitrary real parameter. The metric $d_{\theta}:\Omega_{A}\times\Omega_{A}\to\mathbb{R}^{+}$ is given by:

[TABLE]

Remark 7**.**

The metric $d_{\theta}$ induces the product topology for any $0<\theta<1$, and $\Omega_{A}$ is a compact set..

Definition 8**.**

Whenever $X$ is a Banach space, we denote the set of Lipschitz functions from $\Omega_{A}$ to $X$ by $\mathcal{C}_{\theta}(\Omega_{A};X)\subseteq\mathcal{C}(\Omega_{A};X)$. We consider it to be equipped with the Lipschitz seminorm:

[TABLE]

and the Lipschitz norm:

[TABLE]

Remark 9**.**

When $X$ is finite-dimensional, the Arzelà-Ascoli Theorem ensures that every bounded subset of $\left(\mathcal{C}_{\theta}(\Omega_{A};X),\left\lVert\cdot\right\rVert_{\theta}\right)$ is relatively compact in $\left(\mathcal{C}(\Omega_{A};X),\left|\cdot\right|_{\infty}\right)$.

Definition 10**.**

The shift mapping $\sigma:\Omega_{A}\to\Omega_{A}$ is given by:

[TABLE]

Remark 11**.**

The shift mapping is expanding with respect to the metric $d_{\theta}$ for any $0<\theta<1$. The expanding parameters may be taken $\theta^{-1}>1$ and $\theta>0$. Since $A$ is aperiodic, the shift mapping is also topologically mixing.

Remark 12**.**

Let $\mathcal{A}$ be a C-algebra. Then $\mathcal{C}(\Omega_{A};\mathcal{A})$ is also a C*-algebra. It is commutative if, and only if, $\mathcal{A}$ is. It is unital if, and only if, $\mathcal{A}$ is. Notice it is infinite-dimensional except possibly if $k=1$, when it is isomorphic to $\mathcal{A}$, which may be finite-dimensional. The reader should be mindful of this through the following definitions.*

Definition 13**.**

Let $\mathcal{A}$ be a real C-algebra. We say that an element $a\in\mathcal{A}$ is positive when either of the following equivalent conditions hold:*

$a=b^{\ast}b$* for some $b\in\mathcal{A}$.* 2. 2.

$a=a^{\ast}$* and $\operatorname{spec}(a)\subseteq[0,+\infty)$.*

In such a case, $b\in\mathcal{A}$ can be chosen $b=\left|a\right|=\sqrt{a^{\ast}a}$, which is also positive, taking into account the functional calculus [15, Section 4.6.1]. Then, we write $a\geq 0$.

Definition 14**.**

The set of all $\mathcal{A}\ni a\geq 0$ is called the positive cone of $\mathcal{A}$, and written $\mathcal{A}^{+}$.

Remark 15**.**

The positive cone of $\mathcal{C}(\Omega_{A};\mathcal{A})$, written $\mathcal{C}(\Omega_{A};\mathcal{A})^{+}$, is equal to the set of continuous functions from $\Omega_{A}$ to $\mathcal{A}^{+}$, written $\mathcal{C}(\Omega_{A};\mathcal{A}^{+})$.

Definition 16**.**

Let $\mathcal{A}$ be a unital C-algebra. We write $\mathbb{I}_{\mathcal{A}}$ for the multiplicative unit of $\mathcal{A}$. To simplify the notation, we also write $\mathbb{I}_{\mathcal{C}(\Omega_{A};\mathcal{A})}:=\mathbb{I}_{\mathcal{C}}$. Notice this is merely the constant function $\mathbb{I}_{\mathcal{C}}:\Omega_{A}\to\mathcal{A}$, that maps every point $x\in\Omega_{A}$ to the multiplicative unit of $\mathcal{A}$, that is: $\mathbb{I}_{\mathcal{C}}\equiv\mathbb{I}_{\mathcal{A}}$, or $x\overset{\mathbb{I}_{\mathcal{C}}}{\longmapsto}\mathbb{I}_{\mathcal{A}}$.*

Definition 17**.**

Let $\mathcal{A}$ be a unital C-algebra. We say that an element $a\in\mathcal{A}$ is strictly positive when there exists $\mathbb{R}\ni\varepsilon>0$ such that $a\geq\varepsilon\mathbb{I}_{\mathcal{A}}$. Then, we write $a>0$.*

Remark 18**.**

The set of all $\mathcal{A}\ni a>0$ is the interior of the positive cone of $\mathcal{A}$, and therefore written $\mathcal{A}^{+^{\circ}}$.

Remark 19**.**

Unless $\mathcal{A}=\mathbb{R}$, the conjunction $a\in\mathcal{A}^{+}$ plus $a\neq 0$ does not imply $a\in\mathcal{A}^{+^{\circ}}$.

Remark 20**.**

However, the conjunction $a\in\mathcal{A}^{+}$ plus there exists $a^{-1}$ such that $a^{-1}a=\mathbb{I}_{\mathcal{A}}$ does imply $a\in\mathcal{A}^{+^{\circ}}$. Therefore, the strictly positive elements of a unital C-algebra $\mathcal{A}$ are exactly the positive units444 Units are elements possessing a multiplicative inverse. of $\mathcal{A}$.*

Definition 21**.**

Let $\mathcal{A}$ be a real $C^{*}$-algebra. We denote the set of linear mappings from $\mathcal{A}$ to itself by $\mathfrak{L}(\mathcal{A})$. We consider it to be equipped with the operator norm.

Definition 22**.**

Let $\mathcal{A}$ be a real C-algebra. We say that a linear mapping $\varphi\in\mathfrak{L}(\mathcal{A})$ is positivity-improving when $\varphi(a)>0$ for any $\mathcal{A}^{+}\ni a\neq 0$.*

Remark 23**.**

The composite of positivity-improving linear mappings is a positivity-improving linear mapping.

Remark 24**.**

The sum of positivity-improving linear mappings is a positivity-improving linear mapping.

Definition 25**.**

Let $\Omega_{A}$ be a subshift of finite type and $\mathcal{A}$ a finite-dimensional real C-algebra. We say that a function $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$, written $x\mapsto\varphi_{x}$, is a potential when it takes values in positivity-improving linear mappings and is Lipschitz-continuous with respect to the operator norm.*

Definition 26**.**

Let $\Omega_{A}$ be a subshift of finite type, $\mathcal{A}$ a real finite-dimensional C-algebra, and $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$ a potential. The transfer operator $L_{\varphi}$ acts on continuous functions $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$ by:*

[TABLE]

Definition 27**.**

Let $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$ be a potential. We say that $\varphi$ is normalized if it is such that:

[TABLE]

or, equivalently:

[TABLE]

Remark 28**.**

Equation (2) plus the positivity of every $\varphi_{x}$ allows us to conclude that $0\leq\varphi_{x}(\mathbb{I}_{\mathcal{A}})\leq\mathbb{I}_{\mathcal{A}}$. Also, [42, Corollary 2.9 (Russo-Dye)] applies. Therefore, $\left\lVert\varphi_{x}\right\rVert\leq 1$ for any $x\in\Omega_{A}$.

