gives an alternative example of a positive recurrent graph which is not strongly positive recurrent; the number of the first return loops is given by

Transcription

1 On the Vere-Jones classification and existence of maximal measures for topological Markov chains Sylvie Ruette Abstract We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) entropy. Introduction In this article we are interested in connected oriented graphs and topological Markov chains. All the graphs we consider have a countable set of vertices. If G is an oriented graph, let G be the set of two-sided infinite paths in G and let ff denote the shift transformation. The Markov chain associated to G is the (non compact) dynamical system ( G ;ff). The entropy h(g) of the Markov chain G was defined by Gurevich; it can be computed by several ways and satisfies the Variational Principle [7],[8]. In [4] Vere-Jones classifies connected oriented graphs as transient, null recurrent or positive recurrent according to the properties of the series associated with the number of loops. To a certain extend, positive recurrent graphs resemble finite graphs. In [7] Gurevich shows that a Markov chain admits a measure of maximal entropy (also called maximal measure) if and only if the graph is positive recurrent. In this case, this measure is unique and it is an ergodic Markov measure. When the graph is positive recurrent, its incidence matrix admits a maximal eigenvalue and the associated left and right eigenvectors allow to define a probability Markov measure which is the unique maximal measure; this construction is the same as in the finite case. In [] Salama gives a geometric approach to the Vere-Jones classification. The fact that a graph can (or cannot) be extended" or contracted" without changing its entropy is closely related to its class. Salama first thought that these properties entirely determine whether a graph is transient, null recurrent of positive recurrent. In particular he stated that a graph is positive recurrent if and onlyifithas no proper subgraph of equal entropy but he soon realized that the proof of the only if" part was false (see Theorem 2.3 and Errata in []). The result itself is not valid: in [5] Fiebig gives an example of a locally compact positive recurrent graph without any subgraph of equal entropy. This result shows that the positive recurrent class splits into two subclasses: agraph is called strongly positive recurrent if it has no proper subgraph of equal entropy; it is equivalent to acombinatorial condition (a finite connected graph always satisfies this property). In [] Salama also states that a graphs is transient ifand only if it can be extended to a bigger transient graph of equal entropy. We show that any transient graph G is contained in a recurrent graph of equal entropy, which is positive ornull recurrent depending on the properties of G. We illustrate the two possibilities a transient graph with a positive ornull recurrent extension by an example. It also

2 gives an alternative example of a positive recurrent graph which is not strongly positive recurrent; the number of the first return loops is given by the graph structure in an obvious way, which leads to a straightforward computation. The result of Gurevich entirely solves the question of existence of a maximal measure in term of graph classification. Nevertheless it is not so easy to prove that a graph is positive recurrent and one may wish to have more efficient criteria. In [6] Gurevich and Zargaryan give a sufficient condition for existence of a maximal measure, which we prove to be an equivalent condition for strong positive recurrence; it is formulated in terms of exponential growth of the number of paths inside and outside a finite subgraph. We give a new sufficient criterion based on local entropy. Why consider local entropy? For a compact dynamical system, it is known that a null local entropy implies the existence of a maximal measure ([9], see also [] for a similar but different result). This result may be strengthen in some cases: it is conjectured that, if f is a map of the interval which isc r, r>, and satisfies h top (f) >h loc (f), then there exists a maximal measure [2]. Our initial motivation comes from the conjecture above because smooth interval maps and Markov chains are closely related. If f:[; ]! [; ] is C +ff (i.e. f is C and f is ff-hölder with ff>) with h top (f) > then an oriented graph G can be associated to f, G is connected if f is transitive, and there is a bijection between the maximal measures of f and those of G [2], [3]. We show that a Markov chain is strongly positive recurrent, thus admits a maximal measure, if its local entropy is strictly less that its Gurevich entropy. However this result does not apply directly to interval maps since the isomorphism" between f and its Markov extension does not preserve local entropy. The article is organized as follows. Section contains definitions and basic properties on oriented graphs and Markov chains. In Section 2, after recalling the definitions of transient, null recurrent and positive recurrent graphs and some related properties, we show that any transient graph is contained in a recurrent graph of equal entropy (Proposition 2.5) and we give an example of a transient graph which extends to a positive recurrent (resp. null recurrent) graph. Section 3 is devoted to the problem of existence of maximal measures: Theorem 3.3 shows the converse result of the Theorem of Gurevich-Zargaryan and Theorem 3.6 gives a sufficient condition for the existence of a maximal measure, based on local entropy. The last Section gives a detailed proof of the Theorem of Gurevich-Zargaryan. Background. Graphs and paths Let G be an oriented graph with a countable set of vertices V (G). If u; v are two vertices, there is at most one arrow u! v. A path of length n is a sequence (u ; ;u n ) such that u i! u i+ in G for» i <n. This path is called a loop if u = u n. The graph G is connected (or strongly connected) if for all vertices u; v there exists a path in G from u to v. If H is a subgraph of G, we write H ρ G; if in addition H 6= G, we write H / G and say that H is a proper subgraph. If W is a subset of V (G), the set V (G) n W is denoted by W. We also denote by W the subgraph of G whose vertices are W and whose edges are all edges of G between two vertices in W. Let u; v be two vertices. We define the following quantities. ffl p G uv(n) isthenumber of paths (u ; ;u n ) such that u = u and u n = v; R uv (G) is the radius of convergence of the series P p G uv(n)z n. 2

