STOCHASTIC ADDING MACHINES BASED ON BRATTELI DIAGRAMS. Danilo Antonio Caprio. Ali Messaoudi. Glauco Valle

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1 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS Danilo Antonio Caprio UNESP - Departamento de Matemática do Instituto de iociências, Letras e Ciências Exatas. Rua Cristóvão Colombo, 65, Jardim Nazareth, 554- São José do Rio Preto, SP, rasil. Ali Messaoudi UNESP - Departamento de Matemática do Instituto de iociências, Letras e Ciências Exatas. Rua Cristóvão Colombo, 65, Jardim Nazareth, 554- São José do Rio Preto, SP, rasil. Glauco Valle Universidade Federal do Rio de Janeiro - Instituto de Matemática. Caixa Postal 6853, cep , Rio de Janeiro, rasil. Abstract. In this paper, we dene some Markov Chains associated to Vershik maps on ratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.. Introduction Let g be a homomorphic map on C d, where d is an integer. The set K(g) of z C d such that the forward orbit {g n (z) : n N} is bounded is called the (ddimensional) lled Julia set of g. Filled Julia sets and their boundaries (called Julia sets) were dened independently by Julia and Fatou ([5] and [6], [3] and [4]). The study of Julia sets is connected to many areas of mathematics as dynamical systems, complex analysis, functional analysis and number theory, among others (see for example [6], [8], [9], [], [], [4], [7], [], [5], [7], [8], [33], [37]). There is an important connection between Julia sets and stochastic adding machines. A rst example was given by Killeen and Taylor in [6] as follows: let n be a nonnegative integer and write it in a unique way in base as n = k i= ε i(n) i = ε k... ε, for some k, where ε k = and ε i {, }, for all i {,..., k }. 99 Mathematics Subject Classication. Primary: 37A3, 37F5; Secondary: 6J, 47A. Key words and phrases. Markov chains, stochastic Vershik map, ratteli diagrams, spectrum of transition operators, bered Julia sets. Supported by FAPESP grant 5/66 6. Supported by CNPq grant 37776/5 8 and FAPESP project 3/ Supported by FAPERJ grants E 6/3.48/6 and CNPq grants 3585/5 and 4383/6.

2 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE It is known that the addition of is given by a classical algorithm, namely n + = ε k... ε l+ (ε l + )... where l = min{i : ε i (n) = }. Killeen and Taylor dened the stochastic adding machine assuming that each time a carry should be added, it is added with probability < p < and it is not added with probability p. Moreover, the algorithm stops when the rst carry is not added. So this random algorithm maps n = ε k... ε to n itself with probability p, to n + with probability p l+ and to m = n r + = ε k... ε d+... ε r... with probability p r ( p). With this they obtained a countable Markov chain whose associated transition operator S = (p i,j ) i,j N is a bistochastic innite matrix whose spectrum is equal to the lled Julia set of the quadratic map z ( p), z C. p In [9], [3], [3] and [3], stochastic adding machines based on other systems of numeration have been introduced. They are connected to one-dimensional bered Julia sets (see [9]) and also to Julia sets in dimension greater than one ([7], [3] and [3]). A d-dimensional bered lled Julia set of a sequence (g j ) j of homomorphic maps on C d is the set K((g j ) j ) of z C d such that the forward orbit { g j (z) : j N} is bounded, where g j = g j g j... g for all j. In this paper, we introduce stochastic adding machines associated to Vershik maps on ratteli diagrams. ratteli diagrams are important objects in the theories of operator algebras and dynamical systems. It was originally dened in 97 by O. ratteli [3] for classication of C -algebras. ratteli diagrams turned out to be a powerful tool in the study of measurable, orel, and Cantor dynamics (see [8], [], [8], [35]). The interest on ratteli diagrams is that any aperiodic transformation in measurable, orel, and Cantor dynamics can be realized as a Vershik map acting on the path space of a ratteli diagram (see [], [], [8], [35], [36]). A particular application arises when we use the Vershik map to embed Z + into the set of paths of the associated ratteli diagram. This embedding allows us to consider the restriction of the Vershik map on that copy of Z + as the map n n +. It also allows a representation of systems of numeration through ratteli diagrams, making possible for us to introduce more general stochastic adding machines. Indeed we are able to dene a more general Markov process on the set X of innite paths on the ratelli diagram whose restriction to the copy of Z + is the stochastic adding machine, we call this process the "ratteli-vershik process" or simply V process and the associated Stochastic adding machine the ratteli-vershik stochastic adding machine or simply V stochastic adding machine. We will give necessary and sucient conditions that assure transience or recurrence of the V stochastic adding machines. We will also prove that the spectrum of the V stochastic adding machine transition operator S (acting on l ) is related to bered lled Julia sets in higher dimension. ( For example, ) if the ratteli diagram is a b stationary and its incidence matrix is M = where a, b, c, d are nonnegative c d integers, then the point spectrum of the transition operator of the ratteli-vershik

3 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 3 stochastic adding machine associated to M is related to the Julia set K := {(x, y) C : (g n... g (x, y)) n is bounded}, ( ) where g n (x, y) = p n+ x a y b p n+ p n+, p n+ x c y d p n+ p n+ and < p n+ <, for all n. Just to mention an important connection, the study of these spectra gives information about the dynamical properties of transition operators acting on separable anach spaces (see for instance [] and [9]). For example, if T is topologically transitive, then any connected component of the spectrum intersects the unit circle. However, here we do not aim at the study of the dynamical properties of the transition operators. We will also study topological properties of this spectrum. The paper is organized as follows. In Section we give a background about ratteli diagrams and we dene the Vershik map. In Section 3 we dene the V processes and the V stochastic adding machines giving necessary and sucient conditions for transience, null recurrence and positive recurrence. Section 4 is devoted to provide an exact description of the spectra of the transition operators of V stochastic machines acting on l (N) in the case of ratteli diagrams. Furthermore, we prove some topological properties of this spectrum. Section 5 describes generalization to l l, l 3, ratteli diagrams.. ratteli diagrams.. asics on ratteli diagrams. In this section we introduce the necessary notation on ratteli diagrams. Here we follow [] and [] and we recommend both texts as a reference for the interested reader. Denition.. A ratteli diagram is an innite directed graph (V, E) where the vertex set V and the edge set E can be partitioned into nite sets, i.e V = k=v (k) and E = k=e(k), where #V (k) < and #E(k) < for every k, such that there exist maps s : E V and r : E V such that s restricted to E(k) is a sujective map from E(k) to V (k ) and r restricted to E(k) is a sujective map from E(k) to V (k) for every k. For every e E we call s(e) the source of e and r(e) the range of e (see Figure ). For convenience if #V (k) = l we denote V (k) = {(k, ),..., (k, l)} or simply V (k) = {,..., l} when there is no possibility of misidentication of the value of k. Remark.. It is usual to dene the ratteli diagrams under the condition that V () is one point set, i.e V () = {v()}. Our denition is more suitable to the understanding of stationarity and more appropriated to the discussion of the results in this paper. However we could also use that condition in the denition without any prejudices to the results in this paper.

