A note on transitive topological Markov chains of given entropy and period Jérôme Buzzi and Sylvie Ruette November, 205 Abstract We show that, for every positive real number h and every positive integer p, there exist oriented graphs G, G (with countably many vertices) that are strongly connected, of period p, of Gurevich entropy h, such that G is positive recurrent (thus the topological Markov chain on G admits a measure of maximal entropy) and G is transient (thus the topological Markov chain on G admits no measure of maximal entropy). We also show that any transitive topological Markov chain with infinite entropy carries uncountably many ergodic, invariant probability measures with infinite entropy. Vere-Jones classification of graphs Definition Let G be an oriented graph and let u, v be two vertices in G. We define the following quantities. p G uv(n) is the number of paths u 0 u u n such that u 0 = u and u n = v; R uv (G) is the radius of convergence of the series p G uv(n)z n. f G uv(n) is the number of paths u 0 u u n such that u 0 = u, u n = v and u i v for all 0 < i < n; L uv (G) is the radius of convergence of the series f G uv(n)z n. Definition 2 Let G be an oriented graph and V its set of vertices. The graph G is strongly connected if for all u, v V, there exists a path from u to v in G. The period of a strongly connected graph G is the greatest common divisor of (p G uu(n)) u V,n 0. The graph G is aperiodic if its period is. Proposition 3 (Vere-Jones [8]) Let G be an oriented graph. R uv (G) does not depend on u and v; it is denoted by R(G). If G is strongly connected, If there is no confusion, R(G) and L uv (G) will be written R and L uv. In [8] Vere-Jones gives a classification of strongly connected graphs as transient, null recurrent or positive recurrent. The definitions are given in Table (lines and 2) as well as properties of the series p G uv(n)z n which give an alternative definition. Proposition 4 (Salama [7]) Let G be a strongly connected oriented graph. If G is transient or null recurrent, then R = L uu for all vertices u. Equivalently, if there exists a vertex u such that R < L uu, then G is positive recurrent.
transient null positive recurrent recurrent fuu(n)r G n < n>0 nfuu(n)r G n + + < + n>0 p G uv(n)r n < + + + n 0 lim n + pg uv(n)r n 0 0 λ uv > 0 R = L uu R = L uu R L uu Table : properties of the series associated to a transient, null recurrent or positive recurrent graph G (G is strongly connected); these properties do not depend on the vertices u, v. 2 Topological Markov chains and Gurevich entropy Let G be an oriented graph and V its set of vertices. Then Γ G is the set of two-sided infinite paths in G, that is, Γ G = {(v n ) n Z n Z, v n v n+ in G} V Z. σ is the shift on Γ G. The topological Markov chain on the graph G is the system (Γ G, σ). The set V is endowed with the discrete topology and Γ G is endowed with the induced topology of V Z. The space Γ G is not compact unless G is finite. The topological Markov chain (Γ G, σ) is transitive if and only if the graph G is strongly connected. It is topologically mixing if and only if the graph G is strongly connected and of period. If G is a finite graph, Γ G is compact and the topological entropy h top (Γ G, σ) is well defined (see e.g. [2] for the definition of the topological entropy). If G is a countable graph, the Gurevich entropy [3] of the graph G (or of the topological Markov chain Γ G ) is given by h(g) = sup{h top (Γ H, σ) H G, H finite}. This entropy can also be computed in a combinatorial way, as the exponential growth of the number of paths with fixed endpoints. Proposition 5 (Gurevich [4]) Let G be a strongly connected oriented graph. Then for all vertices u, v h(g) = lim n + n log pg uv(n) = log R(G). Moreover, the variational principle is still valid for topological Markov chains. Theorem 6 (Gurevich [3]) Let G be an oriented graph. Then h(g) = sup{h µ (Γ G ) µ σ-invariant probability measure}. In this variational principle, the supremum is not necessarily reached. The next theorem gives a necessary and sufficient condition for the existence of a measure of maximal entropy (that is, a measure µ such that h(g) = h µ (Γ G )) when the graph is strongly connected. 2