3 ffl f G uv(n) is the number of paths (u ; ;u n ) such thatu = u, u n = v and u i 6= v for <i<n; L uv (G) is the radius of convergence of the series P f G uv(n)z n. In [4] Vere-Jones shows the following solidarity properties on R uv (G) andl uv (G) for connected graphs. Proposition. (Vere-Jones) Let G be an oriented graph. If G is connected, R uv (G) and L uv (G) do not depend on u and v; they are denoted by R(G) and L(G). If there is no confusion, R(G) and L(G) will be written R and L. For a graph G these two radii will be written R and L..2 Markov chains Let G be an oriented graph. G is the set of two-sided infinite paths in G, that is, G = f(v n ) n2z j8n 2 Z;v n! v n+ in Gg ρ(v (G)) Z : ff is the shift on G. The (topological) Markov chain on the graph G is the system ( G ;ff). The set V (G) is endowed with the discrete topology and G is endowed with the induced topology of (V (G)) Z, which has the product topology. The space G is not compact unless G is finite. A compatible distance on G is given by d, defined as follows: fi fi V (G) isidentified with N and the distance D on V (G) isgiven by D(n; m) = fi fi 2 n fi fififi 2 m» : If μu =(u n ) n2z and μv =(v n ) n2z are two elements of G, d(μu; μv) = n2z D(u n ;v n ) 2 jnj» 3: The Markov chain G is transitive if for any nonempty open sets A; B ρ G there exists n> such thatff n (A) B 6= ;. Equivalently, G is transitive if and only if the graph G is connected. In the sequel we will be interested in connected graphs only..3 Entropy If G is a finite graph, G is compact and the topological entropy h top ( G ;ff) is well defined (see e.g. [4] for the definition of the topological entropy in the general case). If G is a countable graph, the Gurevich entropy [7] of G is given by h(g) =supfh top ( H ;ff) j H ρ G; H finiteg: This entropy can also be computed in a combinatorial way, as the exponential growth of the number of paths with fixed endpoints [8]. Proposition.2 (Gurevich) h(g) = Let G be a connected oriented graph. Then for any vertices u; v lim n!+ n log pg uv(n) = log R(G): Another way to compute the entropy is to compactify the space G and then use the definition of topological entropy for compact metric spaces. If G is an oriented graph, denote the one-point 3

4 compactification of V (G) by V (G) [fg and define G as the closure of G in (V (G) [ fg) Z. The distance D (resp. d) naturally extends to V (G) [fg (resp. G ). In [7] Gurevich shows that this gives the same entropy; this means that there is only very little dynamics added in this compactification. Moreover, the Variational Principle is still valid for Markov chains [7]. Theorem.3 (Gurevich) Let G be an oriented graph. Then h(g) =h top ( G ;ff) = supfh μ ( G ) j μff-invariant probability measureg: Moreover, the supremum can be taken on Markov measures only. 2 On the classification of connected graphs 2. Transient, null recurrent and positive recurrent graphs In [4] Vere-Jones gives a classification of non negative irreducible matrices as transient, null recurrent or positive recurrent. This terminology extends to graphs according to the classification of their incidence matrices. If G is a connected oriented graph and v is a vertex of G, Vere-Jones shows that P f G vv(n)r n» and he calls the graph recurrent (resp. transient) if the series P f G vv(n)r n is equal to (resp. is less than ). In the recurrent case, G is null recurrent (resp. positive recurrent) if the series P nf G vv(n)r n is equal to + (resp. is finite). These definitions do not depend on the vertex v. They are recalled in Table, as well as properties of the series P p G uv(n)z n, which give an alternative definition. n> n> n transient null positive recurrent recurrent f G vv(n)r n < nf G vv(n)r n» + + < + p G uv(n)r n < lim n!+ pg uv(n)r n uv > R = L R = L R» L Table : properties of the series associated to a transient, null recurrent or positive recurrent graph G (G is connected and u; v are any two vertices). For a connected oriented graph, it is obvious that R» L. In [] Salama shows that, if R<L, then the graph is positive recurrent. Thus R = L if the graph is transient or null recurrent (see Table ). Moreover, a graph is transient if and only if one can extend" it without changing the entropy (Theorem 2.). In particular, a proper subgraph of equal entropy is necessarily transient. Salama also proves that, if R = L, then there exists a subgraph G / G with the same entropy as G; we prove that the converse is also true (Theorem 2.2). Theorem 2. (Salama) Let G beaconnected oriented graph of finite entropy. (i) P R<Lif and only if f G uu(n)l n >, which implies that G is positive recurrent. 4