4 4 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE It is convenient to give a diagrammatic representation of a ratteli diagram considering V (k) as a "horizontal" level k, and the edges in E(k) heading downwards from vertices at level k to vertices at level k. Also, if #V (k ) = l(k ) and #V (k) = l(k), then E(k) determines a l(k) l(k ) incidence matrix M(k) (see Figure ), where M(k) i,j is the number of the edges going from vertex j in V (k ) to vertex i in V (k). y denition of ratteli diagrams, we have that M(k) has non identically zero lines and columns. LEVEL INCIDENCE MATRICES n- n E(n) s(e) V(n-) V(n) n+ E(n+) e r(e) V(n+) Figure. Diagrammatic representation between the levels n and n + in a ratteli diagram. Let k, k Z + with k < k and let E(k + ) E(k + )... E( k) denote the set of paths from V (k) to V ( k). Specically, E(k + )... E( k) denote the following set: {(e k+,..., e k) : e i E(i), k + i k, r(e i ) = s(e i+ ), k + i k }. The incidence matrix of E(k + )... E( k) is the product M( k)... M(k + ). We dene r(e k+,..., e k) := r(e k) and s(e k+,..., e k) := s(e k+ ). Denition.3. We say that (V, E) is a simple ratteli diagram if for each nonnegative integer k, there exists and integer k > k such that the product M( k)... M(k+) have only non-zero entries... Ordered ratteli diagrams. Denition.4. An ordered ratteli diagram (V, E, ) is a ratteli diagram (V, E) together with a partial order on E such that edges e, e E are comparable if and only if r(e) = r(e ), in other words, we have a linear order on the set r ({v}) for each v V \ V () (see an example in Figure ).

5 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 5 Figure. An order in ratteli diagram of Figure. Remark.. Edges in an ordered ratteli diagram (V, E, ) are uniquely determined by a four dimensional vector e = (k, s, m, r), where k means that e E(k), s = s(e) and r = r(e) are the source and range of e as previously dened and m Z + is the order index means that e = e m r (r(e)) = {e < e <... < e r }. Usually we will write e = e k = (s, m, r) carrying the level index k as a subscript or suppressing it when there is no doubt about the level. Note that if (V, E, ) is an ordered ratteli diagram and k < k in Z +, then the set E(k + ) E(k + )... E( k) of paths from V (k) to V ( k) may be given an induced order as follows: (e k+, e k+,..., e k) > (e k+, e k+,..., e k) if and only if for some i with k + i k, e i > e i and e j = e j for i < j k. Denition.5. A ratteli diagram (V, E) is stationary if there exists l such that l = #V (k) for all k, and (by an appropriate relabelling of the vertices if necessary) the incidence matrices between level k and k + are the same l l matrix M for all k. In other words, beyond level the diagram repeats itself. An ordered ratteli diagram = (V, E, ) is stationary if (V, E) is stationary, and the ordering on the edges with range (k, i) is the same as the ordering on the edges with range ( k, i) for k, k and i =,..., l. In other words, beyond level the diagram with the ordering repeats itself. We still need a denition that will be useful to deal with examples. Denition.6. Let = (V, E, ) be an ordered ratteli diagram. We say that is a consecutive ordering if for all edges e f e with s(e) = s(e ) we have s(f) = s(e) = s(e ). To every ordered ratteli diagram with consecutive ordering = (V, E, ) we associate a sequence of matrices (Q(k)) k called the ordering matrices such that (i) Q(k) is a (l(k)) (l(k )) matrix; (ii) Q(k) i,j = if and only if M(k) i,j = ;

6 6 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE (iii) The non zero entries in each line i of Q(k) form a permutation in #{j : M(k) i,j > } letters. So line i in Q(k) indicates how edges inciding on vertex i V (k) are ordered with respect to its sources in V (k ). The consecutive ordering is said to be canonical if each line of Q(k), k, the permutation in #{j : M(k) i,j > } letters is the identity. For a stationary ordered ratteli diagram, the consecutive ordering is also stationary, i.e Q = Q(k) for every k. As an example consider a stationary ordered ratteli diagram with l = and incidence matrix ( ) a b M =, c with abc >. We have two possible consecutive ordering relative to the ordering matrices ( ) ( ) or, where the rst one is associated to the canonical consecutive ordering..3. The Vershik map. Let = (V, E, ) be an ordered ratteli diagram. Let X denote the associated innite path space, i.e. X = {(e, e,...) : e i E(i) and r(e i ) = s(e i+ ), for all i }. Under the hypotheses of Denition., X is non empty. However X can be a nite set, this only occurs in trivial cases and do not occur for general classes of ratteli diagrams as for instance simple ratelli diagrams with #E(k) > for innitely many k. Hence we require that X is innite for all ratteli diagrams considered here. We endow X with a topology such that a basis of open sets is given by the family of cylinder sets [e, e,..., e k ] = {(f, f,...) X : f i = e i, for all i k}. Each [e,..., e k ] is also closed, as is easily seen. Endowed with this topology, we call X the ratteli compactum associated with = (V, E, ). Let d be the distance on X dened by d ((e j ) j, (f j ) j ) = where k = inf{i : e k i f i }. The topology of the cylinder sets coincide with the topology induced by d. If (V, E) is a simple ratteli diagram, then X has no isolated points, and so is a Cantor space (see [8]). Two paths in X are said to be conal if they have the same tails, i.e. the edges agree from a certain level. Let x = (e, e,...) be an element of X. We will call e k = e k (x) the kth label of x. Recall from Remark. that e k = (s k, m k, r k ) such that r k = s k+ V (k) for every k. We let X max denote those elements x of X such that e k (x) is a maximal edge for all k and X min the analogous set for the minimal edges. It is clear that from any vertex at level k there is an upward maximal path to level, using this we have that

7 X max X min STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 7 is the intersection of non-empty compact sets, so it is non-empty. Analogously is non-empty. From now on we denote X := X \ X max. If = (V, E, ) is an ordered ratteli diagram then it's easy to check that every innite path x X has an unique successor, i.e. the set {y X : y > x} has a smallest element. Indeed let x = (e, e,...) X and ζ(x) be the smallest number such that e ζ is not a maximal edge. Let f ζ = f ζ (x) be the successor of e ζ (and so r(e ζ ) = r(f ζ )). Then the successor of x is V (x) = y = (f,..., f ζ, f ζ, e ζ+,...), where (f,..., f ζ ) = (f (x),..., f ζ (x)) is the minimal path in E() E()... E(ζ ) with range equal to s(f ζ ), i.e. r(f,..., f ζ ) = s(f ζ ). Denition.7. The Vershik map of an ordered ratteli diagram = (V, E, ) is the map V : X X that associates to each x X its successor. We call the resulting pair (X, V ) a ratteli-vershik dynamical system. 3. The ratteli-vershik process and stochastic machine Here we will dene the V process but we need to introduce some new notation before it. Let = (V, E, ) be an ordered ratteli diagram. Recall the denition of ζ(x), for x X, from the previous section and dene A(x) = { i < ζ(x) : e i (x) is not a minimal edge}. Put θ(x) = #A(x) and write A(x) = {k x,,..., k x,θ(x) }, where k x,i < k x,i, for all i {,..., θ(x)}. Since for k A(x) we have that e k (x) is a maximal edge of x which is not minimal which implies that e k (x) is not the only edge arriving at r(e k (x)). Thus if #r (v) > for every v V {v } or equivalently the sum of each line in each incidence matrix is greater than one, then we have that θ(x) = ζ(x) and A(x) = {,..., ζ(x) }. So we have Hypothesis A: For the ordered ratteli diagram = (V, E, ), the sum of each line in each incidence matrix is greater than one. For each j {,..., θ(x)}, let y j (x) X be dened as (3.) y j (x) = (f (j),..., f (j) k x,j, e kx,j +, e kx,j +,...), where (f (j),..., f (j) k x,j ) is the minimal edge in E()... E(k x,j ) with range equal to s(e kx,j +), for each j {,..., θ(x)}.