We will provide a simple example to illustrate the setting we consider here.

Example 29**.**

Let $\mathcal{A}=M_{2\times 2}(\mathbb{R})$, $\Omega=\{1,2\}^{\mathbb{N}}$, and $\sigma$ the full shift. Consider a Lipschitz function $\varphi:\Omega=\{1,2\}^{\mathbb{N}}\to\left[M_{2\times 2}(\mathbb{R})\to M_{2\times 2}(\mathbb{R})\right]=\mathfrak{L}(\mathcal{A})$ , such that there exist strictly positive matrices $A_{1},A_{2}:\Omega\to M_{2\times 2}(\mathbb{R})$, satisfying for any $y\in\Omega$

[TABLE]

[TABLE]

The corresponding transfer operator $L_{\varphi}$ acts on a general continuous function $g:\Omega\to M_{2\times 2}(\mathbb{R})$, in such way that $L_{\varphi}(g)=f$, via

[TABLE]

which means in a short notation

[TABLE]

Note that the potential $\varphi$ is normalized and positivity-improving. In this case, we will be interested in the state $\eta:\mathcal{R}(\Omega_{A};M_{2\times 2}(\mathbb{R}))\to\mathbb{R}$ that plays the role analogous to the eigenprobability for the dual of the Ruelle operator. In (31) and (32) in Example 49 we will elaborate on this issue.

Example 30**.**

Take $\mathcal{A}=\mathbb{R}^{N}$. Therefore, we have to consider potentials of the form $\varphi:\Omega_{A}\to M_{N\times N}(\mathbb{R})=\mathfrak{L}(\mathcal{A})$, which will act on functions $g\in\mathcal{C}(\Omega_{A};\mathbb{R}^{N})$. We present several examples of such kind in Subsection 7.4. The most simple one is to take $\Omega=\{1,2\}^{\mathbb{N}}$, and $\sigma$ the full shift. The multiplicative element in $\mathcal{A}=\mathbb{R}^{2}$ is the vector $(1,1)$, therefore the constant potential, $\forall x$

[TABLE]

is normalized. Given $x\in\Omega\to g(x)=(g_{1}(x),g_{2}(x))$, we get

[TABLE]

More details will be presented in Subsection 7.4.

Our main result is:

Theorem 31**.**

Let $\Omega_{A}$ be a subshift of finite type, $\mathcal{A}$ a finite-dimensional C-algebra, and $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$ a Lipschitz-continuous normalized potential. Then:*

$1\in\mathbb{R}$* is a simple isolated eigenvalue of $L_{\varphi}$.* 2. 2.

All positive eigenfunctions of $L_{\varphi}$ are positive multiples of $\mathbb{I}_{\mathcal{C}}$. 3. 3.

1. (a)

There is a unique state $\eta:\mathcal{C}(\Omega_{A};\mathcal{A})\to\mathbb{R}$ such that $L_{\varphi}^{\prime}(\eta)=\eta$. It is faithful. 2. (b)

For any $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$, the sequence $L_{\varphi}^{n}(g)$ converges uniformly to $\eta(g)\mathbb{I}_{\mathcal{C}}$. 4. 4.

The remainder of the spectrum of $L_{\varphi}$, that is, $\operatorname{spec}(L_{\varphi})\setminus\left\{1\right\}$, is contained in a disk in the complex plane of radius strictly smaller than $1$.

It is natural to call the $\eta$ mentioned above in 3. (a) a Gibbs eigenstate.

We present a few examples in the next section. We believe they will help to better understand the theoretical results being presented. More examples will be described at the end of the paper in Section 7.

3 Some examples

Example 32**.**

Let $\Omega_{A}\subseteq\left\{1,2,\cdots,k\right\}^{\mathbb{N}}$ be any aperiodic subshift of finite type and $\mathcal{A}=M_{d\times d}(\mathbb{R})$. We will define next a Lipschitz potential $\varphi$. First, for each $x\in\Omega_{A}$ consider $\varphi_{x}:M_{d\times d}(\mathbb{R})\to M_{d\times d}(\mathbb{R})$ such that:

[TABLE]

where $P(x)=(P_{i,j})_{i,j=1,...,d}(x)$ is a family of strictly positive matrices (so that each $\varphi_{x}$ is positivity-improving). Suppose that $\varphi$ is normalized, that is:

[TABLE]

for all $x\in\Omega_{A}$.

Now, we consider the potential $\widehat{\operatorname{tr}}\,P$ in the classical sense of [41]. That is, $\widehat{\operatorname{tr}}\,P(x)$ is equal to $\frac{1}{d}\sum_{j=1}^{d}P_{jj}(x).$

Condition (5) implies that the scalar potential $\widehat{\operatorname{tr}}\,P$ is classically normalized. Denoting by $\mu_{\widehat{\operatorname{tr}}\,P}$ the classical equilibrium probability measure for the classical potential $\widehat{\operatorname{tr}}\,P$, we claim that the eigenstate $\eta:\mathcal{C}(\Omega_{A};M_{d\times d}(\mathbb{R}))\to\mathbb{R}$, described in item 3. (a) in Theorem 31, is given by:

[TABLE]

Indeed,

[TABLE]

[TABLE]

As we will see, this example is related to Example 48.

Example 33**.**

The present example takes into account the reasoning of [9, Example 4], [12, Example 8.5] or [32, Section 2.7]. We consider $\Omega=\{1,2\}^{\mathbb{N}}$ and $\mathcal{A}=M_{2\times 2}(\mathbb{R})$.

Before defining the potential $\varphi$, we will consider a certain operator $a\to\varepsilon(a\otimes\operatorname{Id})$, which is a particular case of a quantum channel (see [39]). Later, we will consider the potential $\varphi$ such that $g\to\varepsilon(g\otimes\operatorname{Id})=\varphi(g)$. In summary, the potential $\varphi$ is given by a fixed quantum channel acting on the family of continuous functions

[TABLE]

The explicit expression for the associated eigenstate of the present example is given by (15). The action of the noncommutative Ruelle operator obtained from a general constant potential is described in Subsection 7.1 (see also (24)).

Consider a given line-stochastic matrix $P\in M_{2\times 2}(\mathbb{R})$, such that:

[TABLE]

Define:

[TABLE]

and:

[TABLE]

Notice that $K=K^{*}$.