5 (ii) G is transient if and only if there exists G / G satisfying h(g )=h(g); the graph G can be chosen to betransient. Theorem 2.2 Let G be a connected oriented graph of finite entropy. Then R = L if and only if there exists G / G such that h(g )=h(g). Proof. The only if" part is Theorem 2.2 in []. It is proven as follows. Suppose that R = L. If for all vertices u there is only one arrow starting from u then h(g) = and any proper subgraph G satisfies h(g) = h(g ). Otherwise choose an arrow u! v such that there exists w 6= v with u! w. Define G as the graph G deprived of the arrow u! v and G 2 as the graph G deprived of the arrows u! w for all w 6= v. Since fuu(n) G =f G uu (n)+f G 2 uu (n), there exists i 2f; 2g such that L(G) =L uu (G i ). One has R(G) =L(G) =L uu (G i ) R uu (G i ) R(G); thus R(G) =R uu (G i ), which implies that the connected component ofg i that contains u has the same entropy asg. Conversely, suppose that there exists a subgraph G / G such that h(g) =h(g ), i.e. R = R. By Theorem 2. (ii), the graph G is transient, thus R = L. As G ρ G, one has fvv G (n)» fvv(n) G for any vertex v of G and any integer n, thus L L. This implies R» L» L = R,soR = L. Definition 2.3 Let G be a connected oriented graph. G is called strongly positive recurrent if it has no proper subgraph of equal entropy. Proposition 2.4 are equivalent: Let G be a connected oriented graph of finite entropy. The following properties ffl G is strongly positive recurrent, ffl R<L, ffl f G uu(n)l n > for some vertex u. Proof. This is a straightforward consequence of Theorem 2.2 and Theorem 2.(i). 2.2 Recurrent extensions of equal entropy of transient graphs We show that any transient graph G can be extended to a recurrent graph without changing the entropy by adding a (possibly infinite) number of loops. If the series P nf G vv(n)r n is finite then the obtained recurrent graph is positive recurrent (but not strongly positive recurrent), otherwise it is null recurrent. Proposition 2.5 Let G be a transient graph of finite non zero entropy. Then there exists a recurrent graph G ff G such that h(g) =h(g ). Moreover G can be chosen to be positive recurrent if n> nf G uu(n)r n < + for some vertex u of G, and G is necessarily null recurrent otherwise. 5

6 Proof. The entropy of G is finite and non zero thus <R< and there exists an integer p such that» pr <. Define ff = pr. Let u be a vertex of G and define 2 D = f G uu(n)r n < : One has ff n 2 n =: thus n k+ ff n = ff k ff n ff k : () We build a sequence of integers (n i ) i2i suchthat2 P i2i ffn i = D. For this, we define inductively a strictly increasing (finite or infinite) sequence (n i ) i2i such that for all k 2 I k i= ff n i» D 2 < k i= ff n i + n>n k ff n : Letn be the greatest integer n 2 such that ff k > D 2 (this is possible because ff k 2 > k n k 2 D 2 ). By choice of n one has ff n» D 2,thus ffn» D by(). This is the required property 2 n n + at rank. Suppose that n ; ;n k are already defined. If P k i= ff n i = D then I = f; ;kg and we stop 2 the construction. Otherwise let n k+ be the greatest integer n>n k such that By choice of n k+ and (), one has ff n k+» This is the required property at rank k +. k ff n i + ff j > D 2 : i= j n j n k+ + Define a new graph G ff G by adding2p n i ffl the additional vertices of G are fv i;j k distinct, ffl G has the additional arrows v i;j k n i ). ff j» D 2 k i= ff n i : loops of length n i based at the vertex u, that is, j i 2 I;» j» 2pn i ; <k<n i g, where all the v i;j k are! vi;j, u! k+ vi;j and vn i;j i! u (i 2 I;» j» 2p n i ; <k< Notice that n i 2 so that we do not add the arrow u! u (which may already exists in G), thus the graph G is well defined. Obviously, one has R» R. It is clear that fuu(n G i )=fuu(n G i )+2p n i and fuu(n) G =fuu(n) G ifn62 fn i ;i 2 Ig. Moreover, one has P i2i(pr) n i = D by construction. Therefore 2 fuu(n)r G n = fuu(n)r G n + 2(pR) n i =: (2) i2i 6