8 8 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE First we need to adjust the space where the V process will be dened. This is due to the fact that the successor of x X can be an element of Xmax. To avoid this we max dene X as the set of points x X that are conal with a point on X max. Set max X := X \ X. Note that if x X then V (x) X. Moreover V restricted to X is one to one from X to X X min. Denition 3.. Let (p i ) i be a sequence of non-null probabilities and = (V, E, ) an ordered ratteli diagram. The ratelli-vershik Process is a discrete timehomogeneous Markov Process (Γ n ) n with state space X dened as Γ n = V (n) (Γ ), where V (n) is the n-th interation of V : X X called the random Vershik map and dened as y j (x), with probability p kx,... p kx,j ( p kx,j+ ), for each j {,..., θ(x) }; V (x) = y θ(x) (x), with probability p kx,... p kx,θ(x) ( p ζ(x) ), x, with probability p kx, ; V (x), with probability p kx,... p kx,θ(x) p ζ(x). Thus the transition probabilities of the V process is determined by the random Vershik map. The idea behind the denition is the use of a basic algorithm to obtain V (x) from x by recursively choosing the minimum path from level to level k for k ζ(x) and then at step ζ(x) we nally obtain V (x). Then we impose the rule that step j of the algorithm is performed with probability p j independently of any other step. This transition mechanism is connected to the stochastic adding machines discussed in Section and our next aim is to dene the V stochastic adding machine. Remark 3.. Under Hypothesis A we have that y j (x), with probability p... p j ( p j+ ), for each j {,..., ζ(x) }; V (x) = x, with probability p ; V (x), with probability p... p ζ(x) p ζ(x). Take x X X min and dene X x a bijection between X x V (n) := {x } {V (n) (x ) : n }. Clearly we have and the set of non-negative integers Z + where x and (x ) n for all n. Using the fact that x X min to verify that for every x X x we have V (x) X x To simplify the notation, we put x n := V (n) (x ) and then X x, it is also straightforward with probability one. = {x, x, x,...}.

9 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 9 Denition 3.. Let (p i ) i be a sequence of non-null probabilities, = (V, E, ) be an ordered ratteli diagram and x X X min. The ratelli-vershik stochastic adding machine associated to them is the discrete time-homogeneous Markov chain (Y n ) n on X x dened as Y n = Γ n for n given that Y = x. Let (Y n ) n be a V stochastic adding machine, we will denote the transition matrix of (Y n ) n by S = (S m,n ) m,n N, i.e (3.) S m,n := S(x n, x m ) := P (Y = x n Y = x m ). When X min = {x min } is an unitary set, there is a unique V stochastic adding machine associated to and a given sequence (p i ) i. This stochastic machine is the main object of study in this paper. To simplify notation we write X x min = X. The hypothesis X min = {x min} is a natural one and occurs when the level sets V k are ordered and the order on the edges is endowed by the order on its source level sets. Example 3.3. (The Cantor systems of numeration case) Consider the ordered ratteli diagram represented by the sequence of matrices M j = (d j ) for a sequence d j for every j. In this case we have an unique ordering which is the canonical consecutive ordering. Moreover Hypothesis A is clearly satised. In this case, X min is unitary and given (d j ) j and (p j ) j there is a unique associated V stochastic adding machine. The stochastic adding machines associated to the Cantor systems of numeration were introduced by Messaoudi and Valle [3]. For instance consider d j = j, for all j. Let x = (e, e, e 3, e 4,...) X, where e = (,, ), e = (, 3, ) and e 3 = (, 4, ). A representation of the path (e, e, e 3 ) in the diagram is presented in item (a) of Figure 4. Here we have ζ(x) = 3, because e and e are maximal edges and e 3 is not maximal. Thus V (x) = (f, f, f 3, e 4, e 5,...) where f = (,, ), f = (,, ) and f 3 = (, 5, ). (see the item b) of Figure 4). Moreover, we have A(x) = {, } and y (x) = (f, e, e 3, e 4,...) and y (x) = (f, f, e 3, e 4,...) (see the items (c) and (d) of Figure 4, respectively). We have that x transitions to V (x) with probability p p p 3, x transitions to x with probability p, x transitions to y (x) with probability p ( p ) and x transitions to y (x) with probability p p ( p 3 ). The initial parts of the transition graph and matrix for the chain are represented in Figure 3. Remark 3.. In Example 3.3, if d j = for all j, then we obtain the stochastic adding machine dened by Killeen and Taylor [6]. Example 3.4. Consider as the ( stationary ) ratteli diagram with consecutive ordering and incidence matrix M =. This diagram satises Hypothesis A. 3 Let x = (e, e, e 3, e 4, e 5,...) X be an innite path, where e = (, 3, ), e = (,, ), e 3 = (,, ), e 4 = (,, ) and e j = (,, ) for j 5. The representation of x in the diagram is given by the path in item (a) of Figure 6.

10 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE Figure 3. Initial parts of the transition graph and matrix of the V stochastic adding machine with incidence matrices M j = (d j ) where d =, d = 4 and d 3 = 6. a) b) c) d) E() E() E(3) Figure 4. Representation paths in a ratteli diagram with incidence matrices M j = (d j ) where j, d =, d = 4 and d 3 = 6. Here we have ζ(x) = 3 and V (x) = (f, f, f 3, e 4, e 5,...) where f = (,, ), f = (,, ) and f 3 = (,, ). (see item (b) of Figure 6). Moreover, we have A(x) = {, } and y (x) = ((,, ), e, e 3,...) and y (x) = ((,, ), (,, ), e 3, e 4,...) (see the (c) and (d) of Figure 6, respectively).

11 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS Hence, we have that x transitions to V (x) with probability p p p 3, x transitions to x with probability p, x transitions to y (x) with probability p ( p ) and x transitions to y (x) with probability p p ( p 3 ). Thus, its transition graph and transition operator are represented in Figure 5. Figure 5. Initial parts of the transition graph and matrix of the V stochastic adding machine associated with a stationary ratteli diagram with incidence matrix M. Example 3.5. (The Fibonacci case) Consider the stationary ordered ratteli ( diagram ) with the canonical consecutive ordering and incidence matrix M F =. In this case does not satisfy Hypothesis A. Again X min is unitary and given (p j ) j there is a unique associated V stochastic adding machine. These stochastic adding machines is associated with the Fibonacci system of numeration and have been introduced in [3] Let x = (e, e, e 3, e 4,...) X be an innite path in the ratteli diagram, where e = (,, ), e = (,, ), e 3 = (,, ), and e j = (,, ) for all j 4. The representation of x in the diagram is given by the continuous path in item (a) of Figure 7. We have ζ(x) = 4 and V (x) = (f, f, f 3, f 4, e 5,...) where f 4 = (,, ) and (f, f, f 3 ) is the minimal edge in E() E() E(3) with range equal to s(f 4 ). (see the item (b) of Figure 7). We have A(x) = {, 3} = {n, n } and y n (x) = ((,, ), (,, ), (,, ), e 4,...) and y n (x) = ((,, ), (,, ), (,, ), e 4,...) (see the items (c) and (d) of Figure 7, respectively).

12 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE E() E() E(3) a) b) c) d) 3 E(4) Figure 6. Representation of paths in a stationary ratteli diagram with incidence matrix M. Hence, we have that x transitions to V (x) with probability p p p 3, x transitions to x with probability p, x transitions to y n (x) with probability p ( p ) and x transitions to y n (x) with probability p p ( p 3 ). E() E() E(3) a) b) c) d) E(4) Figure 7. Representation of paths in a stationary ratteli diagram with incidence matrix M F.