Let

[TABLE]

and denote by $\operatorname{Id}$ the identity matrix in $M_{2\times 2}(\mathbb{R})$. We will omit the dependence on $x\in\Omega$ in the next computations. We wish to consider a potential of the form $a\mapsto\varepsilon(a\otimes\operatorname{Id})$, where:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

If $a$ is a positive matrix, then $a_{11},\,a_{22}\geq 0$, and whenever $a\neq 0$, at least one of them must be strictly positive. Since both $P_{1}^{2}$ and $P_{2}^{2}$ are strictly positive, the linear mapping $a\mapsto\varepsilon(a\otimes\operatorname{Id})$ is positivity-improving, for each $x\in\Omega$. Also, since $P$ was taken line-stochastic, the linear mapping $a\mapsto\varepsilon(a\otimes\operatorname{Id})$ is unital.

There exists a unique vector of probability $\pi=(\pi_{1},\pi_{2})$ such that $\pi P=\pi.$ In this way,

[TABLE]

and

[TABLE]

In order to be consistent with the previous notation of Theorem 31, applied to the present case, note that the Ruelle operator acts on functions $g:\Omega\to M_{2\times 2}$, and by abuse of notation let’s write $g(a)(x)=a(x).$ Then, the noncommutative Ruelle operator is

[TABLE]

We want to find the eigenstate $\eta$ for the Ruelle operator $L_{\varphi}$. First, we consider the state $\xi:M_{2\times 2}\to\mathbb{R}$ such that when applied to a matrix $a(x).$

[TABLE]

$\xi:M_{2\times 2}\to\mathbb{R}$* is the eigenstate for $a\in M_{2\times 2}\to\varepsilon(a\otimes\operatorname{Id})$. Indeed, for any matrix $a\in M_{2\times 2}$, from (9) and (10)*

[TABLE]

[TABLE]

It follows that for any $x$,

[TABLE]

In order to be consistent the previous notation of Theorem 31, we write the above expression as

[TABLE]

Now we want to explicit the eigenstate $\eta$ for the Ruelle operator $L_{\varphi}$. From Subsection 7.1 it follows that the eigenstate $\eta$ for the transfer operator associated with the constant potential $g\mapsto\varepsilon(g\otimes\operatorname{Id})$ is given by simply integrating $\xi$ against the maximal entropy measure $\mu$; if $g\in\mathcal{C}(\Omega;M_{2\times 2}(\mathbb{R}))$:

[TABLE]

4 Proof of Theorem 31

It will be of fundamental importance in our reasoning the use of two results, which are stated below for the convenience of the reader. First, the Ionescu-Tulcea and Marinescu Theorem, as given in [51, Theorem B.1.1].

Theorem** (Ionescu-Tulcea and Marinescu).**

Let $E$ be a complex linear (i.e., vector) space and $B$ a linear subspace of $E$. Suppose that $E$ and $B$ are Banach spaces when equipped with the norms $|\cdot|$ and $\|\cdot\|$, respectively. Suppose also that:

• (a1)

If a sequence $(x_{n})_{n=1}^{\infty}$ in $B$ is such that there exists a constant $K>0$ with:

[TABLE]

then:

[TABLE]

Furthermore, let $T:B\to B$ be a bounded linear operator satisfying the following conditions:

• (b1)

There exists a constant $H>0$ such that $|T^{n}|\leq H$ for all $n\in\mathbb{N}$.

• (c1)

There exist $N\in\mathbb{N}$ and constants $0<r<1$ and $R>0$ such that:

[TABLE]

• (d1)

Every bounded subset of $B$ is mapped by $T$ to a relatively compact subset of $E$. That is, the operator $T:(B,\|\cdot\|)\to(E,|\cdot|)$ is compact.

Then:

• (a2)

There are only a finite number of eigenvalues of $T:B\to B$ with modulus equal to $1$. We denote them by $\lambda_{1},\ldots,\lambda_{p}\in\mathbb{S}^{1}\subseteq\mathbb{C}$.

• (b2)

There exist operators $T_{1},\ldots,T_{p},S$, all satisfying properties (b1), (c1), and (d1), such that:

[TABLE]

where $\dim(T_{i}(B))<\infty$ for every $1\leq i\leq p$.

• (c2)

$T_{i}^{2}=T_{i}$,  $T_{i}\circ T_{j}=0$ for $i\neq j$,  and  $T_{i}\circ S=S\circ T_{i}=0$ for each $1\leq i,j\leq p$.

• (d2)

$T\circ T_{i}=T_{i}\circ T=\lambda_{i}T_{i}$* for every $1\leq i\leq p$.*

• (e2)

$r(S)<1$, where $r(S)$ is the spectral radius of $S:B\to B$.

• (f2)

Addendum: $\sigma(T)\setminus\{\lambda_{1},\ldots,\lambda_{p}\}\subset\sigma(S)\subset B(0,r(S))\subset\mathbb{C}$, where $\sigma(T)$ denotes the spectrum of $T$. This property is called the spectral gap of $T$.

Moreover, we will also need Theorem 4.1 from [29].

Theorem** (Glück and Weber).**

Let $X$ be an ordered Banach space with positive cone $X_{+}\neq\{0\}$, let $J=\mathbb{N}_{0}$ or $J=[0,\infty)$, and let $T=(T_{t})_{t\in J}$ be a positive operator semigroup on $X$. Suppose that the following two assumptions are satisfied:

• (a)

For each vector $0\neq x\in X_{+}$, there exists a time $t_{x}\in J$ such that $T_{t_{x}}x$ is an almost interior point of $X_{+}$.

• (b)

The semigroup $T$ is bounded and strongly asymptotically compact.

Then $T_{t}$ converges strongly as $t\to\infty$. If the limit operator $Q:=\lim_{t\to\infty}T_{t}$ is not zero, then it is of the form $Q=y^{\prime}\otimes y$. Here, $y$ is an almost interior point of $X_{+}$ and a fixed point of $T$, and $y^{\prime}\in X^{\prime}$ is a strictly positive functional and a fixed point of the adjoint semigroup $T^{\prime}:=(T_{t}^{\prime})_{t\in J}$ (consisting of the dual operators $T_{t}^{\prime}$).

Remark 34**.**

Recall that $x\in X_{+}$ is called an almost interior point if it is such that $\mu(x)>0$ for every non-zero positive functional $\mu$ [29, Definition 2.4 (ii)]. Also, the operator semigroup $T$ is said to be strongly asymptotically compact if for each $x\in X$ and each sequence $(t_{n})_{n\in\mathbb{N}}\subseteq J$ that converges to $\infty$, the sequence $(T_{t_{n}}x)_{n\in\mathbb{N}}$ in $X$ has a convergent subsequence [29, Definition 3.1]. If convenient, the reader can explore more details in reference [29] to gain familiarity with these definitions and Theorem 4.1 therein.

Remark 35**.**

We want to show that [51, Theorem B.1.1] applies to $E=\mathcal{C}(\Omega;\mathcal{A})$, $B=\mathcal{C}_{\theta}(\Omega;\mathcal{A})$, and $T=L_{\varphi}$. Also, that [29, Theorem 4.1] applies to $X=\mathcal{C}(\Omega;\mathcal{A})$, $X_{+}=\mathcal{C}(\Omega;\mathcal{A})^{+}$, and $T_{t}=L_{\varphi}^{n}$.