7 This implies that R L R thus R = R and h(g) =h(g ). Moreover G is recurrent because of (2) and of the equality L = R. In addition, nfuu(n)r G n = nfuu(n)r G n + n i ff n i i2i and this quantity is finite if and only if P nf G uu(n)r n is finite. In this case the graph G is positive recurrent. If P nf G uu(n)r n =+, let H be a recurrent graph containing G with h(h) =h(g). Then and H is null recurrent. nf H uu(n)r n nf G uu(n)r n =+ Example 2.6 We build a positive (resp. null) recurrent graph G such that P fuu(n)l G n =and then we delete an arrow to obtain a graph G ρ G which is transient and such that h(g )=h(g). First we give a description of G depending on a sequence of integers a(n) thenwe give two different values to the sequence a(n) so as to obtain a positive recurrent graph in one case and a nullrecurrent graph in the other case. Let u be a vertex. The graph G is composed of loops based at the vertex u and there is no arrow between two different loops (see Figure ). The graph is entirely determined by the number of loops of length n. More precisely, choose a sequence a(n) of non negative integers for n, with a() =, and define the set of vertices of G as where the vertices v n;i k V = fug[ +[ n= fv n;i k j» i» a(n);» k» n g; above are distinct. = u for» i» a(n). There is an arrow v n;i k Let v n;i = vn n;i! v n;i for» k» n ;» k+ i» a(n);n and there is no other arrow ing. The graph G is connected and fuu(n) G =a(n) for n. Notice that there is only one arrow u! u because a() = so the graph G is well defined. u Figure : the graphs G and G ; the bold loop (on the left) is the only arrow that belongs to G and not to G, otherwise the two graphs coincide. The sequence (a(n)) n 2 is chosen such that it satisfies a(n)l n =: (3) 7

8 Equation (3) implies that R = L by Theorem 2.(i), thus G is recurrent according to (3). The graph G is defined as follows: the vertices of G are theonesofg and any arrow ing is also an arrow in G except the arrow u! u. Obviously fuu(n) G =fuu(n) G ifn 2, fuu() G = and L = L. Moreover fuu(n)l G n = L<: This implies that R = L by Theorem 2.(i) and G is transient. In addition R = L = R, thatis, h(g) =h(g ). Now we consider two different sequences a(n).. Let a(n 2 )=2 n2 n for n and a(n) =ifn is not a square. It is easy to see that L = 2,and Moreover f G uu(n)l n = nf G uu(n)l n = 2 n2 n = 2 n2 2 n =: n 2 2 n < +; hence the graph G is positive recurrent; it is obviously not strongly positive recurrent. The graph G ρ G is transient and satisfies h(g )=h(g) and P nf G uu(n)r n < Let a() =, a(2 n )=2 2n n for n 2 and a(n) =otherwise. One can compute that L =, 2 and fuu(n)l G n = n n = 2 2n n =: n 2 n 2 Moreover nf G uu(n)l n = 2 + n 2 2 n 2 n =+ hence the graph G is null recurrent. The graph G ρ G is transient and satisfies h(g )=h(g) and P nf G uu(n)r n =+. In both cases, Proposition 2.5 gives a recurrent graph G containing G which is different from G (two loops of length 2 are added to G in order to obtain G ), because in Proposition 2.5 we forbade to add an arrow u! u. Remark 2.7 Example 2.6 shows that the signs»" that appear in Table are to be understood as either = or <", both being possible. Remark 2.8 In the more general setting of thermodynamic formalism for countable Markov chains (introduced in [2]), Sarig puts to the fore a subclass of positive recurrent potentials which he calls strongly positive recurrent[3]; his motivation is different, but the classifications agree. If G is a countable oriented graph, a potential is a continuous map ffi: G! R and the pressure P (ffi) is the analogous of the Gurevich entropy, the paths being weighted by e ffi ; a potential is either transient or null recurrent or positive recurrent. Considering the null potential ffi, we retrieve the case of (non weighted) topological Markov chains. In [3] Sarig introduces a quantity u[ffi]; ffi is transient 8