Theorem 7 (Gurevich [4]) Let G be a strongly connected oriented graph of finite positive entropy. Then the topological Markov chain on G admits a measure of maximal entropy if and only if the graph G is positive recurrent. Moreover, such a measure is unique if it exists. 3 Construction of graphs of given entropy and given period that are either positive recurrent or transient Lemma 8 Let β (, + ). There exist a sequence of non negative integers (a(n)) and positive constants c, M such that a() =, a(n) β n =, n 2, c β n2 n a(n 2 ) c β n2 n + M, n, 0 a(n) M if n is not a square. These properties imply that the radius of convergence of a(n)zn is L = β na(n)ln < +. Proof. First we look for a constant c > 0 such that We have Thus n 2 and that β + c β n2 n =. () β n2 n 2 β n2 n β n2 = β n = n 2 β(β ). () β + c β(β ) = c = (β )2. Since β >, the constant c is positive. We define the sequence (b(n)) by: Then b() :=, b(n 2 ) := cβ n2 n for all n 2, b(n) := 0 for all n 2 such that n is not a square. b(n) β n β + c β n2 n =. β n2 n 2 We set δ := b(n) β [0, [ and k := β 2 δ. Then k β 2 δ < k + < k + β, which n implies that 0 δ k < β 2 β. We write the β-expansion of δ k (see e.g. [, p 5] for the β 2 definition): there exist integers d(n) {0,..., β } such that δ k = β 2 d(n) β. Moreover, n d() = 0 because δ k < β 2 β. Thus we can write δ = n 2 d (n) β n where d (2) = d(2) + k and d (n) = d(n) for all n 3. We set a() := b() and a(n) := b(n) + d (n) for all n 2. Let M := β + k. We then have: 3
a() =, a(n) β n =, n 2, c β n2 n a(n 2 ) c β n2 n + β c β n2 n + M, 0 a(2) β + k = M, n 3, 0 a(n) β M if n is not a square. The radius of convergence L of a(n)zn satisfies log L = lim sup n + Thus L = β. Moreover, log a(n) = n lim n + na(n) β n M n β n + c n 2 β n2 n β n2 n 2 log a(n2 ) = log β because a(n 2 ) cβ n2 n. = M n β n + c n 2 β n < +. Lemma 9 ([5], Lemma 2.4) Let G be a strongly connected oriented graph and u a vertex. i) R < L uu if and only if f G uu(n)l n uu >. ii) If G is recurrent, then R is the unique positive number x such that f G uu(n)x n =. Proof. For (i) and (ii), use Table and the fact that F (x) = f uu(n)x G n is increasing for x [0, + [. Proposition 0 Let β (, + ). There exist aperiodic strongly connected graphs G (β) G(β) such that h(g(β)) = h(g (β)) = log β, G(β) is positive recurrent and G (β) is transient. Remark: Salama proved the part of this proposition concerning positive recurrent graphs in [6, Theorem 3.9]. Proof. This is a variant of the proof of [5, Example 2.9]. Let u be a vertex and let (a(n)) be the sequence given by Lemma 8 for β. The graph G(β) is composed of a(n) loops of length n based at the vertex u for all n (see Figure ). More precisely, define the set of vertices of G(β) as where the vertices v n,i k arrow v n,i k v n,i k+ V = {u} + {v n,i n= k i a(n), k n }, above are distinct. Let v n,i 0 = vn n,i = u for all i a(n). There is an for all 0 k n, i a(n), n 2, there is an arrow u u, and there is no other arrow in G(β). The graph G(β) is strongly connected and f G(β) uu n. By Lemma 8, the sequence (a(n)) is defined such that L = β and (n) = a(n) for all a(n)l n =, (2) 4
