Full Text
UDC 519.53+519.217
MATHEMATICS
B. M. GUREVICH
TOPOLOGICAL ENTROPY OF A COUNTABLE MARKOV CHAIN
(Presented by Academician A. N. Kolmogorov, 2 XII 1968)
1. Introduction. Recently several authors have considered topological analogues of Markov chains (see (¹)). In this note, a topological Markov chain will mean the shift \(T_\Gamma\) in the space \(X(\Gamma)\) of infinite (from \(-\infty\) to \(+\infty\)) paths of a finite or countable directed graph \(\Gamma\). Parry (²) computed the topological entropy \(h_{\mathrm{top}}(T_\Gamma)\) (see (³)) in the case of finite \(\Gamma\) (in (²) the term “absolute entropy” is used), which turned out to be equal to \(\log \lambda(\Gamma)\), where \(\lambda(\Gamma)\) is the maximal eigenvalue of the adjacency matrix of the graph \(\Gamma\). Another result of (²) is that there exists a unique \(T_\Gamma\)-invariant measure \(\mu_0\) for which the metric entropy \(h_{\mu_0}(T_\Gamma)\) coincides with \(h_{\mathrm{top}}(T_\Gamma)\) (for the remaining measures \(h_\mu(T_\Gamma) < h_{\mathrm{top}}(T_\Gamma)\)), and after introducing this measure the topological chain turns into an ordinary Markov chain with a finite number of states*. It later became clear that topological Markov chains, both finite and countable, play an important role in the construction of symbolic dynamics for certain classes of dynamical systems (see (¹, ⁴–⁶)).
Wishing to define the topological entropy of a countable Markov chain, we encounter the noncompactness of the space \(X(\Gamma)\), because of which the usual definition loses its meaning. This difficulty can be overcome by calling the following number the topological entropy:
\[ h(T_\Gamma)=\sup_{\Gamma'}\log\lambda(\Gamma')=\sup_{\Gamma'}h_{\mathrm{top}}(T_{\Gamma'}), \]
where the supremum is taken over all finite subgraphs of the graph \(\Gamma\). The quantity \(h(T_\Gamma)\) is convenient for computation, and it is easy to construct a sequence \(\{\mu_n\}\) of \(T_\Gamma\)-invariant measures for which
\[ h_{\mu_n}(T_\Gamma)\underset{n\to\infty}{\longrightarrow} h(T_\Gamma). \]
Another natural path is to compactify the space \(X(\Gamma)\), allowing one then to apply the usual definition of topological entropy. It will be shown below that both paths lead to one and the same result. Thus, the assertion of (⁶), concerning chains of a special form, turns out to be valid in the general case. Along the way one discovers a fact which, apparently, has not previously been observed: there exists a homeomorphism \(T\) of a compact metric space such that for every \(T\)-invariant measure \(\mu\),
\[ h_\mu(T)<h_{\mathrm{top}}(T). \]
A corresponding example can be provided by the topological Markov chain described in Section 3.
2. Main result. Identify the set of vertices of the graph \(\Gamma\) under consideration with the set \(Z\) of natural numbers. We shall assume that the graph \(\Gamma\) is connected, i.e., for any two of its vertices \(i,j\) there exists a path leading from \(i\) to \(j\). The number of edges entering a path \(\gamma\) is called its length. The adjacency matrix \(M(\Gamma)=(m_{i,j}(\Gamma))\) is defined
* All these results are repeated in (⁴).