9 (resp. recurrent) if u[ffi] < (resp. u[ffi] ). The potential is called strongly positive recurrent if u[ffi] >, which implies it is positive recurrent. A strongly positive recurrent potential ffi is stable under perturbation, that is, any potential ffi + tψ close to ffi is positive recurrent too. For the null potential, u[] = log P f uu(n)l n, thus u[] > if and only if the graph is strongly positive recurrent by Proposition 2.4. Examples of (non null) potentials which are positive recurrent but not strongly positive recurrent can be found in [3]; some of them resemble much the Markov chains of Example 2.6, their graphs being composed of loops as in Figure. 3 Existence of a maximal measure 3. Positive recurrence and maximal measures A Markov chain on a finite graph always has a maximal measure [], but it is not the case for infinite graphs [7]. In [8] Gurevich gives a necessary and sufficient condition for the existence of such a measure. Theorem 3. (Gurevich) Let G be a connected oriented graph of finite non zero entropy. Then the Markov chain ( G ;ff) admits a maximal measure if and only if the graph is positive recurrent. Moreover, such a measure is unique if it exists, and it is an ergodic Markov measure. In [6] Gurevich and Zargaryan give a sufficient condition: if one can find a finite connected subgraph H ρ G such that there are more paths inside than outside H (in term of exponential growth), then the graph G is positive recurrent. For the sake of completeness, we give a detailed proof in the Appendix. It appears in the proof that this sufficient condition implies R < L. We show that it is equivalent. Lemma 3.2 If G is a connected oriented graph and H is a finite subgraph, there exists a finite set of vertices W containing all the vertices of H and such that W is connected. Proof. Let u; v be two vertices of H. As G is connected there exists n = n uv and (w uv; ;wuv n )a path in G such that w uv = u and wn uv = [ v. Define W = i j» i» n uv g: u;v2v fw uv Then H ρ W and W is a finite connected graph. Let G be a connected oriented graph, W asubsetofvertices and u; v two vertices of G. Define t W uv(n) as the number of paths (v ; ;v n ) such that v = u; v n = v and v i 2 W for <i<n,and put fiuv W =limsup n!+ n log tw uv(n). Theorem 3.3 Let G be a connected oriented graph of finite non zero entropy. The following statements are equivalent: (i) R<L. (ii) There exists a finite set of vertices W such that W is connected and for all vertices u; v in W, fi W uv <h(g). 9

10 (iii) There exists a finite set of vertices W such that W is connected and for all vertices u; v in W, fiuv W» h(w ). In this situation, ( G ;ff) admits a maximal measure. Proof. The implication (iii) ) (i) is the result of Gurevich and Zargaryan (see Theorem 4. in Appendix). We prove (i)) (ii) ) (iii). Let G be a connected oriented graph of finite entropy and suppose that R <L. Let W be a finite set of vertices such thatw is connected (such agraph exists by Lemma 3.2). Let u; v 2 W and choose a path (v ; ;v N )inw such that v = v; v N = u and its length N is minimal (which implies that v i is different fromu and v for <i<n); the length N is zero if u = v. Define the subgraph H (uv) ρ G by ffl the set of vertices of H (uv) isw [fv = v; v ; ;v N = ug, ffl H (uv) contains all the arrows of W, the arrows u! w for w 62 W, the arrows w! v for w 62 W and the arrows v i! v i+ for» i<n. u W v H(uv) Figure 2: the graph H(uv). Define H(uv) as the connected component of H (uv) that contains u and v (see Figure 2). If H(uv) is empty, we say byconvention that its entropy is zero. Suppose that H(uv) is not empty. It does not contain any arrow w! v with w in W but such an arrow exists in G because W is connected. Thus H(uv) / G,which implies h(h(uv)) <h(g) by Theorem 2.2. Put C =minfh(h(uv)) j u; v 2 W g <h(g): For all n one has t W uv(n) =fuv H(uv) (n)» p H(uv) uv (n) thus fiuv W» C<h(G) for all u; v 2 W, which is (ii). Now, suppose that (ii) holds. Let C = maxffiuv W j u; v 2 W g < h(g). There exists a finite set of vertices W ff W such that W is connected and h(w ) > C (see the definition of entropy and Lemma 3.2). Let u; v 2 W and choose apath(u ; ;u p ) of minimal length in W such that u 2 W and u p = u. By minimality, u i 62 W for» i» p (if u is already in W then p = ). In the same way, let (v ; ;v q ) be a path such that v = v; v q 2 W and v i 2 W n W for» i» q. If (w ; ;w n ) is a path outside W between u and v (i.e. w = u; w n = v and w i 62 W for <i<n) then (u ; ;u p = w ;w ; ;w n = v ; ;v q ) is a path outside W,which implies that