13 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 3 Remark 3.3. Two distinct ordered ratteli diagrams can generate the same stochastic adding machine. For instance consider two stationary ordered ratteli( diagrams ) with consecutive ordering and incidence matrices M = () and M =. oth diagrams generate a unique V stochastic adding machine that corresponds to the stochastic machine studied by Killeen and Taylor in [6]. efore we discuss the probabilistic properties of the V stochastic adding machines, we present some basic denitions from the theory of Markov chains and we recommend [4] to the unfamiliar reader. Let Y = (Y n ) n be a Markov Chain on a probability space (Ω, O, P ). We denote by E[ ] the expectation with respect to P. We say that Y is irreducible if for any pair of states i and j there exists m such that P (Y m = j Y = i) >. An irreducible Markov chain Y is transient if every state i is transient, i.e. P {Y n = i for some n Y = i} <, is the probability that starting in state i, the process will ever re-enter state i. If an irreducible Markov chain is not transient we say that it is recurrent and this means that every state i is recurrent, i.e. P {Y n = i for some n Y = i} =. Furthermore, a recurrent Markov chain is called positive recurrent if for each state i, the expected return time m i = E[R i Y = i] <, where R i = min{n : Y n = i}. Otherwise, if m i = +, then the Markov chain is called null recurrent. Proposition 3.6. Let (p i ) i be a sequence of non-null probabilities such that #{i : p i < } =. Every V stochastic adding machine associated to (p i ) i is an irreducible Markov chain. Furthermore then stochastic machine is transient if and only if j= p j >. Proof. Let (Y n ) n be a V stochastic adding machine associated to (p i ) i, an ordered ratteli diagram = (V, E, ) and x X X min. We have some special states x n, x n,..., which are conal to x by hypothesis, determined by the following: e k (x nj ) = e k (x ) for k j + and (e (x nj ),..., e j (x nj )) is the maximal edge in E()... E(j) with range equal so s(e j+ (x )). Concerning irreducibility, we just point out that (i) for every n the chain can reach x n with positive probability by making the transitions x x, x x,..., x n x n ; (ii) for j + {i : p i < }, we can make the transition x nj x with probability ( p j+ ) j i= p j >.

14 4 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE y (i) and (ii), it is clear that (Y n ) n is irreducible. Now we consider the transience/recurrence of the chain. We rely on some additional properties of the chain related to the special states x nj, j. We have (iii) Once the chain arrives at x nj +, the successor of x nj, it can only visit x nj again if it visits x rst. (iv) If transition x x is possible with positive probability, then x = x nj. (v) Given that a transition from x nj to x nj + or x occurs, the next state of the chain is x nj + with probability p j+, i.e P ( Y n+ = x nj + Yn = x nj, Y n+ {x, x nj +} ) = p j+. The verication of (iii), (iv), (v) follows directly from the denition of (Y n ) n. y the Markov property and properties (i)-(v) above, the probability that the (Y n ) n never returns to x coincide with the event that (Y n ) n reach x nj before it returns to x for every j which has probability j= p j >. Remark 3.4. Let (Y n ) n be an irreducible V stochastic adding machine. If p < then clearly (Y n ) n is aperiodic since P (Y = x Y = x ) = p >. However, when p = the chain can be periodic or aperiodic depending on the ratteli diagram. Proposition 3.7. Let be an ordered ratteli diagram satisfying Hypothesis A and (p i ) i be a sequence of non-null probabilities such that #{i : p i < } = and j= p j =. Then every V stochastic adding machine associated to (p i ) i is null recurrent. Proof. Let (Y n ) n be a V stochastic adding machine associated to (p i ) i, an ordered ratteli diagram = (V, E, ) and x X X min. Suppose that = (V, E, ) satises Hypothesis A, #{i : p i < } = and j= p j =. y Proposition 3.6 the chain is irreducible and recurrent. Put T = inf{n : Y n = x }, i.e the rst return time to x. We are going to show that the expected value of T, E[T ], is innite and then the chain is null recurrent. To compute E[T ] we need to recall the denition of the special states x nj, j, and their properties from the proof of Proposition 3.6. Also recall the denition of the transition probabilities under Hypothesis A from Remark 3.. Put x n := x and consider the following decomposition T = T I {YT =x nj }, where I W is the indicator function of the event W. We obtain that (3.3) E[T ] = E[T I {YT =x nj }]P (Y T = x nj ). n= n=

15 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 5 Clearly on {Y T = x n } we have T = and P (Y T = x n ) = p. Using item (v) in the proof of Proposition 3.6 we get that ( j (3.4) P (Y T = x nj ) = p j )( p j+ ). We also have that i= (3.5) E[T I {YT =x n }] =. Claim: For every j E[T I {YT =x nj }] + j ( i ) p r Suppose that the claim holds. Then by (3.3) and (3.4) we have that (3.6) (3.7) i= E[T ] ( p ) + ( + ) p ( p ) p ( ) (3.8) p p ( p 3 ) + p p p Rearranging terms and putting p = we obtain (3.9) E[T ] p m...p m+j ( p m+j ) (3.) = m= j= ( m= j m+ p j ) = r= =. Thus the chain is null recurrent. It remains to prove the Claim. We prove it by induction. Suppose the claim holds for j (the case j = is (3.5)). Given {Y T = x nj } write T = T + T where T is the time of the rst visit of the chain to x nj + and T the time spent on {x nj +,..., x nj } until it arrives at x. y the induction hypothesis It remains to prove that m= j ( i ). E[T I {YT =x nj }] + p r i= r= ( j ). E[T I {YT =x nj }] p r Time T is greater or equal to the number of transitions to get to x from x nj, and this is bounded below by the necessary number of trials from j independent ernoulli random variables with parameters p,...,p j to obtain j successes. It is an exercise r=

16 6 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE in probability theory using geometric random variables to prove that this number of trials have expected value equal to ( j r= p r). From the proof of Proposition 3.7 we see that we can drop Hypothesis A if the sequence (p i ) i is constant and the ratteli diagram is stationary. Proposition 3.8. Let be a stationary ordered ratteli diagram. If p i = p (, ) for every i, then every V stochastic adding machine associated to (p i ) i is null recurrent. Although we have Propositions 3.7 and 3.8, a V stochastic adding machine associated to (p i ) i such that j= p j = can be positive recurrent. So Hypothesis A is necessary. In Example 3.5 we describe a stationary V stochastic adding machine associated to an ordered ratteli diagram with consecutive ordering which can be positive recurrent for a suciently fast decreasing sequence (p i ) i. 4. Stochastic machines of stationary ratteli diagrams Let = (V, ( E, ) ) be a stationary simple ordered ratteli diagram with incidence a b matrix M =. c d Since in simple, we have necessarily b > and c >, moreover either a > or d >. We can change the labels of vertices in if necessary and suppose that a >. Therefore a + b > and Hypothesis A is equivalent to c + d >. We start with a Proposition that gives a condition on ratteli diagrams that allows the existence of positive recurrent V stochastic adding machines. Proposition 4.. Let = (V, E, ) be a stationary simple ordered ratteli diagram with a = c =, b > and d =. Then the V stochastic adding machine associated to (p j ) j is positive recurrent if p j decreases to zero suciently fast as j. Proof. Recall the denitions from the proof of Proposition 3.7. In order to prove that the stochastic machine is positive recurrent we have to show that E[T ] <. We claim that there exists (C j ) j depending on b but not on (p j ) j such that (4.) E[T ] C + C j max{p j, p j }. j= From the previous inequality, one simply need to choose p j r j /(C j + C j+ ) with j r j <. To prove (4.) we use (3.3) and (3.4). So we need to bound from above the conditional expectation E[T I {YT =x nj }]. The particular form of x nj is important here. We have that x n = ( (, b, ), (,, ), (,, ),... ),

