A New Definition of t-Entropy for Transfer Operators

Abstract

This article presents a new definition of t-entropy that makes it more explicit and simplifies the process of its calculation.

1. Introduction

The spectral analysis of operators associated with dynamical systems is of considerable importance. In particular, in the series of articles [1,2,3,4,5,6], a relation between t-entropy and spectral radii of the corresponding operators has been established. Here, the authors have uncovered a new dynamical invariant—t-entropy—that is related to the Legendre transform of the spectral exponent of the operators in question. The t-entropy plays a significant role in various nonlinear phenomena. In particular, it serves as a principal object in thermodynamical formalism (see [2,6,7], and the sources quoted therein). The description of t-entropy is not elementary and its calculation is rather sophisticated. In the present article, we give a new definition of t-entropy that makes it more explicit and essentially simplifies the process of its calculation.

The article consists of two sections. In Section 2, we consider t-entropy for the model example. Here, Theorem 2 gives a new definition of t-entropy, that simplifies its calculation. The general situation of arbitrary $C^{*}$-dynamical system is discussed in Section 3. To illustrate similarity and difference between the objects considered in the model and general situations, we present here a number of examples and finally introduce the general new definition of t-entropy in Theorem 3.

2. A New Definition of t-Entropy for Continuous Dynamical Systems

In this Section, we consider a model example. Here, we use definitions, notation, and results from [4,5]. We denote by X a Hausdorff compact space, and by $C\left(X\right)$ we denote the algebra of continuous functions on X taking real values and equipped with the max-norm. Consider an arbitrary continuous mapping $\alpha :X\to X$. The corresponding dynamical system will be denoted by $(X,\alpha )$.

The main object under investigation is a transfer operator $A:C\left(X\right)\to C\left(X\right)$, associated with a given dynamical system. Its definition is given in the following way:

(a)

A is a positive operator (that is it maps nonnegative functions to nonnegative) and

(b)

the following homological identity for A is valid:

$$A\left(g\circ \alpha ·f\right)=gAf,g,f\in C\left(X\right).$$

(1)

The set of linear positive normalized functionals on $C\left(X\right)$ will simply be denoted by M. The Riesz theorem states that elements of M can be identified with regular Borel probability measures on X and henceforth we assume this identification and, therefore, elements of M will be called probability measures.

Let us recall the classical definition of an invariant measure: $\mu \in M$ is α-invariant if $\mu \left(g\right)=\mu (g\circ \alpha )$ for $g\in C\left(X\right)$. The family of $\alpha$-invariant probability measures on X is denoted by $M_{\alpha }$.

A continuous partition of unity in $C\left(X\right)$ is a finite set $G=\{g_1,\cdots ,g_k\}$ consisting of nonnegative functions $g_i\in C\left(X\right)$ satisfying the identity $g_1+\cdots +g_k\equiv 1$.

According to [5], t-entropy is the functional $\tau \left(\mu \right)$ on M which is defined in three steps.

Firstly, for a given $\mu \in M$, each partition of unity $G=\{g_1,\cdots ,g_k\}$, and any $n\in \mathbb{N}$ we set

$${\tau }_n(\mu ,G):=\underset{m\in M}{sup}\sum _{g_i\in G}\mu \left(g_i\right)ln\frac{m\left(A^n{g}_i\right)}{\mu \left(g_i\right)}.$$

(2)

Here, if $\mu \left(g_i\right)=0$ for some $g_i\in G$ then the corresponding summand in (2) is assumed to be zero regardless of the value $m\left(A^n{g}_i\right)$; if $A^n{g}_i=0$ for some $g_i\in G$ and at the same time $\mu \left(g_i\right)>0$, then ${\tau }_n(\mu ,G)=-\infty$.

Secondly, we put

$${\tau }_n\left(\mu \right):=\underset{G}{inf}{\tau }_n(\mu ,G),$$

(3)

here, the infimum is taken over all partitions of unity G in $C\left(X\right)$.

Finally, the t-entropy $\tau \left(\mu \right)$ is defined as

$$\tau \left(\mu \right):=\underset{n\in \mathbb{N}}{inf}\frac{{\tau }_n\left(\mu \right)}{n}.$$

(4)

Let A be a given transfer operator in $C\left(X\right)$. In what follows, we denote by $A_{\phi }$ the family of transfer operators in $C\left(X\right)$, where $\phi \in C\left(X\right)$, given by the formula

$$A_{\phi }f=A\left(e^{\phi }f\right).$$

Next, we denote by $\lambda \left(\phi \right)$ the spectral potential of $A_{\phi }$, namely,

$$\lambda \left(\phi \right)=\underset{n\to \infty }{lim}\frac{1}{n}ln∥A_{\phi }^n∥.$$

The principal importance of t-entropy is clearly demonstrated by the following Variational Principle.

