Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures

Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures

We study finite measures on Bratteli diagrams invariant with respect to the tail equivalence relation. Amongst the proved results on the finiteness of measure extension, we characterize the vertices of a Bratteli diagram that support an ergodic finite invariant measure.

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Дата:2015
Автори: Bezuglyi, S., Karpel, O., Kwiatkowski, J.
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Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2015
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Цитувати:Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures / S. Bezuglyi, O. Karpel , J. Kwiatkowski // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 3-17— Бібліогр.: 9 назв. — англ.

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spelling irk-123456789-1179812017-05-28T10:17:25Z Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures Bezuglyi, S. Karpel, O. Kwiatkowski, J. We study finite measures on Bratteli diagrams invariant with respect to the tail equivalence relation. Amongst the proved results on the finiteness of measure extension, we characterize the vertices of a Bratteli diagram that support an ergodic finite invariant measure. Изучены конечные меры на диаграммах Браттели, инвариантные относительно хвостового отношения эквивалентности. Среди доказанных результатов, касающихся конечности расширения меры, дана характеристика тех вершин диаграммы Браттели, которые относятся к носителю эргодической конечной инвариантной меры. Вивчаються скінченні міри на діаграмах Браттелі, інваріантні відносно хвостового відношення еквівалентності. Серед доведених результатів, що стосуються скінченності розширення міри, надано характеристику тим вершинам діаграми Браттелі, які відносяться до носія ергодичної скінченної інваріантної міри. 2015 Article Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures / S. Bezuglyi, O. Karpel , J. Kwiatkowski // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 3-17— Бібліогр.: 9 назв. — англ. 1812-9471 DOI: 10.15407/mag11.01.003 MSC2000: 37A05, 37B05 (primary); 28D05, 28C15 (secondary) http://dspace.nbuv.gov.ua/handle/123456789/117981 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We study finite measures on Bratteli diagrams invariant with respect to the tail equivalence relation. Amongst the proved results on the finiteness of measure extension, we characterize the vertices of a Bratteli diagram that support an ergodic finite invariant measure.
format Article
author Bezuglyi, S.
Karpel, O.
Kwiatkowski, J.
spellingShingle Bezuglyi, S.
Karpel, O.
Kwiatkowski, J.
Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
Журнал математической физики, анализа, геометрии
author_facet Bezuglyi, S.
Karpel, O.
Kwiatkowski, J.
author_sort Bezuglyi, S.
title Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
title_short Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
title_full Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
title_fullStr Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
title_full_unstemmed Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures
title_sort subdiagrams of bratteli diagrams supporting finite invariant measures
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/117981
citation_txt Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures / S. Bezuglyi, O. Karpel , J. Kwiatkowski // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 3-17— Бібліогр.: 9 назв. — англ.
series Журнал математической физики, анализа, геометрии
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AT karpelo subdiagramsofbrattelidiagramssupportingfiniteinvariantmeasures
AT kwiatkowskij subdiagramsofbrattelidiagramssupportingfiniteinvariantmeasures
first_indexed 2025-07-08T13:06:45Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2015, vol. 11, No. 1, pp. 3–17 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures S. Bezuglyi and O. Karpel B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv, 61103, Ukraine E-mail: bezuglyi@ilt.kharkov.ua, helen.karpel@gmail.com J. Kwiatkowski Kotarbinski University of Information Technology and Management Olsztyn, Poland E-mail: jkwiat@mat.umk.pl Received November 13, 2013, revised April 18, 2014 We study finite measures on Bratteli diagrams invariant with respect to the tail equivalence relation. Amongst the proved results on the finiteness of measure extension, we characterize the vertices of a Bratteli diagram that support an ergodic finite invariant measure. Key words: Bratteli diagrams, ergodic invariant measures, measure sup- port, tail equivalence relation, Cantor set. Mathematics Subject Classification 2010: 37A05, 37B05 (primary); 28D05, 28C15 (secondary). 