UDC 519.217+519.53

MATHEMATICS

B. M. GUREVICH

ENTROPY OF A SHIFT AND MARKOV MEASURES IN THE PATH SPACE OF A COUNTABLE GRAPH

(Presented by Academician A. N. Kolmogorov, 4 XII 1969)

Let \(G\) be a directed graph with a countable set of vertices, which henceforth will be identified with the natural numbers. Denote the adjacency matrix of the graph \(G\) by \(\Pi(G)=\Pi=(\pi_{ij})\). We shall assume that the graph is connected, i.e., any two of its vertices can be joined by a path. The length of a path is the number of edges in it. If the lengths of all closed paths (cycles) of the graph \(G\) are relatively prime, then the graph is called aperiodic. An infinite two-sided sequence of vertices \(\{i_k,\ -\infty<k<\infty\}\) forms an infinite path if, for every \(k\), there is an edge leading from \(i_k\) to \(i_{k+1}\), i.e. \(\pi_{i_k i_{k+1}}=1\). Denote by \(\Omega(G)\) the totality of all infinite paths of the graph \(G\).

Let \(M\) be the set of numbers of the form \(0,1/n\) \((n=1,2,\ldots)\), and let \(\Omega(M)\) be the set of infinite two-sided sequences of elements of \(M\). If \(M\) is regarded as a compact subset of the line, then \(\Omega(M)\) with the topology of the direct product will be a compact topological space. The mapping \(n\mapsto 1/n\) induces an embedding of \(\Omega(G)\) into \(\Omega(M)\) and thereby defines a topology in \(\Omega(G)\). The same topology can be obtained if one introduces the discrete topology on the set \(N\) of natural numbers and regards \(\Omega(G)\) as a subset of the space \(\Omega(N)\) of two-sided sequences of natural numbers with the topology of the direct product. Obviously, \(\Omega(G)\) is closed in \(\Omega(N)\) and not closed in \(\Omega(M)\). Denote its closure in \(\Omega(M)\) by \(\overline{\Omega}(G)\).

In each of the spaces mentioned one can define the shift, which takes the sequence \(\omega=\{\omega_k\}\) to \(\omega'=\{\omega'_k\}\), where \(\omega'_k=\omega_{k-1}\), and which is a homeomorphism. The shift \(T(G)\) in the space \(\Omega(G)\) is called a topological Markov chain (see \((^1,^2)\)).

From the theorem proved in \((^2)\) it follows that the topological entropy (see \((^3)\)) of the shift in the compact space \(\overline{\Omega}(G)\) coincides with the upper bound of the metric entropies over all normalized invariant Borel measures concentrated on \(\Omega(G)\).

Denote by \(h(G)\) the common value of these two quantities and define \(\lambda(G)\) by the condition \(h(G)=\log \lambda(G)\) (the base of the logarithms is the same as in the definition of entropy). A measure \(\mu\) in the space \(\Omega(G)\) for which the metric entropy of the shift \(h_\mu(T(G))\) is equal to \(h(G)\) will be called a measure with maximal entropy. The example contained in \((^2)\) shows that, in the case under consideration, unlike the case of a finite graph (see \((^4,^5)\)), there need not exist a measure with maximal entropy. In this note we give necessary and sufficient conditions for the existence and uniqueness of such a measure.

1. Special graphs. First consider a graph \(G\) whose adjacency matrix \(\Pi(G)=\Pi=(\pi_{ij})\) is determined by a nondecreasing sequence of natural numbers \(\{k_s,\ s=1,2,\ldots\}\) according to the rule: \(\pi_{ij}=1\) if

\[ i\ne \sum_{s=1}^{n} k_s,\quad j=i+1 \]

or

\[ i=\sum_{s=1}^{n} k_s,\quad j=1+\sum_{s=1}^{m} k_s,\quad n\ge 1,\ m\ge 0 \]

(we agree that \(\sum_{s=1}^{0} k_s=0\)), and \(\pi_{ij}=0\) in all other cases. Graphs of the described—

graphs of this form will be called special. Such graphs, in connection with the problem of interest to us, were first considered by E. I. Dinaburg \((^6)\)*.

Theorem 1. If \(G\) is a special graph defined by the sequence \(\{k_s\}\), and \(r(G)=r\) is the radius of convergence of the series

\[ \varphi(t)=\sum_{s=1}^{\infty} t^{k_s}, \]

then \(\lambda(G)=r^{-1}\) for \(\varphi(r)\leq 1\) and \(\lambda(G)=t_0^{-1}\) for \(\varphi(r)>1\), \(\varphi(t_0)=1\), \(t_0>0\).