17 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 7 thus the time to get to x n + from x given Y T = x nj is equal to one plus a negative binomial distribution with parameters b and p because the chain uses one unit of time to leave x and then spend a geometric time of parameter p on each of the last b edges of E() with range V (). Therefore and E[T I {YT =x n }] = + b p E[T I {YT =x n }]P (Y T = x n ) + b = C. efore we can use induction on j we still need to deal with E[T I {YT =x n }] and we need to compute the mean time to get to x n + from x n +. We have x n + = ( (,, ), (,, ), (,, ), (,, )... ), where the rst edge is the unique edge in E() with range. So from x n + we only need to change b edges in E() to get to x n + and on each of these edges we spend a geometric time of parameter p. Therefore E[T I {YT =x n }] = E[T I {YT =x n }] + b p = + b p + b p, and E[T I {YT =x n }]P (Y T = x n ) is bounded above by p p + bp + bp ( + b) max{p, p } = C max{p, p }. Analogous estimates allow us to show that E[T I {YT =x n3 }]P (Y T = x n3 ) is bounded above by Now Suppose that p p p 3 + p p 3 b + p p 3 b + p b ( + b + b ) max{p, p 3 }. E[T I {YT =x nj }]P (Y T = x nj ) C j max{p j, p j }, and we are going to estimate E[T I {YT =x nj+ }]. Using the fact that a = c = to go from x nj+ + to x nj+ + we need to change b edges in E(j + ) without change the edge (j +,,, ) E(j + ) but considering all edges in E()... E(j) with range V (j). Thus b E[T I {YT =x nj+ }] E[T I {YT =x nj }] + (E[T I {YT =x nj+ }] ). p j+ Thus E[T I {YT =x nj+ }]P (Y T = x nj+ ) is bounded above by C j max{p j, p j }p j+ + C j max{p j, p j }p j+ C j max{p j+, p j+ }. So we just need to take C j+ = C j.

18 8 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE From the proof of proposition 4. we can also see that it is enough to have j p j summable to obtain a positive recurrent stochastic machine from the hypothesis of the proposition. As a corollary we get the result from [7] about the existence of positive recurrent Fibonacci stochastic adding machines. Corollary 4.. The Fibonacci stochastic adding machines associated to (p j ) j are positive recurrent if p j decreases to zero suciently fast as j. To continuing the study of V stochastic machines of ratteli diagrams, we need to introduce some notation related to systems of numeration associated to the ratteli diagrams. Let us denote M n by ( ) M n an b = n, c n d n for all n, where M = I is the identity matrix. For each n, put F n = a n + b n, G n = c n + d n n. This gives F = G =, F = a + b and G = c + d. Remark 4.3. For each nonnegative integer n, F n is the number of paths from V () to the vertex (n, ) at the ratteli diagram. Respectively, G n is the number of paths from V () to the vertex (n, ). Lemma 4.4. We have F n+ = (a + d)f n (ad bc)f n and G n+ = (a + d)g n (ad bc)g n, for all n. ( ) ( ) Fn Proof. It comes to the fact that = M G n for all n and the characteristic polynomial of M is p(x) = x + (a + d)x (ad bc). n 4.. case under Hypothesis A and consecutive ordering. From now on we assume that abc >, c + d > and that is endowed with the consecutive ordering. Thus is simple and satises Hypothesis A. Moreover X min = {x } is a unitary set and for each x X we have A(x) = {,..., ζ(x) }. The aim of this section is the study of the spectrum of V stochastic machines under these conditions. We rst need to establish a proper notation to deal with the possible transitions of the chain in X = X x. Dene j as the minimum edge of E(j) with range, i.e. j = (j,,, ). For convenience we will sometimes not write the level index j simply writing = (,, ). Let x = (e j ) j = ((s j, m j, r j )) j X. Recall that x = ( j ) j and x x is conal with x, thus there exists N N such that x N = V N(x ) = x. Put ξ(x) = min{j : e l = for all l > j}.

19 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 9 The reader should recall the denition of ζ(x) and note that ζ(x) and ξ(x) play a dierent role. 4.. Numeration systems associated to ratteli diagrams. Denition 4.5. Let x X. For each j {,..., ξ(x)}, dene δ j (x) = δ j and γ j (x) = γ j according to the following four cases: (i) If s j = and r j = then δ j = m j {,..., a } and γ j = ; (ii) If s j = and r j = then δ j = a and γ j = m j a {,..., b }; (iii) If s j = and r j = then δ j = m j {,..., c } and γ j = ; (iv) If s j = and r j = then δ j = c and γ j = m j c {,..., d }. For x N = V N (x ) = x we also denote δ j = δ j (N) and γ j = γ j (N). Observe that m j = δ j + γ j, for all j. Moreover, if d =, then (s j, r j ) (, ), for all j. Example 4.6. Consider ( the) consecutive ordering ratteli diagram reprensented 3 by the matrix M =. y Lemma 4.4, we have 4 F =, F = 4, F = 9, F 3 = 9,... G =, G = 5, G = 4, G 3 = 5,... Consider x, y where x = (x j ) j = ((, 3, ), (, 4, ), (,, ),,,...) and y = (y j ) j = ((,, ), (,, ), (, 3, ),,,...). The representation of x and y in the ratteli diagram is given respectively in the items (a) and (b) of Figure 8. y Denition 4.5, we have that δ (x) =, γ (x) = ; δ (x) =, γ (x) = 3; δ 3 (x) =, γ 3 (x) = ; δ i (x) = γ i (x) =, for all i 4. and δ (y) =, γ (y) = ; δ (y) =, γ (y) = ; δ 3 (y) =, γ 3 (y) = ; δ i (y) = γ i (y) =, for all i 4. Proposition 4.7. Let N be a nonnegative integer and x X such that V N(x ) = x. Then, N = ξ(x) j= δ j+f j + γ j+ G j, where δ i (N) = δ i and γ i (N) = γ i are dened in Denition 4.5, for all i. Proof. Fix a nonnegative integer N and let x = V N (x ) = (e, e, e 3,...), with e i = (s i, m i, r i ) for all i. From Remark 4.3, we have that for each nonnegative integer k (4.) V F k (x ) = (,...,, ẽ, }{{} f,,,...) (k ) times where either ẽ = and f = (,, ) if a > or ẽ = (,, ) and f = (,, ) if a =. Thus, since e k = (s k, m k, r k ), it follows that if s k = and r k =, then a >,

20 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE a) b) E() E() E(3) E(4) m k {,..., a } and (4.3) Figure 8. Representation of a ratteli diagram. V δ kf k +γ k G k (x ) = V m kf k (x ) = (,...,, ẽ, (, m }{{} k, ),,,...), (k ) times and if s k = and r k =, then m k {a,..., a + b } and (4.4) V δ kf k +γ k G k (x ) = V af k +(m k a)g k (x ) = (,...,, ẽ, (, m }{{} k, ),,,...). (k ) times Now, consider k = ξ(x) = min{j : e l = for all l > j} and put N k = δ k F k + γ k G k. For each j {,..., k }, let N j = δ j F j + γ j G j + N j+ and x(j +) := (,...,, ẽ, e }{{} j+, e j+,..., e k,,,...). Suppose that V N j+ (x ) = x(j +), j times for some j {,..., k }. Here, we need to consider four cases: i) s j = and r j = ; iii) s j = and r j = ; ii) s j = and r j = ; iv) s j = and r j =.