11 t W uv (n)» t W u v q (n + p + q). Consequently, This is (iii). fi W uv = lim sup n!+ log twuv (n)» lim sup n n!+ n log tw u v q (n + p + q)» C <h(w ): Finally, ifr<lthen G is (strongly) positive recurrent, thus ( G ;ff) admits a maximal measure by Theorem Local entropy and maximal measures For a compact system, the local entropy is defined according to a distance but does not depend on it. One may wish to extend this definition to non compact metric spaces although the notion obtained in this way is not intrinsic. Definition 3.4 Let be a metric space, d its distance and let T :! be a continuous map. The Bowen ball of centre x, of radius r and of order n is defined as E is a (ffi;n)-separated set if B n (x; r) =fy 2 j d(t i x; T i y) <r;» i<ng: 8y; y 2 E;y 6= y ; 9» k<n; d(t k y; T k y ) ffi: The maximal cardinality ofa(ffi;n)-separated set contained in Y is denoted by s n (ffi;y). The local entropy of(; T) is defined as h loc () = lim "! h loc (; "), where h loc (; ") = lim lim sup ffi! n!+ n sup log s n (ffi;b n (x; ")): x2 If the space is not compact, these notions depend on the distance. When = G, we use the distance d introduced in Section.2. It can be proven that the local entropy of G does not depend on the identification of the vertices with N and it is equal to h loc ( G ;ff). We prove that, if h loc ( G ) < h(g), then G is strongly positive recurrent. First we introduce some notations. Let G be an oriented graph. If W is a subset of vertices, H a subgraph of G and μu =(u n ) n2z 2 G, define C H (μu; W )=f(v n ) n2z 2 H j8n 2 Z; (u n 2 W, v n 2 W )g: If S ρ G and p; q 2 Z [f ; +g, define [S] q p = f(v n ) n2z 2 G j9(u n ) n2z 2 S; 8p» n» q; u n = v n g: Lemma 3.5 Let G be an oriented graph on the set of vertices N. (i) If W fff; ;p+2g then for all μu 2 G and all n, C G (μu; W ) ρ B n (μu; 2 p ): (ii) If μu =(u n ) n2z and μv =(v n ) n2z are twopaths in G such that (u ; ;u ) 6= (v ; ;v ) and u i ;v i 2 f; ;q g for» i» n then (μu; μv) is (2 q ;n)-separated, that is, there exists» i» n, such that d(ff i μu; ff i μv) 2 q.

12 Proof. (i) Let μu = (u n ) n2z 2 G. If μv = (v n ) n2z 2 C G (μu; W ), then D(u j ;v j )» 2 (p+2) for all j 2 Z (see Section.2 for the definition of D). Consequently for» i<n d(ff i (μu);ff i (μv)) = k2z» k2z D(u i+k ;v i+k ) 2 jkj 2 (p+2) 2 jkj» 3:2 (p+2) < 2 p : (ii) Let» i» n such that u i 6= v i. By hypothesis, u i ;v i» q. Suppose u i <v i. Then d(ff i (μu);ff i (μv)) D(u i ;v i )=2 u i ( 2 (v i u i ) ) 2 (u i+) 2 q : Theorem 3.6 Let G be a connected oriented graph of finite entropy on the set of vertices N. If h loc ( G ) <h(g), then the graph G is strongly positive recurrent and the Markov chain ( G ;ff) admits a maximal measure. Proof. Fix C such that h loc ( G ) <C<h(G) and "> such that h loc ( G ;") < C. Let p be an integer such that 2 (p ) < ". By definition of h(g) there exists a finite subgraph G such that h(g ) > C. By Lemma 3.2 there exists a finite subset of vertices V such that V contains the vertices of G and the vertices f; ;pg and V is connected. Define W = V, V q = fn» qg and W q = V q n V = W V q for all q. Our aim is to bound t W uu (n) =tv uu (n). Choose u; u 2 V and let (w ; ;w n ) be a path between u and u with w i 2 V for» i» n. Fix n. For any path (v ; ;v n )ing, there exists q such that v ; ;v n» q, thus t W uu (n) = lim q!+ twq uu (n): Fix ffi > such that 8ffi» ffi ; lim sup n!+ n sup μv2 G log s n (ffi;b n (μv; ")) <C: Take q arbitrarily large and ffi» ffi such that ffi» 2 (q+). Choose N (depending on ffi) such that 8n N;8μv 2 G ; n log s n(ffi;b n (μv; ")) <C: (4) If t Wq uu (n) 6=,choose a path (v ; ;v n ) such that v = u; v n = u and v i 2 W q for <i<q. Define μv (n) =(v (n) i ) i2z as the periodic path of period n + n satisfying ffl v (n) i = v i for» i» n, ffl v (n) n+i = w i for» i» n. 2

13 IN W q u v (n) V u u u u u u -n n n+n 2n+n 2(n+n ) i Figure 3: The set E q (n; k) (k = 2 on the picture): μv (n) (in solid) is a periodic path, μu (in dashes) is a element ofe q (n; k). Between the indices and k(n + n ), μv (n) and μu coincide when v (n) i is in V and μv (n) and μu are in W q at the same time. Before or after k(n + n ), the two paths coincide. Define the set E q (n; k) as follows (see Figure 3): h i C Vq (μv (n) k(n+n ) ;V) E q (n; k) = i i + hμv hμv (n) (n) : k(n+n ) The paths in E q (n; ) are exactly the paths counted by t Wq uu (n) which are extended outside the k. indices f; ;ng like the path μv (n),thus #E q (n; ) = t Wq uu (n). Similarly, #E q (n; k) = t Wq uu (n) By definition, E q (n; k) ρc G (μv (n) ;V) and f; ;pg ρ V thus E q (n; k) ρ B k(n+n )(μv (n) ;") by Lemma 3.5(i). Moreover, if (w i ) i2z and (wi ) i2z are two distinct elements of E q (n; k), there exists» i<k(n + n ) such that w i 6= wi and w i;wi» q, thus E q(n; k) isa(ffi;k(n + n ))-separated set by Lemma 3.5(ii). Choose k such that k(n + n ) N. Then by (4) #E q (n; k)» s k(n+n )(ffi;b k(n+n )(μv (n) ;")) <e k(n+n)c : k, As #E q (n; k) = t Wq W uu (n) one gets t q uu (n) <e (n+n)c. This is true for all q, thus and t W uu (n) = lim q!+ twq uu (n)» e (n+n )C fiuu W = fi uv V» C<h(V ): Consequently, we are in case (iii) of Theorem 3.3, which is enough to conclude. Remark 3.7 Define the entropy atinfinity of the graph G as h (G) = lim n!+ h(g n G n) where (G n ) n is any increasing sequence of finite graphs such that S n G n = G. The local entropy satisfies h loc ( G ) h (G) but in general these two quantities are not equal and the condition h (G) <h(g) does not imply that G is strongly positive recurrent. This is illustrated by Example 3