The proof is obtained by comparing the results of \((^2,^6)\).

Let us note that the shift in the path space of a special graph serves as a topological analogue of a special automorphism (see, for example, \((^7)\)), constructed from a Bernoulli automorphism with a countable number of states and a function depending only on the state at time zero and equal to \(k_n\) on the \(n\)-th state.

Theorem 2. Let, in the notation of the preceding theorem, \(r>0\). Then, for the existence of a measure with maximal entropy, it is necessary and sufficient that one of the following two conditions hold:

\[ 1)\ \varphi(r)>1; \qquad 2)\ \varphi(r)=1,\quad \varphi'(r)<\infty . \]

If a measure with maximal entropy exists, then it is unique, and after the introduction of this measure the shift \(T(G)\) becomes a special automorphism constructed from a Bernoulli automorphism with states \(E_1,E_2,\ldots\) and probabilities \(p(E_n)=(\lambda(G))^{-k_n}\), \(n=1,2,\ldots\), and the function \(f(E_n)=k_n\).

In the proof the following simple

Lemma. Let

\[ \psi(x)=\sum_{n=1}^{\infty} a_n x^n,\qquad \text{where } a_n\geq 0. \]

Then

\[ x[\psi'(x)]^2-\psi(x)\psi'(x)-x\psi(x)\psi''(x)\leq 0, \]

and equality is possible only in the case when \(\psi(x)=cx^k\).

2. The general case. We now pass to the study of an arbitrary graph \(G\), assuming only that \(\lambda(G)<\infty\). From this assumption it follows that for any vertex \(i\) and any \(n>0\) there is only a finite number of cycles of length \(n\) passing through \(i\) (see \((^2)\), Lemma 3). Fixing \(i\), enumerate, in increasing order of their lengths, all cycles passing through \(i\) exactly once (cycles of the same length may be numbered in an arbitrary order). Let \(k_s(i)\) be the length of the \(s\)-th cycle in our enumeration, and let \(G(i)\) be the special graph defined by the sequence \(\{k_s(i)\}\). We shall call \(G(i)\) the special representation of the graph \(G\) corresponding to the vertex \(i\).

Using the special representation and the results of Section 1, one can establish the following facts:

Theorem 3. Let \(G\) be a connected graph and \(\lambda(G)<\infty\). Then: 1) if a measure with maximal entropy exists, then it is unique; 2) if \(i\) is an arbitrary vertex of the graph \(G\) and \(G(i)\) is the special representation corresponding to it, then \(T(G)\) and \(T(G(i))\) possess measures with maximal entropy simultaneously; 3) if \(\mu_0\) is a measure with maximal entropy, then \(\mu_0\) is a Markov measure positive on every open set (the latter means that the random variables \(x_n(\omega)=\omega_n\), \(\omega=(\ldots,\omega_{-1},\omega_0,\omega_1,\ldots)\in\Omega(G)\), form a Markov chain).

Corollary. If \(G\) is a connected aperiodic graph, \(\lambda(G)<\infty\), and \(\mu_0\) is a measure with maximal entropy, then \((T(G),\mu_0)\) is a \(K\)-automorphism.

Let us give one more fact, useful for applications and following from the results of \((^2)\).

Theorem 4. If \(G\) is a connected graph, then for every \(c>0\) there exists an ergodic Markov measure, positive on every open set,

* Taking the occasion, I note at the request of the author of \((^6)\) that, for the validity of assertion 4) from that paper, the condition \(f(k)>c\ln k\) must be replaced by \(f(k)<c(\ln k)^a,\ a>1\).

measure \(\mu_c\) such that \(h_\mu\bigl(T(G)\bigr)>h(T(G))-c\) when \(\lambda(G)<\infty\), and \(h_{\mu_c}(T(G))>c\) when \(\lambda(G)=\infty\).

3. Relation with the matrix \(\Pi(G)\). As in the case of a finite graph \(G\) (see \((^4,^5)\)), the entropy \(h(G)\) and the measure with maximal entropy, if it exists, can be directly related to the matrix \(\Pi(G)\). Suppose that \(\lambda(G)<\infty\). Then all powers \(\Pi^n=(\pi_{ij}^{(n)})\), \(n>0\), of the matrix \(\Pi=\Pi(G)\) are defined (see \((^2)\), Lemma 3), and the radius of convergence \(R(G)\) of the series

\[ \sum_n \pi_{ij}^{(n)} t^n \]

does not depend on \(i\) and \(j\) (see \((^8)\)).