Theorem 1.

([5], Theorem 5.6) Let $A:C\left(X\right)\to C\left(X\right)$ be a transfer operator for a continuous mapping $\alpha :X\to X$ of a compact Hausdorff space X. Then,

$$\lambda \left(\phi \right)=\underset{\mu \in M_{\alpha }}{max}\left(\mu \left(\phi \right)+\tau \left(\mu \right)\right),\phi \in C\left(X\right).$$

The next principal result of the article presents a new definition of t-entropy.

Theorem 2.

For α-invariant measures $\mu \in M_{\alpha }$, the following formula is true

$$\tau \left(\mu \right)=\underset{n,G}{inf}\frac{1}{n}\sum _{g\in G}\mu \left(g\right)ln\frac{\mu \left(A^{n}g\right)}{\mu \left(g\right)}.$$

(5)

In other words, in the definition of t-entropy, one should not calculate the supremum in (2) but can simply put $m=\mu$ there. Thus, expression (2) is changed for

$${\tau }_n^{\prime }(\mu ,G)=\sum _{g\in G}\mu \left(g\right)ln\frac{\mu \left(A^{n}g\right)}{\mu \left(g\right)}.$$

(6)

Remark 1.

In connection with Theorem 2, it is worth mentioning the results of [7], where for a special case of transfer operator similar formulae are obtained and their relation to thermodynamical formalism is studied.

To prove Theorem 2, we need the next

Lemma 1.

Let G be a partition of unity in $C\left(X\right)$. Then, for any pair of numbers $n\in \mathbb{N}$, $\epsilon >0$ there exists a partition of unity E in $C\left(X\right)$ such that for each pair of functions $g\in G$ and $h\in E$ the oscillation of $A^{n}g$ over $\mathrm{supp}h:=\{x\in X\mid h(x)>0\}$ is less than ε:

$$sup\left\{A^{n}g\left(x\right)|h\left(x\right)>0\right\}-inf\left\{A^{n}g\left(x\right)|h\left(x\right)>0\right\}<\epsilon .$$

(7)

Proof. 

For any $g\in G$ and $n\in \mathbb{N}$, the function $A^{n}g$ belongs to $C\left(X\right)$. Therefore, its range is contained in a certain segment $[a,b]$.

Evidently, there exists a partition of unity $\{f_1,\cdots ,f_k\}$ in $C[a,b]$ such that the support of every one of its elements is contained in a certain interval of the length less than $\epsilon$. Then, the family $E_g=\{f_1\circ A^{n}g,\dots ,f_k\circ A^{n}g\}$ forms a partition of unity in $C\left(X\right)$ and the oscillation of $A^{n}g$ is less than $\epsilon$ on the support of each of its elements. Now all the products ${\prod }_{g\in G}{h}_g$, where $h_g\in E_g$, form the desired partition of unity E.  ☐

Now let us prove Theorem 2. Comparing (2) and (6), one sees that

$${\tau }_n^{\prime }(\mu ,G)\le {\tau }_n(\mu ,G).$$

Therefore, to prove (5), it is enough to verify the inequality

$${\tau }_n\left(\mu \right)\le {\tau }_n^{\prime }(\mu ,G).$$

Since in the case when ${\tau }_n\left(\mu \right)=-\infty$ the latter inequality is trivial, in what follows we assume that ${\tau }_n\left(\mu \right)>-\infty$.

Let us fix some $n\in \mathbb{N}$, a partition of unity G in $C\left(X\right)$ and $\epsilon >0$. For these objects, there exists a continuous partition of unity E mentioned in Lemma 1. Consider one more partition of unity in $C\left(X\right)$ that consists of the functions $g·h\circ {\alpha }^n$, here $g\in G$ and $h\in E$. For this partition, by the definition of ${\tau }_n\left(\mu \right)$ (see (2) and (3)), there exists a probability measure $m\in M$ for which the next inequality holds:

$${\tau }_n\left(\mu \right)-\epsilon \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{m\left(A^n(g·h\circ {\alpha }^n)\right)}{\mu (g·h\circ {\alpha }^n)}.$$

From the homological identity, it follows that $A^n(g·h\circ {\alpha }^n)=h{A}^{n}g$. Therefore, the latter inequality is equivalent to

$${\tau }_n\left(\mu \right)-\epsilon \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{m\left(h{A}^n\left(g\right)\right)}{\mu (g·h\circ {\alpha }^n)}.$$