21 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS For example, in the case ii) ( we have ) ẽ = (,, ), m j {a,..., a + b } and V N j (x ) = V af j +(m j a)g j V N j+ (x ) = V af j +(m j a)g j (x(j + )) = x(j). In the same way, we can check that V N j (x ) = x(j) for the other cases. y induction we have V N (x ) = x() = x and since δ i = γ i =, for all i > k, it k + follows that N = (δ i F i + γ i G i ) = (δ i F i + γ i G i ) = N. i= Remark 4.8. We believe that the last proposition is another formulation of Lemma 4 in [5], which gives a formula of the rst entrance time map. i= Remark 4.9. We call ((δ, γ ), (δ, γ ),...) the (F,G)-representation of N and we put N = ξ(x) j= δ j+f j + γ j+ G j = ((δ, γ ), (δ, γ ),...). The set of (F, G)-representations is recognized by a nite graph called automaton (see Figure 9). Figure 9. Automaton related to the (F, G)-representation of N = ((δ, γ ), (δ, γ ),...), where δ i {x a, x c } and γ i {y b, y d }, for all i with x a {,..., a }, x c {,..., c }, y b {,..., b } and y d {,..., d }. Remark 4.. In Example 4.6, it follows by Proposition 4.7 that L(N x ) = x and L(N y ) = y where N x = F + G + F + 3G + F + G = 65 and N y = F + G + F + G = 69. Example 4.. If M = (d) for d, by Proposition 4.7, we obtain the numeration in base d, with digits {,,..., d }. Remark 4.. We can dene the sequences (δ i (x)) i and (γ i (x)) i for all x X as done in Denition 4.5, in the case where the ( ratteli ) diagram does not satises a b Hypothesis A, i.e. the incidence matrix M =, where ab >. Furthermore, in this case δ i {,..., a} and γ i {,..., b }, for all i. Moreover, by Lemma 4.4, we have that G n = F n, for all n. y Proposition 4.7, the (F, G)- representation of N is given by the automaton represented in Figure. Observe in Figure that when b =, the representation of N is equal to ((δ, )(δ, ),...), with δ i δ i < lex a, for all i.

22 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE Figure. Automaton related to the (F, G)-representation, where x a {,..., a } and y b {,..., b } Spectrum of the stochastic machines of ratteli diagrams. We are nally in position to compute the spectrum of the transition operator (acting in l ) of the V stochastic adding machines associated to a stationary ratteli diagram endowed with the consecutive ordering. We denote the spectrum, point spectrum and approximate point spectrum of the transition operator S respectively by σ(s), σ pt (S) and σ a (S). Recall that λ belongs to σ(s) (resp. σ pt (S)) if S λi is not bijective (resp. not one-to-one). Also, λ σ a (S) if there exists a sequence (v n ) n such that v n =, for all n and (S λi)v n converges to when n goes to innity. For each λ C, let (u Fn (λ)) n = (u Fn ) n and (w Fn (λ)) n = (w Fn ) n be the sequences dened by u F = w F = λ ( p ) p and for all n. (4.5) u Fn = p n+ u a F n w b F n p n+ p n+, w Fn = p n+ u c F n w d F n p n+ p n+. From this, let (v n ) n be the sequence dened by v n = ξ(n) i= u δ i+ F i w γ i+ F i, where δ j = δ j (n) and γ j = γ j (n), j {,..., ξ(n)}, are given in denition 4.5. Since v Fn = u Fn, for all n, we will denote v n by u n. Theorem 4.3. Let S be the transition operator of a V stochastic machine associated to a ratteli diagram. Then, acting in l (N), we have that the set of eigenvalues of S is σ pt (S) = {λ C : (u n (λ)) n is bounded}. Remark 4.4. From Theorem 4.3, we deduce that σ pt (S) {λ C : (u Fn (λ)) n is bounded}. Moreover, if det M, we can show (see Proposition 4.7) that σ pt (S) E := {λ C : (u Fn (λ), w Fn (λ)) is bounded}. Since, g n... g (u F, u F ) = (u Fn, w Fn ), for all n, where g n : C C are polynomials dened by ( g n (x, y) = p n+ x a y b p n+ p n+, p n+ x c y d p n+ p n+ ), for all n, if follows that σ pt (S) is contained in the set {λ C : ( λ +p p, λ +p p K := {(x, y) C : (g n... g (x, y)) n is bounded}. ) K} where

23 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 3 This set is the -dimensional bered lled Julia set associated to (g n ) n (for more on bered Julia sets see [34] and references therein). In particular, if (p i ) i is constant, then K is a -dimensional lled Julia set. For the proof of Theorem 4.3, we need the following lemma. Lemma 4.5. For all z = (z i ) i l, ( ζn ) (Sz) N = p j z N+ + ( p )z N + j= ζ N r= ( r j= p j ) ( p r+ )z N r j= δ j+f j +γ j+ G j, if ζ N and (Sz) N = p z N+ + ( p )z N if ζ N =, where δ i, γ i are given in Denition 4.5, for all i {,..., ζ N }. Proof. Let N N and V N(x ) = x = (e, e, e 3,...). All we need to do is identify S N, Ñ for Ñ N. Let ξ(x) = k and ζ(x) = ζ N. Thus, x = (e,..., e ζn, e ζn, e ζn +,..., e k,,,...) and under Hypothesis A, we have that A(x) = {,..., ζ N }. From Denition 3. and Remark 3., we have that S N,N = p, S N,N+ = ζ N j= p j and S N, Ñ = if Ñ / {N, N +}. Thus, if ζ N =, we are done. Suppose that ζ N. For each i A(x), consider y i (x) dened by relation (3.). We can check that y i (x) = (,...,, ẽ, e }{{} i+, e i+,..., e ζn, e ζn, e ζn +,..., e k,,,...), i times where ẽ = (,, ) = if s i+ = and ẽ = (,, ) if s i+ =. For each i A(x), let N i N, such that V N i (x ) = y i (x). Thus, from Proposition 4.7, we have that N i = N i j= δ j+f j + γ j+ G j. Hence, from Remark 3., we have that S N,Ni = i j= p j( p i+ ). Furthermore S N, Ñ = if Ñ / {N, N +, N i, i A(x)} and the proof is nished. Our next step is to prove Theorem 4.3. The proof uses the same idea of the case M = (d), for d done in [9]. However, the extension is far from elementar. Proof of Theorem 4.3. Let z = (z N ) N be a sequence of complex numbers such that (Sz) N = λz N for every N. We shall prove that z N = u N z for all N. For this we need to have in mind the representation of N as a path in X, i.e. x = V N (x ) = (e,..., e ξ(x),,,...) where e j = (s j, m j, r j ), j ξ(x). The proof is based on the representation of Lemma 4.5. We use induction on N N. For N = we have by denition that δ =, γ = and δ j = γ j = for all j. Furthermore, λz = ( p )z + p z z = λ + p p z = u z.