14 2.6 (see Figure ). However, if for all vertices u the number of arrows starting from and ending at u is finite (not necessarily bounded for all u) then the condition h (G) <h(g) implies that R<L. Indeed, let W be a finite set of vertices, let u; u 2 W and let v ; ;v p (resp. v ; ;v q) be the set of vertices such that u! v i and v i 62 W (resp. vi! u and vi 62 W ). Then t W uu (n) = p q i= j= p W v i v j (n 2) thus fi uu (W ) = maxf R vi vj (W ) j» i» p;» j» qg»h(w ): For " small and W large enough h(w )» h (G)+"<h(G), thus fi uu (W ) <h(g), and Theorem 3.3 gives the conclusion. 4 Appendix We recall that t W uv(n) is the number of paths (v ; ;v n ) such that v = u; v n = v and v i 62 W for <i<n,andfi W uv = lim sup n log tw uv(n). Theorem 4. (Gurevich-Zargaryan [6]) Let G be a connected oriented graph of finite nonzero entropy. Suppose there exists a finite subset of vertices W such that W is connected and for all vertices u; v 2 W one has h(w ) fiuv W. Then R <Land the Markov chain ( G;ff) admits a measure of maximal entropy. Proof. Let W as in the Theorem and fix a vertex w in W. Let u; v be two vertices in W. We define the following quantities. ffl a uv (n) isthenumber of paths (v ; ;v n )suchthatv = u; v n = v, and there exists» k<n such thatv i 2 W nfwg for <i<kand v i 62 W for k» i<n. We have a uv () = a uv () =. ffl A uv (z) = P a uv (n)z n is the associated series. Its radius of convergence is denoted by r uv. ffl If V is a subset of vertices, ρ uv (V ) is the radius of convergence of the series T V uv(z) = P t V uv (n)z n, that is fi V uv = log ρ uv (V ). In the sequel, we omit to write the graph in exponent inf ss (n) and F ss (x) when the graph is G; L(G) andr(g) are denoted by L and R. One checks that thus for all x f uw (n) =f W uw(n)+a uw (n)+ F uw (x) =F W uw(x)+a uw (x)+ v2w nfwg k=2 v2w nfwg a uv (k)f vw (n k); A uv (x)f vw (x): (5) Since the coefficients are non negative, all the considered series are defined for all x (they can be infinite). In (5), both left and right quantities are finite at the same time. Let J be the set of x such that F ww (x) < +; J is equal to the interval [;L[ or [;L] according to F ww (L) =+ or F ww (L) < +. Let u be a vertex of G and let N be the length of a 4

15 minimal path joining w to u. It is easy to see that f uw (n)» f ww (n + N), thus x N F uw (x)» F ww (x) and F uw (x) < + for all x 2 J. If x 2 J nfg then A uv (x)f vw (x) < + by (5)and F vw (x) > thus A uv (x) < +. Hence for all x 2 J, all the series that appear in (5) are finite. For x 2 J define the matrix M(x) = (m uv (x)) u;v2w by m uv (x) = A uv (x) if v 6= w and m uw (x) =. Equation (5) gives a system of linear equations whose variables are (F uw (x)) u2w. For x 2 J the system can be rewritten as B (I uw (x) C A u2w where I is the identity matrix on the set of indices W. We can also write f uw (n) as n k=2 f uw (n) =f W uw(n)+a uw (n)+ n + + n k = n v ; ;v k 2 W v i 6= w Hence for all x a uv (n )a v v 2 (n 2 ) a vk 2 v k (n k ) h B uw(x)+a W uw (x) C A a vk w(n k )+f W v k w(n k ) F uw (x) =F W uw(x)+a uw (x)+ (7) k=2 v ; ;v k 2 W v i 6= w u2w A uv (x)a v v 2 (x) A vk 2 v k (x)[a vk w(x)+f W v k w(x)]: For x 2 J define the matrix Q(x) =(q uv (x)) u;v2w by q uv (x) = v ; ;v k 2W nfwg A uv (x)a v v 2 (x) A vk 2 v k (x) if v 6= w and q uw =. Both left and right quantities in (7) are finite at the same time and F uw (x) < + for x 2 J, thus the coefficients of Q(x) are finite for all x 2 J. For x 2 J, Equation (7) can be rewritten as uw (x) C A u2w B = (I + uw(x)+a W uw (x) For x 2 J, all the coefficients of M(x) andq(x) are finite, and an easy computation shows that (I M(x))(I + Q(x)) = I, that is I + Q(x) =(I M(x)) and det(i M(x)) 6= for all x 2 J: (8) Let S = minfl(w );r uv j u; v 2 W g. For» x < S and u 2 W, one has Fuw(x) W < +, A uw (x) < + and all the coefficients of the matrix M(x) are finite. The map x 7! (x) = det(i M(x)) is continuous and () =. If there exists x < S such that (x) =, put 5 C A i u2w : (6)