(8)

Now for each pair $g\in G$, $h\in E$ choose a number $y_{gh}$ satisfying two conditions

$$\begin{array}{c}m\left(h{A}^{n}g\right)=m\left(h\right)y_{gh},\end{array}$$

(9)

$$\begin{array}{c}inf\left\{A^{n}g\left(x\right)|h\left(x\right)>0\right\}\le y_{gh}\le sup\left\{A^{n}g\left(x\right)|h\left(x\right)>0\right\}.\end{array}$$

(10)

Then, inequality (8) takes the form

$${\tau }_n\left(\mu \right)-\epsilon \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{m\left(h\right)y_{gh}}{\mu (g·h\circ {\alpha }^n)},$$

(11)

which is equivalent to

$${\tau }_n\left(\mu \right)-\epsilon \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{y_{gh}}{\mu (g·h\circ {\alpha }^n)}+\sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)lnm\left(h\right).$$

(12)

Let us consider separately the second summand in the right-hand side of (12):

$$\sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)lnm\left(h\right)=\sum _{h\in E}\mu (h\circ {\alpha }^n)lnm\left(h\right)=\sum _{h\in E}\mu \left(h\right)lnm\left(h\right).$$

(13)

Here, in the left-hand equality, we have exploited the fact that G is a partition of unity and in the right-hand equality we have used $\alpha$-invariance of $\mu$. If we treat $m\left(h\right)$ in (13) as independent nonnegative variables satisfying the condition ${\sum }_{h\in E}m\left(h\right)=1$, then the routine usage of the Lagrange multipliers principle shows that the function ${\sum }_{h\in E}\mu \left(h\right)lnm\left(h\right)$ attains its maximum when $m\left(h\right)=\mu \left(h\right)$. Evidently, the same is true for the right-hand sides in (12) and (11). Therefore,

$${\tau }_n\left(\mu \right)-\epsilon \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{\mu \left(h\right)y_{gh}}{\mu (g·h\circ {\alpha }^n)}.$$

(14)

Observe that estimates (7) and (10) imply

$$\mu \left(h\right)y_{gh}\le \mu \left(h(A^{n}g+\epsilon )\right).$$

(15)

Observing that the logarithm is a concave function, and using (14), (15), and the fact that E is a partition of unity in $C\left(X\right)$, we conclude that

$$\begin{array}{cc}\hfill {\tau }_n\left(\mu \right)-\epsilon & \le \sum _{g\in G}\sum _{h\in E}\mu (g·h\circ {\alpha }^n)ln\frac{\mu \left(h(A^{n}g+\epsilon )\right)}{\mu (g·h\circ {\alpha }^n)}\hfill \\ \hfill & =\sum _{g\in G}\mu \left(g\right)\sum _{h\in E}\frac{\mu (g·h\circ {\alpha }^n)}{\mu \left(g\right)}ln\frac{\mu \left(h(A^{n}g+\epsilon )\right)}{\mu (g·h\circ {\alpha }^n)}\hfill \\ \hfill & \le \sum _{g\in G}\mu \left(g\right)ln\sum _{h\in E}\frac{\mu \left(h(A^{n}g+\epsilon )\right)}{\mu \left(g\right)}=\sum _{g\in G}\mu \left(g\right)ln\frac{\mu (A^{n}g+\epsilon )}{\mu \left(g\right)}.\hfill \end{array}$$

By the arbitrariness of $\epsilon$, this implies

$${\tau }_n\left(\mu \right)\le \sum _{g\in G}\mu \left(g\right)ln\frac{\mu \left(A^{n}g\right)}{\mu \left(g\right)}={\tau }_n^{\prime }(\mu ,G)$$

and finishes the proof of Theorem 2.

Now let us proceed to the general $C^{*}$-dynamical setting.

3. The General Case of $C^{*}$-Dynamical Systems

The general notion of t-entropy involves the so-called base algebra and a transfer operator for a $C^{*}$-dynamical system. Let us recall definitions of these objects (see [5]).

Let $\mathcal{B}$ be a commutative $C^{*}$-algebra with an identity $\mathbf{1}$ and $\mathcal{C}$ be its selfadjoint part, that is,

$$\mathcal{C}=\{b\in \mathcal{B}\mid b^{*}=b\}.$$

In this situation, we call $\mathcal{C}$ a base algebra.

A $C^{*}$-dynamical system is a pair $(\mathcal{C},\delta )$, where $\delta$ is an endomorphism of $\mathcal{C}$ satisfying the equality $\delta \left(\mathbf{1}\right)=\mathbf{1}$.