24 4 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE Now x N = ((δ, γ ), (δ, γ ),...) and suppose that z j = u j z for all j {,..., N}. Suppose that ζ N =. Since z N = z ξ(x) i= u δ i+ F i w γ i+ F i = u δ F w γ F z ξ(x) i= u δ i+ F i w γ i+ F i, (Sz) N = p z N+ + ( p )z N = λz N and (4.6) u F = w F = λ ( p ) p, we have z N+ = u F z N = u δ +γ + F z ξ(x) i= u δ i+ F i w γ i+ F i. From here, we need to consider two cases: Case : if s =, then δ < a if r = and δ < c if r =. Furthermore, γ =. Thus, N + = ((δ +, γ ), (δ, γ ),...) and ξ(x) u N+ = u δ + F w γ F i= ξ(x) u δ i+ F i w γ i+ F i = u δ +γ + F i= u δ i+ F i w γ i+ F i. Case : if s =, then δ = a and γ < b if r = and δ = c and γ < d if r =. Thus, N + = ((δ, γ + ), (δ, γ ),...) and ξ(x) u N+ = u δ F w γ + F i= ξ(x) u δ i+ F i w γ i+ F i = u δ +γ + F i= u δ i+ F i w γ i+ F i. Hence, in both cases we have that z N+ = u N+ z. Now for ζ N we consider separately the cases d > and d =. Case d>: First, suppose that ζ N = (i.e. e = (s, m, r ) is a maximal edge and e is not maximal). Thus, by Lemma 4.5 and the fact that (Sz) N = λz N, we have Hence, z N+ = p p ((λ ( p ))z N p ( p )z N δ F γ G ). z N+ z ξ(x) = (λ ( p δ ))u F w γ F u δ F w γ F p u δ r= u δ r+ w γ r+ F p F p p w γ F r Since e is a maximal edge, it follows that s =. If r =, then δ = a and γ = b and if r = then δ = c and γ = d. Thus, z N+ z ξ(x) = r= u δ r+ w γ r+ λ ( p ) p λ ( p ) p ua F w b F u δ F w γ F p p p u δ F w γ F, if r =, uc F w d F u δ F w γ F p p p u δ F w γ F, if r =.

25 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 5 y (4.6), we deduce z N+ z ξ(x) = r= u δ r+ w γ r+ and so (4.7) z N+ = Since it follows that { ( u a+b F p p ) ( p w c+d F p p p z u δ + F w γ z u δ F w γ + F N = ((δ, γ )(δ, γ )...) = N + = and from (4.7), we have that z N+ = u N+ z. Finally we have to consider ζ N 3. ξ(x) r= u δ r+ w γ r+ ξ(x) F r= u δ r+ w γ r+ u δ F w γ F = u δ + F w γ F, if r =, ) u δ F w γ F = u δ F w γ + F, if r =,, if r =,, if r =. { ((a, b )(δ, γ )...), if r =, ((c, d )(δ, γ )...), if r =, { ((, )(δ +, γ )...), if r =, ((, )(δ, γ + )...), if r =, In this case, since (e,..., e ζn ) is a maximal element of E() E()... E(ζ N ) and d >, it follows that s j = r j = for all j {,..., ζ N }. Therefore, m j = c + d (i.e δ j = c and γ j = d ) for all j {,..., ζ N }. Furthermore, we have two subcases: () if r ζn = then m ζn = a + b (i.e δ ζn = a and γ ζn = b ), () if r ζn = then m ζn = c + d (i.e δ ζn = c and γ ζn = d ). Thus, by Lemma 4.5 and Denition 4.5, since (Sz) N = λz N, we have that (4.8) (4.9) [ ζn 3 [λ ( p )] [ ζn 3 ( p ) z N+ z ξ(x) r=ζ N u δ r+ w γ r+ = ] r= u c w d ] r= u c w d u δ ζ N F ζn wγ ζ N F ζn uδ ζ N F ζn wγ ζ N F ζn j= p j u δ ζ N F ζn wγ ζ N F ζn uδ ζ N F ζn wγ ζ N F ζn j= p j... ( p ζn )u δ ζ N F wγ ζ N ζn F uδ ζ N ζn F wγ ζ N ζn F ζn j=ζn p j y (4.6), the rst term in (4.9) is equal to p ζ N p ζn u δ ζn F ζn wγ ζ N F ζn.

26 6 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE w c+d F [ ζn ] 3 r= u c w d Summing with the the second term, we get [ ζn ] w c+d 3 F ( p ) r= u c w d p which is equal to w F [ ζn ] 3 r= u c w d u c F w d F [ ζn ] 3 r= u c w d u δ ζ N F wγ ζ N ζn F uδ ζ N ζn F ζn wγ ζ N F ζn j= p j. u δ ζ N F wγ ζ N ζn F uδ ζ N ζn F wγ ζ N ζn F ζn j=3 p j u δ ζ N F wγ ζ N ζn F uδ ζ N ζn F wγ ζ N ζn F ζn j=3 p j = u δ ζ N F wγ ζ N ζn F uδ ζ N ζn F wγ ζ N ζn F ζn j=3 p j. y induction, we have that the sum of the rst ζ N terms in (4.9) is equal to u δ ζ N F wγ ζ N + ζn F ζn u δ ζ N F ζn wγ ζ N F ζn p ζn. Finally, summing the previous expression with the last term in (4.9) we have that (4.8) is equal to Therefore, u a F ζn wb F ζn ( p ζ N ) p ζn u c F ζn wd F ζn ( p ζ N ) z N+ = p ζn { z u δ ζ N + F wγ ζ N ξ(x) ζn F ζn z u δ ζ N F ζn wγ ζ N + F ζn u δ ζn F ζn wγ ζ N F ζn = uδ ζ N + F ζn wγ ζ N F ζn, if r ζ N = ; u δ ζn F ζn wγ ζ N F ζn = uδ ζ N F ζn wγ ζ N + F ζn, if r ζ N =. r=ζ N u δ r+ w γ r+, if r ζn = ξ(x) r=ζ N u δ r+ w γ r+, if r ζn = = u N+z, where the next equality comes from the fact that δ i (N + ) = γ i (N + ) =, for all i {,..., ζ N }. Case d = : Suppose that r = and ζ N is an odd number (the proof for the cases r = or ζ N even can be dealt in the same way). Thus, since (e,..., e ζn ) is a maximal element of E() E()... E(ζ N ), we have that r i =, r i =, s i = and s i =, for all i {,..., ζ N }. Therefore, m i = a + b (i.e δ i = a and γ i = b ) and m i = c (i.e δ i = c and γ i = ) for all i {,..., ζ N }.

27 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 7 For each i {,..., ζ N }, let P i be the product dened by P i := ζ N r=i Thus, we have that P i = u a F i w b F i P i+ if i is an even number and P i = u c an odd number. y Lemma 4.5, since (Sv) N = λv N, we have that F i u δr+ w γ r+ P i+ if i is. (4.) (4.) v N+ v ξ(n+) r=ζ N u δ r+ w γ r+ = [λ ( p )]P u δ ζ N F wγ ζ N ζn F ζn ( p )P u δ ζ N F wγ ζ N ζn F ζn j= p ζn j j= p j ( p 3 )P u δ ζ N F wγ ζ N ζn F ζn ( p ζn )P ζn 3u δ ζ N... j=3 p ζn j j=ζn p j F wγ ζ N ζn F ζn ( p ζn )P ζn u δ ζ N j=ζn p j F wγ ζ N ζn F ζn p ζ N p ζn u δ ζn F ζn wγ ζ N F ζn. Since u F = w F = λ + p, the rst term in (4.) is equal to p u F P u δ ζ N F wγ ζ N ζn F ζn = j= p j u a+b F Summing with the the second term, we get u a+b F ( p ) p δ ζn P u F wγ ζ N ζn F ζn j=3 p j = u F Summing with the the third term, we get u c F ( p 3 ) p 3 δ ζn P u F wγ ζ N ζn F ζn j=4 p j = w F P u δ ζ N P u δ ζ N P u δ ζ N F wγ ζ N ζn F ζn j= p j. F wγ ζ N ζn F ζn j=3 p j = u c F F wγ ζ N ζn F ζn j=4 p j = u a F w b F P u δ ζ N F wγ ζ N ζn F ζn j=3 p j. P 3 u δ ζ N F wγ ζ N ζn F ζn j=4 p j. y induction we have that the sum of the rst ζ N terms in (4.) is equal to u a F ζn 3 wb F ζn 3 ( p ζ N ) p ζn P ζn u δ ζ N F wγ ζ N ζn F ζn j=ζn p j = u c F ζn u δ ζ N F ζn wγ ζ N F ζn p ζn. Finally, summing the previous expression with the last term in (4.) we have that (4.) is equal to u c F ζn ( p ζ N ) p ζn u δ ζn F ζn wγ ζ N F ζn = uδ ζ N F ζn wγ ζ N + F ζn.