16 =minfx j (x) =g, otherwise put = S. If x<, then (x) 6= thus the system (6) is invertible, which implies that F uw (x) < + for any u 2 W. Hence» L. We split the proof into two cases depending on whether L R(W )ornot. ffl Suppose L R(W ). For all n, f ww (n) f W ww(n) and, since G is infinite, there exists n such that f ww (n) > f W ww(n). Hence F ww (L) > F W ww(l) F W ww(r(w )). Since W is finite, W is positive recurrent andf W ww(r(w )) =, thus F ww (L) > andr<lby Theorem 2. (i). ffl Suppose L<R(W ). Since» L this implies <R(W ). Fact: When W checks the asymptions of Theorem 4., one has R(W )» r uw for all u; v 2 W. Proof of the fact. One has thus a uv (n) =t W uv(n)+ A uv (x) =T W uv (x)+ s 2 W s 6= w k=2 s 2 W s 6= w t W us nfwg (n k)t W sv (k) Both left and right quantities in (9) are finite at the same time, thus r uv minfρ sv (W );ρ us (W nfwg) j s 2 W g: T W nfwg us (x)t W sv (x): (9) ρ us (W nfwg) R(W ) because t W sv nfwg (n)» p W sv (n) for all n. Moreover ρ sv (W ) R(W ) by hypothesis of the Theorem. Hence r uw R(W ) and this ends the proof of the fact. By the fact above, R(W )» minfr uv j u; v 2 W g. In addition, R(W )» L(W ), thus R(W )» S according to the definition of S. Since <R(W ) this implies that <Sthus ( ) =. If F ww (L) < + then J =[;L]; this is impossible because (x) 6= for all x 2 J by (8) and» L. Consequently, F ww (L) =+, which implies R<Lby Theorem 2. (i). In both cases, R < L thus the graph G is positive recurrent and the Markov chain ( G ;ff) admits a maximal measure by Theorem 3.. References [] R. Bowen. Entropy-expansive maps. Trans. Amer. Math. Soc., 64:323 33, 972. [2] J. Buzzi. Intrinsic ergodicity of smooth interval maps. Israel J. Math., :25 6, 997. [3] J. Buzzi. On entropy-expanding maps. Preprint, 2. [4] M. Denker, C. Grillenberger, and K. Sigmund. Ergodic theory on compact spaces. Springer- Verlag, Berlin, 976. Lecture Notes in Mathematics, Vol [5] U. Fiebig. Symbolic dynamics and locally compact markov shifts, 996. Habilitationsschrift. 6

17 [6] B.M. Gurevich and A.S. Zargaryan. Conditions for the existence of a maximal measure for a countable symbolic Markov chain. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 5:4 8, 3, 988. [7] B.M. Gurevi»c. Topological entropy of enumerable Markov chains. Soviet. Math. Dokl., (no. 4):9 95, 969. [8] B.M. Gurevi»c. Shift entropy and Markov measures in the path space of a denumerable graph. Soviet. Math. Dokl., : , 97. [9] S.E. Newhouse. Continuity properties of entropy. Annals of Mathematics, 29:25 235, 989. Corrections, 3:49 4, 99. [] W. Parry. Intrinsic Markov chains. Trans. Amer. Math. Soc., 2:55 66, 964. [] I.A. Salama. Topological entropy and recurrence of countable chains. Pacific J. Math., 34(no. 2):325 34, 988. Errata, 4(no. 2):397, 989. [2] O.M. Sarig. Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam. Systems, 9(6): , 999. [3] O.M. Sarig. Phase transitions for countable Markov shifts. CMP, to appear. [4] D. Vere-Jones. Geometric ergodicity in denumerable Markov chains. Quarterly J. Math., 3:7 28, 962. Sylvie Ruette, Institut de Mathématiques de Luminy - CNRS - case avenue de Luminy Marseille cedex 9 - France ruette@iml.univ-mrs.fr 7