Definition of a transfer operator A (for $(\mathcal{C},\delta )$) is given in the following way:

(a)

A is a linear positive operator in $\mathcal{C}$ and

(b)

the homological identity for A is valid:

$$A\left(\left(\delta g\right)f\right)=gAf,g,f\in \mathcal{C}.$$

(16)

Let $M\left(\mathcal{C}\right)$ be the family of all linear positive normalized functionals on $\mathcal{C}$. A functional $\mu \in M\left(\mathcal{C}\right)$ is δ-invariant if $\mu \left(\delta f\right)=\mu \left(f\right)$ for all $f\in \mathcal{C}$. By $M_{\delta }\left(\mathcal{C}\right)$, we denote the family of all $\delta$-invariant functionals from $M\left(\mathcal{C}\right)$.

By a partition of unity in the algebra $\mathcal{C}$, we mean any finite collection $G=\{g_1,\cdots ,g_k\}$ consisting of nonnegative elements $g_i\in \mathcal{C}$ satisfying the identity $g_1+\cdots +g_k=\mathbf{1}$.

The formulae (2)–(4) from the previous section naturally lead to a definition of t-entropy for $C^{*}$-dynamical systems. Namely, the t-entropy $\tau \left(\mu \right)$ for $\mu \in M\left(\mathcal{C}\right)$ is defined in three steps as follows:

$$\begin{array}{c}{\tau }_n(\mu ,G):=\underset{m\in M\left(\mathcal{C}\right)}{sup}\sum _{g\in G}\mu \left(g\right)ln\frac{m\left(A^{n}g\right)}{\mu \left(g\right)},\end{array}$$

(17)

$$\begin{array}{c}{\tau }_n\left(\mu \right):=\underset{G}{inf}{\tau }_n(\mu ,G),\end{array}$$

(18)

and

$$\begin{array}{c}\hfill \tau \left(\mu \right):=\underset{n\in \mathbb{N}}{inf}\frac{{\tau }_n\left(\mu \right)}{n}.\end{array}$$

(19)

The infimum in (18) is taken over all the partitions of unity G in $\mathcal{C}$.

The t-entropy just defined is of principal importance in spectral analysis of abstract transfer and weighted shift operators in $L^p$-type spaces (see [5], Theorems 6.10, 11.2, 13.1 and 13.6).

The similarity and essential difference between the objects considered in this and the previous sections are discussed in ([5], Section 7).

We now present the $C^{*}$-dynamical analogue to Theorem 2.

Theorem 3.

For δ-invariant functionals $\mu \in M_{\delta }\left(\mathcal{C}\right)$, the following formula is true

$$\tau \left(\mu \right)=\underset{n,G}{inf}\frac{1}{n}\sum _{g\in G}\mu \left(g\right)ln\frac{\mu \left(A^{n}g\right)}{\mu \left(g\right)}.$$

(20)

Proof. 

This theorem can be derived from Theorem 2.

By means of the Gelfand transform, one can establish an isomorphism between the algebra $\mathcal{C}$ and the algebra $C\left(X\right)$ of continuous functions on X with real values (where X is the compact space of maximal ideals in $\mathcal{C}$).

Moreover, under the identification of $\mathcal{C}$ and $C\left(X\right)$ the endomorphism $\delta$ mentioned in the definition of the $C^{*}$-dynamical system $(\mathcal{C},\delta )$ takes the form

$$\left[\delta f\right]\left(x\right)=f\left(\alpha \left(x\right)\right)$$

(for details, see [5], Theorem 6.2). Thus, the $C^{*}$-dynamical system $(\mathcal{C},\delta )$ is completely defined by the corresponding dynamical system $(X,\alpha )$.

In terms of $(X,\alpha )$, the homological identity (16) for the transfer operator A can be rewritten as (1).

By the Riesz theorem, the identification between measures $\mu$ on X and functionals $\mu \in \mathcal{C}$ is given by

$$\mu \left(g\right)={\int }_{X}gd\mu ,g\in \mathcal{C}=C\left(X\right).$$

(21)

Finally, if $\mu \in M_{\delta }\left(\mathcal{C}\right)$ is a $\delta$-invariant functional, then the corresponding measure $\mu$ in (21) is $\alpha$-invariant, that is

$$\mu \left(g\right)=\mu (g\circ \alpha ),g\in C\left(X\right).$$

In this manner, one identifies the set $M_{\delta }\left(\mathcal{C}\right)$ with $M_{\alpha }$ mentioned in Section 2.

Under all these identifications, the desired result follows from Theorem 2.  ☐