by the equality
\[ m_{ij}(\Gamma)= \begin{cases} 1, & \text{if there exists a path of length 1 from } i \text{ to } j,\\ 0 & \text{otherwise.} \end{cases} \]
The set \(X(\Gamma)\) consists of all infinite (in both directions) sequences \(\{i_n\}\) for which \(m_{i_n i_{n+1}}=1\) for every \(n=0,\pm 1,\pm 2,\ldots\). From the definition it is clear that \(X(\Gamma)\) is invariant with respect to the shift taking the sequence \(\{i_n\}\) into \(\{i_n'\}\), where \(i_n'=i_{n-1}\). Complete the set \(Z\) with the discrete topology by the point \(\infty\) so as to obtain the compact space \(\overline Z\). Then the space \(\overline X=\prod_{-\infty<n<\infty}\overline Z_n\), where \(\overline Z_n=\overline Z\), is also compact and \(X(\Gamma)\subset \overline X\). On \(\overline X\) one can introduce a metric by setting (see (6))
\[ \rho(\{i_n\}\{j_n\})= \sum_{-\infty<n<\infty} \frac{1}{2^{|n|}}\, \frac{1-\delta_{i_n,j_n}}{\min(i_n,j_n)}; \qquad i_n,j_n=1,2,\ldots,\infty \]
(it is assumed that \(\min(i,\infty)=i,\ \min(\infty,\infty)=\infty,\ 1/\infty=0/\infty=0\)).
The closure \(\overline{X(\Gamma)}\) of the set \(X(\Gamma)\) in \(\overline X\) is a compact metric space, and the shift \(T_\Gamma\) is its homeomorphism.
Theorem. \(h(T_\Gamma)=h_{\mathrm{top}}(T_\Gamma)\).
The proof of the theorem is based on several auxiliary assertions.
Lemma 1. For any connected graph \(\Gamma\) there exists a sequence \(\{\Gamma_k\}\) of its finite connected subgraphs such that: 1) \(\Gamma_k\) is a subgraph of the graph \(\Gamma_{k+1}\), \(k=1,2,\ldots\); 2) if \(i\le k^2\), then \(i\) belongs to the set \(G(\Gamma_k)\) of vertices of the graph \(\Gamma_k\). *
Lemma 2. Let \(G\) be an arbitrary finite set of vertices of the connected graph \(\Gamma\). Then there exist a connected subgraph \(\Gamma'(G)\) of the graph \(\Gamma\) and a number \(l'(G)\) such that, for any path in \(\Gamma\) of length \(\ge l'(G)\) with beginning and end in \(G\), there is a path in \(\Gamma'(G)\) of the same length and with the same beginning and end.
Remark. From this formulation it is clear that \(G\) is contained in the set of vertices of the graph \(\Gamma'(G)\).
Lemma 3. If \(G\) is any finite set of vertices of a graph \(\Gamma\) satisfying the condition \(h(T_\Gamma)<\infty\), and \(l\) is any positive number, then in \(\Gamma\) there can be found only finitely many paths of length \(\le l\) beginning and ending in \(G\).
Lemma 4. Let \(P_n(r)\) be the number of sequences of length \(n\) of zeros and ones in which every run of zeros has length \(\ge r\). Then
\[ \lim_{r\to\infty}\lim_{n\to\infty}\frac{1}{n}\log P_n(r)=0. \]
Sketch of the proof of the theorem. Let \(\xi_k\) be the partition of the space \(\overline{X(\Gamma)}\), defined as follows: the points \(\{i_n\}\) and \(\{j_n\}\) belong to one element \(C_\xi\) of the partition \(\xi_k\) if and only if \(i_0\notin G(\Gamma_k),\ j_0\notin G(\Gamma_k)\), or \(i_0\in G(\Gamma_k),\ j_0\in G(\Gamma_k),\ i_0=j_0\). The partition \(\xi_k\) is at the same time an open cover. Put
\[ \eta_k=\bigvee_{i=-k}^{k} T_\Gamma^{i}\xi_k. \]
By Lemma 1,
\[ \eta_k\le \eta_{k+1},\qquad \max_{C_{\eta_k}}(\operatorname{diam} C_{\eta_k})\xrightarrow[k\to\infty]{}0. \]
It follows from this (see (3)) that
\[ \lim_{k\to\infty} h_{\mathrm{top}}(T_\Gamma,\xi_k) = \lim_{k\to\infty} h_{\mathrm{top}}(T_\Gamma,\eta_k) = h_{\mathrm{top}}(T_\Gamma). \tag{1} \]
* Any subgraph of a given graph is uniquely determined by the set of its vertices.