For the proof of our main result, let us begin by proving hypothesis (b1) of [51, Theorem B.1.1]. Since each $\varphi_{ix}$ is positivity-improving, and particularly positive, then so is their sum. But beware: this only ensures $\sum_{i}\varphi_{ix}$ to be positivity-improving for every $x\in\Omega_{A}$, which is not enough to ensure the linear operator $L_{\varphi}:\mathcal{C}(\Omega_{A};\mathcal{A})\to\mathcal{C}(\Omega_{A};\mathcal{A})$ to be positivity-improving.

We will see later that $L_{\varphi}$ is merely “eventually” positivity-improving, in a sense to be made precise then; this will be sufficient. For now, it is enough to know that each $\varphi_{x}$ being positive for every $x\in\Omega_{A}$, the linear operator $L_{\varphi}$ follows to be positive. Therefore, we may employ [42, Corollary 2.9 (Russo-Dye)] to conclude that:

[TABLE]

Now, we pass to the issue of showing $L_{\varphi}$ is “weakly contracting for Lipschitz functions”, in the sense that inequality (17) holds. This corresponds to showing $L_{\varphi}$ satisfies hypothesis (c1) of [51, Theorem B.1.1]. To do so, first, we prove the basic inequality [41, Proposition 2.1]. We consider two cases:

Case 1. Say $d(x,y)>\theta$. If this holds, then:

[TABLE]

Case 2. Say $d(x,y)\leq\theta$. If this holds instead, then $x_{1}=y_{1}$, so that:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Define $C_{1}:=C_{1}(\varphi,\theta):=\max\left\{k\theta\left|\varphi\right|_{\theta},\frac{2}{\theta}\right\}$. Combining both cases, it follows that:

[TABLE]

Proceeding by induction, assume $\left|L_{\varphi}^{n}g\right|_{\theta}\leq C_{n}\left|g\right|_{\infty}+\theta^{n}\left|g\right|_{\theta}$. Then:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $\theta<1$, we may recursively choose $C_{n+1}$ as:

[TABLE]

[TABLE]

Therefore:

[TABLE]

Second, we infer the Ionescu-Tulcea and Marinescu Inequality [51, Corollary 13.8.16] by observing that:

[TABLE]

[TABLE]

The hypothesis (d1) of [51, Theorem B.1.1] follows from [51, Remark B.1.2] plus Remark 9.

Finally, the hypothesis (a1) of [51, Theorem B.1.1] can be verified directly: let $(g_{n})_{n\in\mathbb{N}}$ be a sequence of Lipschitz-continuous functions $g_{n}\in\mathcal{C}_{\theta}(\Omega_{A};\mathcal{A})$ such that $\left\lVert g_{n}\right\rVert_{\theta}\leq K$ for every $n\in\mathbb{N}$ and some $K\in\mathbb{R}$. Suppose $g_{n}$ converges uniformly to $g$, that is:

[TABLE]

Observe that $g\in\mathcal{C}_{\theta}(\Omega_{A};\mathcal{A})$ and that the Lipschitz norm of $g$ is bounded above by $K$, because:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This means we are all set to apply [51, Theorem B.1.1]. So, we already know $1\in\mathbb{R}$ is an isolated eigenvalue of $L_{\varphi}$. To see that it is simple, suppose there is some $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$ such that $L_{\varphi}g=g$, that is, $g\neq 0$ is an eigenfunction corresponding to the eigenvalue $1$. By uniquely decomposing each $a\in\mathcal{A}$ (and, simultaneously, each $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$) into positive elements (and, simultaneously, into positive functions) as per [15, Lemma 4.38], and noticing that $L_{\varphi}$ leaves the set of positive functions invariant, we may assume, without loss of generality, that $g\geq 0$.

Being a fixed point of $L_{\varphi}$, we conclude that $g>0$. This happens because:

[TABLE]

and the possibility that $g(ix)=0$ for any (some) $ix\in\Omega$ can be ruled out because it implies $g(y)=0$ for every $y\in\cup_{n\in\mathbb{N}}\sigma^{-n}(ix)$, and by continuity, for every $x\in\Omega_{A}$; contradicting $g\neq 0$. It follows that $\varphi_{ix}g(ix)>0$ for every $ix\in\Omega$, and, adding these, that $g>0$.

Consider the finite constant $\mathbb{R}\ni s>0$ given by:

[TABLE]

Notice that $s$ is bounded above, for instance, by any finite upper bound for $g$, and bounded below, for instance, by any positive lower bound for $g$. Notice also that $g-s\mathbb{I}_{\mathcal{C}}\geq 0$, but not $g-s\mathbb{I}_{\mathcal{C}}>0$. But, being a linear combination of two fixed points of $L_{\varphi}$, $g-s\mathbb{I}_{\mathcal{C}}$, it is also a fixed point of $L_{\varphi}$. We have just seen that any non-zero positive fixed points of $L_{\varphi}$ must be strictly positive. Thus, $g-s\mathbb{I}_{\mathcal{C}}$ must be zero, that is, $g=s\mathbb{I}_{\mathcal{C}}$.

Regarding the dual space $\mathcal{C}(\Omega_{A};\mathcal{A})^{\prime}$, and the corresponding dual operator $L_{\varphi}^{\prime}$, we remark that the set:

[TABLE]

and its subset:

[TABLE]

where $\sigma_{\sharp}$ denotes the push-forward mapping, defined by:

[TABLE]

are both weak*-compact and invariant by $L_{\varphi}^{\prime}$. Therefore, there is a $\sigma$-invariant state $\eta_{\varphi}$ such that $L_{\varphi}^{\prime}(\eta_{\varphi})=\eta_{\varphi}$. We now see that it is unique and characterized by item 3b of Theorem 31 by employing [29, Theorem 4.1] as follows:

The basic inequality (16) shows the sequence $(L_{\varphi}^{n}(g))_{n\in\mathbb{N}}$ to be equicontinuous for every $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$. Since $\left\lVert L_{\varphi}\right\rVert=1$, the sequence $(L_{\varphi}^{n}(g))_{n\in\mathbb{N}}$ is also bounded for every $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$. Combining all these with the Arzelà-Ascoli Theorem, it follows that hypothesis (b) of [29, Theorem 4.1] is satisfied.

Now say that $g\in\mathcal{C}(\Omega_{A};\mathcal{A})^{+}$ and $g\neq 0$. This means there is some $x\in\Omega_{A}$ such that $g(x)\neq 0$. By continuity, there is an open set $U\subseteq\Omega_{A}$ for which $0\notin g(U)$. Since $A$ is aperiodic, there exists an $N=N(U)\in\mathbb{N}$ such that $\sigma^{-n}(x)\cap U\neq\emptyset$, for any $x\in\Omega_{A}$, and $n\geq N$. Then, $L_{\varphi}^{n}(g)>0$, since, for any $x\in\Omega_{A}$:

[TABLE]

and $g(y)\neq 0$ for at least one $y\in\sigma^{-n}(x)$. This shows that hypothesis (a) of [29, Theorem 4.1] is also satisfied. It follows that $L_{\varphi}^{n}(g)$ converges uniformly to $\eta_{\varphi}(g)\mathbb{I}_{\mathcal{C}}$, and also that $\eta_{\varphi}$ is a faithful state [15, Definition 4.61], that is, strictly positive [29, Definition 2.4].