28 8 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE Therefore, v N+ = v u δ ζ N F wγ ζ N + ξ(n+) ζn F ζn r=ζ N u δ r+ w γ r+ = u N+ v, where the last equality comes from the fact that δ i (N + ) = γ i (N + ) =, for all i {,..., ζ N }. Proposition 4.6. Let F := {λ C : (u Fn (λ)) n is bounded}. Then σ pt (S) F σ a (S). Proof. y Theorem 4.3, σ pt (S) F and we only have to prove that F σ a (S). Let λ F and suppose that λ / σ pt (S). We will prove that λ σ a (S). In fact, for each k, consider x (k) = (x (k), x (k), x (k),..., x (k) k,,,...) = (, u (λ), u (λ),..., u k (λ),,,...), where (u n (λ)) n = (u n ) n is the sequence dened in relation (4.5). Dene y (k) := x(k) x (k). Claim: lim (S n + λi)y(fn) =. In fact, for all i {,..., k }, we have ((S λi)y (k) ) i = and y i =, for all i > k. Hence, note that (S λi)y (k) + = sup (S λi) ij y (k) j i = sup k j= (S λi) ijx (k) j i k x (k). j= Let n >, k = F n and i k. We consider two cases: Case a > : If i = F n, then since a >, by relation (4.) we have that V Fn (x ) = (,...,, (,, ),,,...). Since n >, it follows that S }{{} i,j =, for all n j {,..., F n } and S i,i = p. Therefore, F n j= (S λi) ijx (Fn) j = p λ u Fn. If F n < i F n, then since a >, by the proof of Proposition 4.7, we have that V i (x ) = (e,..., e n, (,, ),,,...). Hence S i,j =, for all j {,..., F n }. Furthermore, since S i,i = p and S is a stochastic matrix, it follows that S i,j p, for j = F n. Therefore, F n j= (S λi) ijx (Fn) j p u Fn.

29 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 9 If i F n, then V i (x ) = (e,..., e l,,,...), with e l and l n+. Since a >, we have m l > and so S i,j =, for all j {,..., F n }. Furthermore, { p... p S i, = l ( p l+ ), if (e,..., e l ) is a maximal way; p, if is not.. Therefore, F n j= (S λi) ijx (Fn) j p x = p. Case a = : If i = F n then S i,j =, for all j {,..., F n }, and S i,i = p. Therefore, F n j= (S λi) ijx (Fn) j = p λ u Fn. If F n < i < F n +G n then S i,j =, for all j {,..., F n }, and S i,j p, for j = F n. Therefore, F n j= (S λi) ijx (Fn) j p u Fn. If i = F n +G n then S i,j =, for all j {,..., F n }, S i,j p for j = F n, S i, p if b = and S i, = if b >. Therefore, F n j= (S λi) ijx (Fn) j p + p u Fn. If i F n + G n then S i,j =, for all j {,..., F n }, and S i,j p, for j =. Therefore, F n j= (S λi) ijx (Fn) j p. Hence, from both cases it follows that (4.) (S λi)y (Fn) p λ u Fn + p u Fn + p x (Fn). Since λ F and λ / σ pt (S) it follows that (u Fn ) n is a bounded sequence and (u n ) n is not. Therefore, we have lim n + x (Fn) = +, which implies from relation (4.) that lim (S n + λi)y(fn) =. Therefore, λ σ a (S) σ(s). Proposition 4.7. If det M = ad bc, then (u Fn ) n is bounded if and only if (w Fn ) n is bounded. Proof. Let R n = p n+ u Fn + p n+ and S n = p n+ w Fn + p n+, for all n. y (4.5), we have that Rn+ c = Sn+w a bc ad F n and Sn+ b = Rn+u d bc ad F n. Since ad bc and (p n ) n is bounded, we obtain the result. Question. If det M >, is (u Fn ) n bounded equivalent (w Fn ) n bounded? Remark 4.8. From Remark 4.4 and Propositions 4.6 and 4.7, we have that if det M, then σ pt (S) E = {λ C : ( λ +p ) K} σ a (S). p, λ +p p

30 3 D. A. CAPRIO, A. MESSAOUDI AND G. VALLE Remark 4.9. If e := a + b = c + d, then we have F n = G n = e n, for all n. In this case, the Vershik map is related to addition of in base e, see Remark 3.3 and Example 3.3. For this class, it was proved in [9] that the ( point spectrum ) of S is equal to the bered lled Julia set of f n (x) = p n+ x a+b p n+. In the next Proposition we will prove the same result cited below. Proposition 4.. If a + b = c + d then σ pt (S) = E. Furthermore, ( ) E = {λ C : (f n... f (u F )) n is bounded}, where f n (x) = p n+ x a+b p n+, for all n. Proof. From Theorem 4.3 and Remark 4.4 we have that σ pt (S) E. Let λ E. Since a + b = c + d, it follows from (4.5) that u Fn (λ) = w Fn (λ), for all λ C and n. Thus, it follows that u Fn (λ), w Fn (λ) for all n, indeed let R > be a real number such that u Fk = w Fk > R. Since abc > and c + d >, it follows that min{a + b, c + d} >. Thus, u Fk+ = w Fk+ > p k+ R a+b p k+ p k+ > R a+b R. y induction we obtain that u Fk+i = w Fk+i > R i, for all i. Since R >, it follows that (u Fn ) n and (w Fn ) n are unbounded and λ / E which yields a contradiction. Therefore, if λ E and then u Fn (λ), w Fn (λ) for all n, by (4.5), we have that u n (λ), for all n, i.e. λ σ pt (S). To prove that E = {λ C : (f n... f (u F )) n is bounded}, we just need to observe that f n... f (u F (λ)) = u Fn (λ) = w Fn (λ), for all n. Remark 4.. In Proposition 4. we have proved that min{ u Fi, w Fi } > for some integer i, then min{ u Fn, w Fn } goes to when n goes to innity. Question. If det M, can we prove that σ pt (S) = E = σ(s)? Example 4.. Consider ( the ) consecutive ordering ratteli diagram represented 3 by the matrix M =. elow, we present some pictures describing the set E = {λ C : (u Fn (λ), w FN (λ)) n is bounded} for some choices of (p i ) i.

31 STOCHASTIC ADDING MACHINES ASED ON RATTELI DIAGRAMS 3 Example 4.3. Consider ( the ) consecutive ordering ratteli diagram represented by the matrix M =. elow, we present some pictures describing the set 3 E = F = {λ C : (u Fn (λ)) n is bounded} for some choices of (p i ) i. Example 4.4. Consider ( the ) consecutive ordering ratteli diagram represented 5 by the matrix M =. elow, we present some pictures describing the set 9 E = F = {λ C : (u Fn (λ)) n is bounded} for some choices of (p i ) i. Remark 4.. It will be interesting to compute the dierent parts of the spectrum of S acting on other anach spaces like c, c, l q, with q as done for base in [3] and for Cantor systems of numeration in [3] Some topological properties of the set E. Let us suppose for simplicity that p i = p ], [, for all i. Theorem 4.5. Assume that det M = ad bc < and bc > (ad bc), then the set E satises the following properties: () C \ E is a connected set. () If p <, then E is not connected.