Let us further note that the sequence \(\{\Gamma_k\}\) described in Lemma 1 necessarily has the property that any finite subgraph \(\Gamma'\) of the graph \(\Gamma\) is a subgraph of some \(\Gamma_k\). In this case \(\lambda(\Gamma') \leq \lambda(\Gamma_k)\) and, consequently,
\[ \lim_{k\to\infty} \log \lambda(\Gamma_k)=h(T_\Gamma). \tag{2} \]
Let now \(N_n(\Gamma_k)\) be the number of paths of length \(n\) in the graph \(\Gamma_k\), and let \(N_n(\xi_k)\) be the number of elements of the partition (covering)
\[ \xi_k^n=\bigvee_{i=0}^{-n} T_\Gamma^i \xi_k . \]
From (1), (2), using the obvious inequality \(N_n(\xi_k)\geq N_n(\Gamma_k)\) and the fact that
\[ h_{\mathrm{top}}(T_\Gamma,\xi_k)=\lim_{n\to\infty}\frac{1}{n+1}\log N_n(\xi_k),\qquad \log\lambda(\Gamma_k)=\lim_{n\to\infty}\log\frac{1}{n+1}N_n(\Gamma_k), \]
we obtain
\[ h_{\mathrm{top}}(T_\Gamma)\geq h(T_\Gamma). \]
In proving the reverse inequality, it suffices to restrict ourselves to the case \(h(T_\Gamma)<\infty\). Fix an arbitrary \(k\) and, applying Lemma 2 to the set \(G_k=G(\Gamma_k)\), obtain a subgraph \(\Gamma'(G_k)\) and a number \(l'(G_k)\). Next fix any number \(r\geq l'(G_k)\) and, using Lemma 3 with \(G=G_k\), \(l=r\), extend the subgraph \(\Gamma'(G_k)\) to a finite connected subgraph \(\Gamma''(G_k)=\Gamma''\), possessing the following properties: 1) every path in \(\Gamma\) of length \(\leq r\) with beginning and end in \(G_k\) is a path in \(\Gamma''\); 2) for every path in \(\Gamma\) of length \(>r\) with beginning and end in \(G_k\), there is a path of the same length in \(\Gamma''\) with the same beginning and end.
These two properties of the graph \(\Gamma''\) make it possible to prove the fundamental inequality
\[ N_n(\xi_k)\leq 2r(r+1)N_n(\Gamma'')P_{n+1}(r), \tag{3} \]
where \(N_n(\Gamma'')\) is the number of paths of length \(n\) in the graph \(\Gamma''\), and \(P_{n+1}(r)\) is defined in the condition of Lemma 4. From (3) it follows that
\[ \begin{aligned} h_{\mathrm{top}}(T_\Gamma,\xi_k) &\leq \lim_{n\to\infty}\frac{1}{n+1}\log(2r(r+1)) +\lim_{n\to\infty}\frac{1}{n+1}\log N_n(\Gamma'') \\ &\quad +\lim_{n\to\infty}\frac{1}{n+1}\log P_{n+1}(r) \leq \log\lambda(\Gamma'')+\lim_{n\to\infty}\frac{1}{n+1}\log P_{n+1}(r) \\ &\leq h(T_\Gamma)+\lim_{n\to\infty}\frac{1}{n+1}\log P_{n+1}(r). \end{aligned} \tag{4} \]
Using Lemma 4, one can choose \(r\) from the very beginning so that the second term on the right-hand side of inequality (4) does not exceed a preassigned \(\varepsilon\). This means that
\[ h_{\mathrm{top}}(T_\Gamma)\leq h(T_\Gamma) \]
and, consequently,
\[ h_{\mathrm{top}}(T_\Gamma)=h(T_\Gamma). \]
- Example. Consider the topological Markov chain corresponding to a graph \(\Gamma\) in whose adjacency matrix the only nonzero elements are