To see that there is no other unitary eigenvalue of $L_{\varphi}$, suppose the contrary: that there is $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$ such that $g\neq 0$, $L_{\varphi}g=\lambda g$, $\lambda\in\mathbb{R}$, $\left|\lambda\right|=1$, $\lambda\neq 1$. Notice:

[TABLE]

so that $\eta_{\varphi}(g)=0$. It follows that $\lim_{n}L_{\varphi}^{n}g=\lambda^{n}g\equiv 0$, which, together with $\left|\lambda\right|=1$, implies that $g=0$, a contradiction.

Finally, to see that any positive eigenfunction of $L_{\varphi}$ must be a positive multiple of $\mathbb{I}_{\mathcal{C}}$, suppose the contrary: that there is $g\in\mathcal{C}(\Omega_{A};\mathcal{A})^{+}$ such that $g\neq 0$, $L_{\varphi}g=\lambda g$, $\lambda\in\mathbb{R}$, $\left|\lambda\right|<1$. Repeating the reasoning of (18), we again conclude that $\eta_{\varphi}(g)=0$. Since $\eta_{\varphi}$ is faithful (strictly positive) and $g$ is positive, it follows that $g=0$, a contradiction.

5 Entropy for a certain family of potentials

In the classical thermodynamic formalism, the entropy $h_{\mu}(\sigma)$ of an equilibrium state $\mu$ for a normalized potential $B:\Omega\to\mathbb{R}$ admits the expression:

[TABLE]

where $J_{\mu}:\Omega\to(0,1)$ denotes the so called Jacobian of $\mu$ with respect to the shift dynamics $\sigma$ (see [53], [21] or Section 3 in [33]). This follows from the variational principle of pressure and the normalization condition, and $J_{\mu}$ can be interpreted as the point-wise expansion factor of the measure (or as a kind of Radon-Nikodym derivative in inverse branches, as described in [21] or in Section 7 in [33]).

Definition 36**.**

In our $\mathcal{A}$-valued setting, for a normalized potential $\varphi\in\mathcal{C}(\Omega_{A};\mathfrak{L}(\mathcal{A}))$, it is natural to define the point-wise Jacobian $J$ as the $\mathcal{A}$-valued function:

[TABLE]

It is a generalization of the previuous concept of Jacobian.

Remark 37**.**

Because of [42, Corollary 2.9 (Russo-Dye)], this definition implies the equality $\left|J(x)\right|:=\left\lVert\varphi_{x}\right\rVert$.

Definition 38**.**

Given an eigenstate $\eta$ of the transfer operator $L_{\varphi}$, we propose the following ansatz for the entropy of $\eta$:

[TABLE]

Remark 39**.**

Since $\varphi_{x}$ is positivity-improving, the logarithm expression $\log J$ is point-wise well defined. The property of it being continuous follows from the continuity of $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$, plus the compactness of $\Omega_{A}$. Therefore, it makes sense to evaluate $\eta$ at $\log J$.

Remark 40**.**

In the classical (scalar) setting, the Kolmogorov-Sinai entropy is a functional of the invariant measure alone. In our noncommutative framework the quantity $h(\varphi,\eta,\sigma)$ depends on both the eigenstate $\eta$ and the choice of normalized $\mathfrak{L}(\mathcal{A})$-valued potential $\varphi$. Distinct normalized potentials may share the same $\eta$ but yield different $J_{\varphi}$ and hence different entropies.

Example 41**.**

Let $\Omega_{A}=\left\{1,2\right\}^{\mathbb{N}}$, $\mathcal{A}=M_{2\times 2}$, $\mathfrak{L}(\mathcal{A})=\mathfrak{L}(M_{2\times 2}(\mathbb{R}))$, and consider potentials $\varphi_{x}(g)=(\widehat{\operatorname{tr}}\,g)P(x)$, where $\widehat{\operatorname{tr}}=\frac{1}{2}\operatorname{Tr}$. Normalization in this situation means $P(1y)+P(2y)=\operatorname{Id}$ for all $y\in\left\{1,2\right\}^{\mathbb{N}}$. Note that $\widehat{\operatorname{tr}}\,(\operatorname{Id})=1$, so $J_{\varphi}(x)=\varphi_{x}(\operatorname{Id})=P(x)$.

As explained in Example 32, the associated eigenstate for the transfer operator is $\eta$ with $\eta(g)=\int\widehat{\operatorname{tr}}\,(g(x))\operatorname{d}\!\mu(x)$, where $\mu$ denotes the equilibrium measure for the scalar potential $x\mapsto\widehat{\operatorname{tr}}\,P(x)$. In the next two examples, $\widehat{\operatorname{tr}}\,P(\cdot)\equiv\frac{1}{2}$, so $\mu$ is the Bernoulli $(\frac{1}{2},\frac{1}{2})$ measure.

1. (i)

Constant case.

Take $P(x)\equiv\begin{bmatrix}\frac{1}{2}&0\\ 0&\frac{1}{2}\end{bmatrix}$. Then $\log J_{\varphi}=\log P=\begin{bmatrix}\log\frac{1}{2}&0\\ 0&\log\frac{1}{2}\end{bmatrix}$, hence:

[TABLE] 2. (ii)

First-coordinate dependent case.

Let $p\in(0,1)$. Take:

[TABLE]

so that $P(1y)+P(2y)=\operatorname{Id}$. Then, for either the first symbol:

[TABLE]

and therefore:

[TABLE]

[TABLE]

This quantity is minimized at $p=\frac{1}{2}$, where $h(\varphi,\eta,\sigma)=\log 2$, and:

[TABLE]

In particular, both potentials yield the same eigenstate $\eta$ but produce different entropies $h(\varphi,\eta,\sigma)$, illustrating that our entropy is a function of the pair $(\eta,\varphi)$, not of $\eta$ alone. Admittedly, $\eta$ depends on $\varphi$, and in that sense, this quantity we are labeling as entropy, would be a function of the potential.

Now, in order to justify more properly our ansatz, consider the case of Example 32. $\Omega_{A}$ is an aperiodic subshift of finite type, $\mathcal{A}=M_{d\times d}(\mathbb{R})$, and the potential $\varphi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$ is characterized by

[TABLE]

where $P(x)$ is a (Lipshitz continuous) family of strictly positive matrices such that:

[TABLE]

for all $x\in\Omega_{A}$.