u Figure : the graphs G(β) and G (β); the bold loop (on the left) is the only arrow that belongs to G(β) and not to G (β), otherwise the two graphs coincide. where L = L uu (G(β)) is the radius of convergence of the series a(n)z n. If G(β) is transient, then R(G(β)) = L uu (G(β)) by Proposition 4, but Equation (2) contradicts the definition of transient (see the first line of Table ). Thus G(β) is recurrent, and R(G(β)) = L by Equation (2) and Lemma 9(ii). Moreover na(n)l n < + by Lemma 8, and thus the graph G(β) is positive recurrent (see Table ). By Proposition 5, h(g(β)) = log R(G(β)) = log β. The graph G (β) is obtained from G(β) by deleting a loop u u of length n 0 for some n 0 2 such that a(n 0 ) (such an integer n 0 exists because L < + ). Obviously one has L uu (G (β)) = L and f G (β) uu (n)l n = L n 0 <. Since R(G (β)) L uu (G (β)), this implies that G (β) is transient. Moreover R(G (β)) = L uu (G (β)) by Proposition 4, so R(G (β)) = R(G(β)), and hence h(g (β)) = h(g(β)) by Proposition 5. Finally, both G(β) and G (β) are of period because of the loop u u. Corollary Let p be a positive integer and h (0, + ). There exist strongly connected graphs G, G of period p such that h(g) = h(g ) = h, G is positive recurrent and G is transient. Proof. For G, we start from the graph G(β) given by Proposition 0 with β = e hp. Let V denote the set of vertices of G(β). The set of vertices of G is V {,..., p}, and the arrows in G are: (v, i) (v, i + ) if v V, i p, (v, p) (w, ) if v, w V and v w is an arrow in G(β). According to the properties of G(β), G is strongly connected, of period p and positive recurrent. Moreover, h(g) = p h(g(β)) = p log β = h. For G, we do the same starting with G (β). According to Theorem 7, the graphs of Corollary satisfy that the topological Markov chain on G admits a measure of maximal entropy the topological Markov chain on G admits no measure of maximal entropy ; both are transitive, of entropy h and carried by a graph of period p. 5
4 Topological Markov chains with infinite entropy Proposition 2 Any transitive topological Markov chain with infinite entropy carries uncountably many ergodic, invariant probability measures with infinite entropy. Proof. Let (Γ G, σ) be a topological Markov chain, where G is a strongly connected graph such that h(g) = +. Let v be a vertex of G and choose a sequence (n k ) k such that fvv(n G k ) 2 k2k n k (such a sequence exists because h(g) = + ). Set a k := fvv(n G k ) and define f = k= a kz n k. The loop shift σ f embeds into the topological Markov chain (Γ G, σ). For all k, define q k = (2 k a k n k ) and p 0 = k= q k. Consider σ f as a vertex shift. Let µ be the Markov measure assigning the base vertex v probability p 0, with transition probability q k /p 0 from v to each vertex following v and lying in a first return loop of length n k. Then µ is an ergodic, atomless invariant probability measure of σ f and h µ (σ f ) = +. By use of other subsequences, we easily get uncountably many distinct ergodic measures with infinite entropy. References [] K. Dajani and C. Kraaikamp. Ergodic Theory of Numbers. Number v. 29 in Carus Mathematical Monographs. Mathematical Association of America, 2002. [2] M. Denker, C. Grillenberger, and K. Sigmund. Ergodic theory on compact spaces. Lecture Notes in Mathematics, no. 527. Springer-Verlag, 976. [3] B. M. Gurevich. Topological entropy of enumerable Markov chains (Russian). Dokl. Akad. Nauk SSSR, 87:75 78, 969. English translation Soviet Math. Dokl, 0(4):9 95, 969. [4] B. M. Gurevich. Shift entropy and Markov measures in the path space of a denumerable graph (Russian). Dokl. Akad. Nauk SSSR, 92:963 965, 970. English translation Soviet Math. Dokl, (3):744 747, 970. [5] S. Ruette. On the Vere-Jones classification and existence of maximal measures for topological Markov chains. Pacific J. Math., 209(2):365 380, 2003. [6] I. A. Salama. Topological entropy and recurrence of countable chains. Pacific J. Math., 34(2):325 34, 988. Errata, 40(2):397, 989. [7] I. A. Salama. On the recurrence of countable topological Markov chains. In Symbolic dynamics and its applications (New Haven, CT, 99), Contemp. Math, 35, pages 349 360. Amer. Math. Soc., Providence, RI, 992. [8] D. Vere-Jones. Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. (2), 3:7 28, 962. 6