Theorem 5. If \(G\) is a connected graph, \(\lambda(G)<\infty\), and \(R=R(G)\), then

\[ h(G)=\log \lambda(G)=-\log R. \]

The number \(R(G)\) is called the convergence parameter of the matrix \(\Pi(G)\), and \([R(G)]^{-1}\) serves as an analogue of the maximal eigenvalue (see \((^9)\)). If \(R(G)=R\) and \(\pi_{ij}^{n}R^n\to 0\), i.e. the matrix \(\Pi(G)\) is, by definition, \(R\)-recurrent and \(R\)-positive, then \(R^{-1}\) is a genuine eigenvalue: there exist sequences of positive numbers \(x(G)=x=(x_1,x_2,\ldots)\), \(y(G)=y=(y_1,y_2,\ldots)\), such that

\[ \sum_j \pi_{ij}x_j=(1/R)x_i, \]

\[ \sum_i \pi_{ij}y_i=(1/R)y_j, \]

and the vectors \(x,y\) are determined uniquely up to a factor, and

\[ \sum_i x_i y_i<\infty \]

(see \((^9)\)).

Theorem 6. If \(G\) is a connected graph and \(\lambda(G)=1/R<\infty\), then for the existence of a measure \(\mu_0\) with maximal entropy it is necessary and sufficient that the matrix \(\Pi(G)\) be \(R\)-recurrent and \(R\)-positive. Under these conditions \(\mu_0\) is the Markov measure corresponding to transition probabilities

\[ p_{ij}=x_j\pi_{ij}/\lambda(G)x_i,\quad i,j=1,2,\ldots, \]

and stationary probabilities

\[ p_i=x_i y_i, \]

where the vectors \(x(G)=x,\ y(G)=y\) are normalized so that

\[ \sum_i x_i y_i=1. \]

4. Application to \(Y\)-systems. The results of § 2 find application in the theory of classical dynamical systems. In particular, with their help the following can be proved.

Theorem 7. Let \(T\) be a \(C^2\)-diffeomorphism of a closed \(n\)-dimensional Riemannian manifold \(M\), possessing an integral invariant and satisfying the condition \(Y\) of D. V. Anosov \((^{10})\). Then on \(M\) there exists a unique invariant, relatively \(T\)-normalized, Borel measure \(\mu_0\), for which the metric entropy \(h_{\mu_0}(T)\) assumes the maximum of the possible values, equal to the topological entropy (see \((^3,^{11})\)). The transformation \(T\) of the space \(M\) with measure \(\mu_0\) is a \(K\)-automorphism.

Remark. The existence of a measure with maximal entropy for a \(Y\)-diffeomorphism was earlier proved by R. Bowen \((^{12})\).

The author thanks E. I. Dinaburg for useful conversations.

Moscow State University
named after M. V. Lomonosov

Received
21 XI 1969

CITED LITERATURE

1. V. M. Alekseev, UMN, 23, no. 2 (1968).
2. B. M. Gurevich, DAN, 187, no. 4 (1969).
3. R. L. Adler, A. G. Konheim, M. H. McAndrew, Trans. Am. Math. Soc., 114, no. 2 (1965).
4. W. Parry, ibid., 112, no. 1 (1964).
5. R. L. Adler, B. Weiss, Proc. Nat. Acad. Sci. U.S.A., 57, no. 6 (1967).
6. E. I. Dinaburg, UMN, 23, no. 4 (1968).
7. B. M. Gurevich, DAN, 153, no. 4 (1963).
8. D. Vere-Jones, Quart. J. Math. (Oxford), Ser. 13 (1962).
9. D. Vere-Jones, Pacific J. Math., 22, no. 2 (1967).
10. D. V. Anosov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 90 (1967).
11. E. I. Dinaburg, DAN, 190, no. 1 (1970).
12. R. Bowen, Preprint. Univ. of Warwick, Coventry, 1969.

* In the presence of an integral invariant, each leaf of the expanding and contracting foliations corresponding to \(T\) is everywhere dense in \(M\) (see \((^{10})\)). Only this is needed for the proof of the last assertion of Theorem 7.