Definition 42**.**

In case $\varphi$ is defined via the previous paragraph (see (19) and (20)), we say it is of trace-type with matrix-factor $P$.

We have seen that when $\varphi$ is of trace-type, the eigenstate $\eta$ for the $\mathcal{A}$-valued transfer operator $L_{\varphi}$ is given by integrating the normalized trace against the eigenprobability $\mu_{\widehat{\operatorname{tr}}\,P}$, for the classical transfer operator $L_{\widehat{\operatorname{tr}}\,P}$.

Note that $\eta$ is invariant for the action of the shift $\sigma$ in the sense that for any $g$ we get

[TABLE]

Moreover, we claim that the entropy of such an eigenstate $\eta$ is related to the one for $\mu$. In fact, by employing our definition of entropy, we have the following inequality:

[TABLE]

[TABLE]

where $h_{\mu_{\widehat{\operatorname{tr}}\,P}}(\sigma)$ denotes the Kolmogorov-Sinai entropy of $\mu_{\widehat{\operatorname{tr}}\,P}$ with respect to $\sigma$. If we let $d=1$, we are back to the classical setting. Then the trace becomes trivial, and the inequality becomes an equality, as expected.

We believe this is a satisfactory justification for the definition of entropy introduced above.

Our definition of entropy also satisfies an inequality that is somehow associated with the variational principle in the case the pressure (as defined in [41]) is zero. More precisely, if $J:\Omega\to(0,1)$ is normalized and Holder we get that

[TABLE]

The maximal value [math] is attained when $\tilde{\mu}=\mu_{J}$, where $\mathcal{L}_{J}^{*}(\mu_{J})=\mu_{J}$.

Here we can show the following:

Theorem 43**.**

Let $\varphi$ be a normalized trace-type potential, with eigenstate $\eta$ and jacobian $J$. Then,

[TABLE]

where $\tilde{\eta}$ is the eigenstate for a normalized trace-type potential $\psi$. The value [math] in (22) is obtained when $\tilde{\eta}=\eta$ and $\psi=\varphi$.

Proof.

By definition, $h(\varphi,\eta,\sigma)+\eta(\log J)=0$. Furthermore, if $\tilde{J}$ denotes the Jacobian for $\psi$, we have, also by definition, $h(\psi,\tilde{\eta},\sigma)+\tilde{\eta}(\log\tilde{J})=0$. Therefore, (22) may be rewritten as $-\tilde{\eta}(\log\tilde{J})+\tilde{\eta}(\log J)\leq 0$. To prove this inequality holds, we use the following standard lemma [41, Lemma 3.3]:

Lemma 44**.**

Let $q_{i}$, $p_{i}$, $1\leq i\leq n$ be two families of strictly positive real numbers such that $\sum q_{i}=\sum p_{i}=1$. Then:

[TABLE]

Let $Q$ be the matrix-factor of $\psi$ (recall Definition 42). Employing Jensen’s inequality to the normalized trace, plus Lemma 44, it follows that, for all $x\in\Omega_{A}$:

[TABLE]

[TABLE]

[TABLE]

or, equivalently:

[TABLE]

Because the eigenmeasure for $\widehat{\operatorname{tr}}\,Q$ is positive:

[TABLE]

and because $\tilde{\eta}$ is the eigenstate for $L_{\psi}$:

[TABLE]

∎

The above results suggest that the concept $h(\varphi,\eta,\sigma)$ of Definition may capture the same structural role as entropy in the scalar case, and it is consistent with the interpretation of $J$ as a generalized Jacobian. We also remark that because both inequalities hold for any trace-type potentials inducing the given states, they would also hold for their infima over such potentials, which would in turn be a function of the states themselves. However, Example 46 and the ones in Subsection 7.2 show that it is possible for a non-trace-type potential ($p\neq 1$) to share eigenstates with a potential of trace-type ($p=1$).

6 Relation to previous literature

The results of the present work touch upon some topics also addressed in [16] and [55], both of which consider transfer operators associated with matrix-valued potentials under given dynamical frameworks, but they are not included there. Our main theorem is much more akin to the main theorems of [55]. We will elaborate on that.

In [16] results are framed in the context of smooth dynamical systems on compact differentiable manifolds. There, the base map is assumed to be a covering map, which ensures that each point has finitely many local inverses—analogous to inverse branches in symbolic dynamics. In contrast, [55] considers an iterated function system (IFS) acting on a compact attractor via contractive mappings, where the symbolic space is replaced by a compact metric space.

In [55] it is assumed that the potential is positive (assumption (H1)) and moreover that, for each point, the family of matrix-valued functions given by the potential along its preimages satisfy a primitivity condition (assumption (H2)). On the other hand, [16] assumes invertibility of the potential matrices and does not require primitivity.

Our work is closer to [55] (but different), in the sense that our functions also take values in a finite-dimensional space, and our potentials are required to be positivity-improving, which is stronger than the combination (H1) plus (H2). However, the positivity aspect of our hypothesis refers to the intrinsic cone of positive elements in the C*-algebra $\mathcal{A}$, which is fundamentally different from the coordinate-wise positive cone. For example, in $M_{2\times 2}(\mathbb{R})$ a matrix is positive if and only if it is Hermitian and has non-negative eigenvalues, a condition that is nonlinear in the matrix entries and richer than positivity in $\mathbb{R}^{4}$ (which would just mean positivity of each coordinate viewed as an element of $\mathbb{R}^{4}$).

As such, the positivity-improving hypothesis in our setting implies distinct structural constraints that are not captured by simply embedding the algebra into a vector space. Consequently, our results cover a genuinely new class of examples. This distinction, although subtle, also has implications for the structure of eigenstates and the behavior under iteration of the associated transfer operators. The examples we present here evidence such a distinction.

Besides this distinction, it is also worthwhile to emphasize that, in the case of a noncommutative C*-algebra $\mathcal{A}$, our eigenstate is not necessarily a (vector-valued) eigenmeasure, as in [55]. Rather, it is an “$\mathcal{A}$-state-valued” measure. Not only is the positivity condition different, but the normalization condition is different as well (and accordingly).

We believe that a general formulation in the setting of order-unit spaces could clarify these distinctions, potentially allowing one to isolate the minimal structure needed for our results. Within such a framework, C*-algebras would appear as natural examples where one still manages to handle richer cone geometries explicitly. We leave such generalizations for further work.

7 Some more Examples

7.1 The constant potential and the maximal entropy measure

Assume $\Gamma:\mathcal{A}\to\mathcal{A}$ is positivity-improving, $\mathcal{A}=M_{n\times n}(\mathbb{R})$, and satisfies $\Gamma(\mathbb{I})=\mathbb{I}$. Moreover, assume that trace $\Gamma(a)=$ trace $a$, for any matrix $a\in M_{n\times n}(\mathbb{R})$. We denote $\xi(a)=$ trace $a$, in this way for any $a\in M_{n\times n}(\mathbb{R})$ we get

[TABLE]

Set $\Omega_{A}=\left\{1,2,\cdots,k\right\}^{\mathbb{N}}$ the full shift on $k$ symbols.