\[ m_{11}(\Gamma),\ m_{i,i+1}(\Gamma),\ m_{i+1,i}(\Gamma),\quad i=1,2,\ldots . \]
In this case the closure \(\bar X(\Gamma)\) is obtained from \(\dot X(\Gamma)\) by adjoining a single point \(x_\infty=(\ldots,\infty,\infty,\infty,\ldots)\). If \(\Gamma_k\) is the subgraph with vertices \(1,2,\ldots,k\), then, as is not hard to verify,
\[ 2-\frac{1}{k}\leq \lambda(\Gamma_k)\leq 2. \]
On the basis of the theorem of Section 2, it follows from this that
\[ h_{\mathrm{top}}(T_\Gamma)=h(T_\Gamma)=\log 2. \]
We shall show that for any measure \(\mu\) invariant with respect to \(T_{\Gamma}\),
\[ h_{\mu}(T_{\Gamma}) < \log 2. \]
If this is not so, then there exists a measure \(\mu_{0}\) such that \(h_{\mu_{0}}(T_{\Gamma})=\log 2\). Without loss of generality, one may assume that \(\mu_{0}(x_{\infty})=0\). The well-known theorems of the entropy theory of dynamical systems (see (7)) lead to the relations
\[ \lim_{k\to\infty} h_{\mu_{0}}(T_{\Gamma},\xi_k) = \lim_{k\to\infty} H_{\mu_{0}}(\xi_k\mid \xi_{\bar{k}}) = \lim_{k\to\infty}\int_{X(\Gamma)} H_{\mu_{0}}(\xi_k\mid C_{\xi_{\bar{k}}}(x))\,d\mu_{0} = \log 2. \tag{5} \]
(Here \(C_{\xi_{\bar{k}}}(x)\) is the element of the partition
\[
\xi_{\bar{k}}=\bigvee_{i<0} T^{i}\xi_k
\]
that contains the point \(x\).)
Since any element \(C_{\xi_{\bar{k}}}\) can intersect only two elements of the partition \(\xi_k\), i.e. \(H_{\mu_{0}}(\xi_k\mid C_{\xi_{\bar{k}}}(x))\leqslant \log 2\), it follows from (5) that
\[ H_{\mu_{0}}(\xi_k\mid C_{\xi_{\bar{k}}}(x)) \xrightarrow[\text{in measure}]{} \log 2. \tag{6} \]
Let \(E_s\) be the set of points \(x=\{i_k\}\) for which \(i_0=s\). The collection of sets \(E_s\), \(s=1,2,\ldots\), forms a certain partition \((\bmod\,0)\xi\) of the space \(X(\Gamma)\). Put
\[ p_1(x)= \begin{cases} 1/2, & \text{if } x\in T_{\Gamma}^{-1}(E_1\cup E_2),\\ 0, & \text{otherwise;} \end{cases} \qquad p_s(x)= \begin{cases} 1/2, & \text{if } x\in T_{\Gamma}^{-1}(E_{s-1}\cup E_{s+1}),\\ 0, & \text{otherwise;} \end{cases} \]
\[ s=2,3,\ldots \]
Using standard techniques, from (6) it is easy to derive that for almost all \(x\in X(\Gamma)\)
\[ \mu_{0}(E_s\mid C_{\xi^-}(x)) = \lim_{k\to\infty}\mu_{0}(E_s\mid C_{\xi_{\bar{k}}}(x)) = p_s(x), \qquad s=1,2,\ldots . \]
But this means that \(\mu_{0}\) is a Markov measure corresponding to the transition matrix \(P=(p_{ij})\), where
\[ p_{ij}= \begin{cases} 1/2, & \text{if } i=j=1 \text{ or } |i-j|=1,\\ 0, & \text{otherwise.} \end{cases} \]
At the same time it is known (see, for example, (8)) that a Markov chain with such a transition matrix has no finite invariant measure.
Moscow State University
named after M. V. Lomonosov
Received
27 XI 1968
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