In most of our examples, $\xi=$ trace will help to define the eigenstate; but, for instance, when $\mathcal{A}=\mathbb{C}^{d}$, this is not exactly the case.

Now, we define the potential $\Phi:\Omega_{A}\to\mathfrak{L}(\mathcal{A})$ in such way that for any $x$ we get that $\Phi(x)=\Gamma.$ We call such $\Phi$ of the constant potential. In this case $\Phi(\mathbb{I}_{\mathcal{A}})=\mathbb{I}_{\mathcal{A}}$. We denote by $L_{\frac{1}{k}\Phi}$ the associated Ruelle operator.

Under our assumptions, for any $g:\Omega\to M_{n\times n}(\mathbb{R})$ we get that

[TABLE]

We claim that the eigenstate $\eta$ given by Theorem 31 acting on continuous functions $g\in\mathcal{C}(\Omega_{A};\mathcal{A})$ is given by the expression:

[TABLE]

where $\mu$ denotes the maximal entropy measure.

This is so because:

[TABLE]

[TABLE]

[TABLE]

where $L_{\frac{1}{k}}$ denotes the classical normalized transfer operator associated with the maximal entropy measure.

7.2 An example related to Pauli matrices

Example 45**.**

Let $\Omega_{A}=\{1,2\}^{\mathbb{N}}$ be the full shift on two symbols and $\mathcal{A}=M_{2\times 2}(\mathbb{R})$, and $0<p\leq 1$. We want to define, for each value $p$ a potential $\varphi^{p}.$ Consider the constant potential $\varphi^{p}:M_{2\times 2}(\mathbb{R})\to M_{2\times 2}(\mathbb{R})$ valued in the depolarizing channel, that is:

[TABLE]

[TABLE]

where $X$, $Y$, and $Z$ are the Pauli matrices (see [39]):

[TABLE]

The traces of all these matrices is equal to zero, then the complex nature of the Pauli matrices will not interfere in our reasoning. The depolarizing channel present such feature (see [39]).

(25) is the normalized qubit depolarizing channel, and it defines a completely positive, unital, and positivity-improving map for every $0<p\leq 1$ [39, Subsection 8.3.4]. The action of its associated transfer operator is:

[TABLE]

[TABLE]

By iteration, one finds for each value $n$ the general expression:

[TABLE]

where:

[TABLE]

Notice how $\overline{\widehat{\operatorname{tr}}\,g}^{(n)}$ coincides with the $n^{\operatorname{th}}$ iterate of the scalar valued function $\widehat{\operatorname{tr}}\,g$ with respect to the classical transfer operator associated with the maximal entropy measure. Therefore, as $n\to+\infty$, we have $(1-p)^{n}\to 0$ and $\overline{\widehat{\operatorname{tr}}\,g}^{(n)}(y)\to\int\widehat{\operatorname{tr}}\,g(x)\operatorname{d}\!\mu(x)$. Hence:

[TABLE]

Item 3. (b) in Theorem 31 then ensures this convergence is uniform and that the eigenstate $\eta$ is given by:

[TABLE]

7.3 Potentials taking values in positivity-improving

linear mappings of $d\times d$ complex matrices

In this section, we consider the case when $\mathcal{A}=M_{d\times d}(\mathbb{R})$; some states that are in principle non-commutative in nature will be related to classical equilibrium measures as in [41].

Now we are going to focus only on potentials taking values in positivity-improving linear mappings of the form:

[TABLE]

where $\widehat{\operatorname{tr}}:=\frac{1}{d}\operatorname{tr}:M_{d\times d}(\mathbb{R})\to\mathbb{R}$ denotes the normalized trace functional, and $\mathcal{C}(\Omega_{A};\mathbb{R})\ni f_{1},f_{2},\cdots,f_{d}>0$.

For one such potential, the action of the transfer operator on a given $g\in\mathcal{C}(\Omega_{A};M_{d\times d}(\mathbb{R}))$ becomes:

[TABLE]

Next, we will present a series of specific examples that will demonstrate the novelties that follow from our theoretical results.

Example 46**.**

Let $k=2$, $\mathcal{A}=M_{2\times 2}(\mathbb{R})$, and the shift of finite type is defined via $A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}$; that is the full shift. In this case, $\Sigma=\left\{1,2\right\}$ and $\Omega_{A}=\left\{1,2\right\}^{\mathbb{N}}$ is the full shift on two symbols. Also, let $f_{1}=f_{2}=\frac{1}{2}$. Then, the positivity-improving linear self-mapping of $M_{2\times 2}(\mathbb{R})$ in question is (half) a very well-known quantum channel: the (degenerate) depolarizing channel (where, in the language of [39, Subsection 8.3.4], $p=1$). More explicitly, $\varphi_{x}$ does not depend on $x\in\Omega_{A}$ and is such that:

[TABLE]

We denote the corresponding potential by $\frac{\operatorname{Id}}{2}\widehat{\operatorname{tr}}$. The action of its associated transfer operator is: for $y\in\Omega$

[TABLE]

Notice that after the first iteration, we (essentially) recover the classical transfer operator with constant normalized potential. In some sense, it is as if such a transfer operator only “sees” the trace, which is a scalar function of $x$. However, there is some differences; in the classical case (i.e., when $\mathcal{A}=\mathbb{R}$, or $d=1$), the eigenstate of the dual of the transfer operator is a probability measure, and acts on scalar functions $f\in\mathcal{C}(\Omega_{A};\mathbb{R})$ simply by integration, while, in the present example, our observables are matrix-valued functions $g\in\mathcal{C}(\Omega_{A};M_{2\times 2}(\mathbb{R}))$, and the eigenstate $\eta$ is no longer a measure, but rather a linear functional $\eta:\mathcal{C}(\left\{1,2\right\}^{\mathbb{N}};M_{2\times 2}(\mathbb{R}))\to\mathbb{R}$ defined by:

[TABLE]

where $\mu$ denotes the maximal entropy measure associated with the classical full shift.

In other words, while the measure $\mu$ still plays a role, it is no longer the eigenstate itself. Instead, the eigenstate is the composition of integration against $\mu$ with evaluation via the (normalized) trace. This reflects the operator-valued nature of the observables and the fact that scalar probability measures are insufficient to describe equilibrium in the noncommutative setting.

Example 47**.**

Again, let $\Omega_{A}$ be the full shift on two symbols and also $d=2$, $\mathcal{A}=M_{2\times 2}(\mathbb{R})$. Consider the functions $f_{1}$ and $f_{2}$ depending only on the first coordinate, such that:

[TABLE]

Then, the eigenstate $\eta:\mathcal{C}(\left\{1,2\right\}^{\mathbb{N}};M_{2\times 2}(\mathbb{R}))\to\mathbb{R}$ is:

[TABLE]

where $\mu_{\operatorname{avg}}$ is the eigenmeasure for the classical transfer operator associated with the scalar potential $\frac{f_{1}+f_{2}}{2}$. Notice that this case generalizes example 46.

Example 48**.**

Let $\Omega_{A}\subseteq\left\{1,2,\cdots,k\right\}^{\mathbb{N}}$ be any aperiodic subshift of finite type, $\mathcal{A}=M_{d\times d}(\mathbb{R})$, and $\mathcal{C}(\Omega_{A};\mathbb{R})\ni f_{1},f_{2},\cdots,f_{d}>0$ a $d$-tuple of normalized scalar potentials. Notice that their average is also a normalized scalar potential. Then, the corresponding potential $\varphi_{f_{1},f_{2},\cdots,f_{d}}:\Omega_{A}\to\mathcal{L}(M_{d\times d}(\mathbb{R}))$ is also normalized. We claim that the eigenstate $\eta:\mathcal{C}(\Omega_{A};M_{d\times d}(\mathbb{R}))\to\mathbb{R}$ mentioned in Theorem 31 is:

[TABLE]

where $\mu_{\operatorname{avg}}$ is the eigenmeasure for the classical transfer operator associated with the scalar potential $\frac{1}{d}\sum_{i=1}^{d}f_{i}$. Notice how this generalizes example 47.

Proof.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Example 49**.**

Let $\mathcal{A}=M_{2\times 2}(\mathbb{R})$ and $\Omega=\{-1,1\}^{\mathbb{N}}$ . We can consider that $-1$ is associated with the spin $-$ and $1$ with the spin $+$.

Consider the class of functions $\varphi:\Omega=\{-1,1\}^{\mathbb{N}}\to\mathcal{L}(M_{2\times 2}(\mathbb{R}))$ such that there exist strictly positive matrices $A_{-1},A_{1}\in M_{2\times 2}(\mathbb{R})$, satisfying for any $y\in\Omega$,

[TABLE]

[TABLE]

Defining:

[TABLE]

it follows that for any $y\in\Omega$, $J_{\varphi}(-1,y)+J_{\varphi}(1,y)=1$. Denote by $\mu_{J_{\varphi}}$ the (classical) equilibrium probability associated with the normalized potential $J_{\varphi}$.

We denote by $g$ a general continuous function $g:\Omega\to M_{2\times 2}(\mathbb{R})$.

The action of the transfer operator is:

[TABLE]

Note that the potential $\varphi$ is positivity-improving.

We claim that $\eta_{\varphi}$ such that

[TABLE]

is the eigenstate for such a Ruelle operator. Indeed, given $g$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We may call each $\eta_{\varphi}$ obtained from a $\varphi$ as in (29) of Gibbs eigenstate (see Section 5).

In consonance with Section 5, for each $\varphi$ the corresponding entropy of $\eta_{\varphi}$ is:

[TABLE]

Example 50**.**

We will define a potential $\varphi$ acting on the family of continuous functions $g:\Omega\to M_{2\times 2}(\mathbb{R})=\mathcal{A}$, where $\Omega=\Omega_{A}=\{1,2\}^{\mathbb{N}}$ is the full shift space. Consider a Lipschitz continuous map $x\in\Omega_{A}\to P(x)$, where $P(x)$ is double stochastic with positive entries, and the corresponding matrices

[TABLE]

and

[TABLE]

where $(\pi_{1}(x,),\pi_{2}(x))=(\frac{1}{2},\frac{1}{2})$ is the stationary vector for the double stochastic matrix $P(x)$.

Consider the transformation $W$ such that

[TABLE]

For each $x$, consider potentials of the form

[TABLE]

We want to determine the eigenstate for $x\to\varepsilon_{x}(W(a(x))\otimes\operatorname{Id})$

Denote by

[TABLE]

a general continuous map on $x\in\Omega$. For simplification, we will sometimes omit the dependence on $x$.

For the fixed family $P(x)$ consider the variable function $a$. In this case

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the similar way as in Example 33 (see also (28) in Example 46) we set for any $g$

[TABLE]

[TABLE]

The eigenstate is

[TABLE]

Notice that this eigenstate is the same as the one coming from the potential of Example 46.

7.4 Potentials taking values in positive matrices

In this section we consider the case when $\mathcal{A}=\mathbb{R}^{N}$. This means we are concerned with vector-valued functions $g\in\mathcal{C}(\Omega_{A};\mathbb{R}^{N})$. The multiplicative element in $\mathcal{A}=\mathbb{R}^{N}$ is the vector $(1,1,...,1)$. The positivity-improving linear self-mappings of $\mathbb{R}^{N}$ are precisely the ones corresponding to $N\times N$ square-matrices with strictly positive entries. Therefore, the potentials we will be interested in are functions of the form $\varphi:\Omega_{A}\to M_{N\times N}(\mathbb{R})$ such that:

[TABLE]

For one such potential, the action of the transfer operator becomes:

[TABLE]

[TABLE]

Example 51**.**

Let $k=2$, and $A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}$. That is, $\Sigma=\left\{1,2\right\}$ and $\Omega_{A}=\left\{1,2\right\}^{\mathbb{N}}$ is the full shift on two symbols. Also, let $\varphi_{x_{\ell m}}=\frac{1}{2N}$, that is, $\varphi_{x}=\frac{1}{2N}J$, where $J$ is the $N$ by $N$ matrix with all entries equal to $1$; therefore does not depend on $x$. Then, the action of the associated normalized transfer operator becomes: given $g=(g_{1},g_{2},...,g_{N})$

[TABLE]

and the corresponding eigenstate $\eta:\mathcal{C}(\Omega_{A};\mathbb{R}^{N})\to\mathbb{R}$ is given by:

[TABLE]

where $\mu$ denotes the maximal entropy measure.

Example 52**.**

Again, let $\Omega_{A}=\left\{1,2\right\}^{\mathbb{N}}$ be the full shift on two symbols. This time, let $\varphi_{x}=\frac{1}{2}\left(\frac{p}{N}J+(1-p)\operatorname{Id}\right)$, not depending on $x$. The $n^{\operatorname{th}}$ power of $\left(\frac{p}{N}J+(1-p)\operatorname{Id}\right)$ is equal to $\left(\frac{1-(1-p)^{n}}{N}J+(1-p)^{n}\operatorname{Id}\right)$. Therefore, the $n^{\operatorname{th}}$ iterate of the corresponding transfer operator is:

[TABLE]

Taking the limit $n\to+\infty$, we see the action of the above operator becomes the action of $L_{\frac{1}{2N}J}$. Therefore, they determine the same eigenstate $\eta$.

.

 

.

William M. M. Braucks (braucks.w@gmail.com)

Artur O. Lopes (arturoscar.lopes@gmail.com)

Inst. Mat. Est. - UFRGS - Porto Alegre, Brazil