1. Introduction and Background In this note, we continue the study of ergodic measures on the path space XB of a Bratteli diagram B started in [1–3]. Recall that, given a minimal (or even aperiodic) homeomorphism T of a Cantor set X, one can construct a refining sequence (ξn) (beginning with ξ0 = X) of clopen partitions such that every ξn is a finite collection of T -towers (X(n) v : v ∈ Vn) [7–9]. This fact is in the base of the very fruitful idea: (X, T ) can be realized as a homeomorphism ϕ (Vershik map) acting on the path space of a Bratteli diagram. By definition, a Bratteli diagram B is represented as an infinite graph with the set of vertices V partitioned into the levels Vn, n ≥ 0, such that the edge set En between the levels n − 1 and n is determined by the intersection of towers of partitions ξn−1 and ξn (the c© S. Bezuglyi, O. Karpel, and J. Kwiatkowski, 2015 S. Bezuglyi, O. Karpel, and J. Kwiatkowski detailed definition and references are given below). Every T -invariant (hence, ϕ-invariant) measure µ on X is completely defined by its values µ(X(n) v ) on all towers where v ∈ Vn and n ≥ 0. In [1] and [3], the cases of stationary and finite rank Bratteli diagrams (i.e., |Vn| ≤ d for all n) were studied. We notice that while studying ϕ-invariant measures, we can ignore some rather subtle questions about the existence of a Vershik map on the path space (see [4, 5]) and work with the measures invariant with respect to the tail equivalence relation E (cofinal equivalence relation, in other words). Our interest and motivation for this work arise from the following result proved in [3]: for any ergodic probability measure µ on a finite rank diagram B there exists a subdiagram B of B defined by a sequence W = (Wn), where Wn ⊂ Vn, such that µ(X(n) w ) is bounded from zero for all w ∈ Wn and n. It was also shown that µ can be obtained as an extension of an ergodic measure on the subdiagram B, in other words, B supports µ (the detailed definitions can be found below). What is an analogue of the above result for general Bratteli diagrams? Sup- pose we take a subdiagram B = B(W ) of a Bratteli diagram B and consider an ergodic probability measure ν on B. Then this measure can be naturally ex- tended (by E-invariance) to a measure ν̂ defined on the E-saturation X̂B of the path space XB. If the cardinality of Wn is growing, then we cannot expect that the measures of the towers corresponding to the vertices from Wn are bounded from below. But we do expect that the rate of changes of ν̂(X(n) v ) is essentially different for v ∈ Wn and v /∈ Wn. We prove that if the measure ν̂ is finite and the ratio |Wn| |V \Wn| is bounded, then the minimal value of {ν̂(X(n) v ) : v /∈ Wn} is much smaller than the maximal value {ν̂(X(n) v ) : v ∈ Wn}. We also get the results for the ratio of the tower heights corresponding to Wn and V \Wn. Another assertion that is proved in the paper is a modification of [3, Theo- rem 6.1]. We also give a criterion and a sufficient condition for the finiteness of the extended measure ν̂, using the condition on entries of the incidence matrices. A number of examples related to this issue is also considered in the paper. Most of definitions and notation used in this paper are taken from [3]. Since the concept of Bratteli diagrams has been studied in a great number of recent research papers devoted to various aspects of Cantor dynamics, we give here only some necessary definitions and notation referring to the pioneering articles [7, 8], (see also [3, 6]) where the reader can find more detailed definitions and the widely used techniques, for instance, the telescoping procedure. A Bratteli diagram is an infinite graph B = (V, E) such that the vertex set V = ⋃ i≥0 Vi and the edge set E = ⋃ i≥1 Ei are partitioned into disjoint subsets Vi and Ei where (i) V0 = {v0} is a single point; 4 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures (ii) Vi and Ei are finite sets; (iii) there exists a range map r and a source map s, both from E to V , such that r(Ei) = Vi, s(Ei) = Vi−1, and s−1(v) 6= ∅, r−1(v′) 6= ∅ for all v ∈ V and v′ ∈ V \ V0. Given a Bratteli diagram B, the n-th incidence matrix Fn = (f (n) v,w), n ≥ 0, is a |Vn+1| × |Vn| matrix such that f (n) v,w = |{e ∈ En+1 : r(e) = v, s(e) = w}| for v ∈ Vn+1 and w ∈ Vn. Here the symbol | · | denotes the cardinality of a set. For a Bratteli diagram B = (V, E), the set of all infinite paths in B is de- noted by XB. The topology defined by finite paths (cylinder sets) turns XB into a 0-dimensional metric compact space. We will consider only those Bratteli dia- grams for which XB is a Cantor set. The tail equivalence relation E on XB says that two paths x = (xn) and y = (yn) are tail equivalent if and only if xn = yn for n sufficiently large. Let W = {Wn}n>0 be a sequence of (proper, non-empty) subsets Wn of Vn. Set W ′ n = Vn \Wn. The (vertex) subdiagram B = (W, E) is defined by the vertices W = ⋃ i≥0 Wn and the edges E that have their source and range in W . In other words, the incidence matrix Fn of B is defined by those edges from B that have their source and range in vertices from Wn and Wn+1, respectively. We use the following notation for an E-invariant measure µ on XB and n ≥ 1 and v ∈ Vn: • X (n) v ⊂ XB denotes the set of all paths that go through the vertex v; • h (n) v denotes the cardinality of the set of all finite paths (cylinder sets) between v0 and v; • p (n) v denotes the µ-measure of the cylinder set e(v0, v) corresponding to a finite path between v0 and v (since µ is E-invariant, the value p (n) v does not depend on e(v0, v)). If B is a subdiagram defined by a sequence W = (Wn), then we use the notation X (n) w and h (n) w to denote the corresponding objects of the subdiagram B. Take a subdiagram B and consider the set XB of all infinite paths whose edges belong to B. Let X̂B := E(XB) be the subset of the paths in XB that are tail equivalent to the paths from XB. In other words, the E-invariant subset X̂B of XB is the saturation of XB with respect to the equivalence relation E (or XB is a countable complete section of E on X̂B). Let µ be a probability measure on XB invariant with respect to the tail equivalence relation defined on B. Then µ can be canonically extended to the measure µ̂ on the space X̂B by the invariance with respect to E [3]. If we want to extend µ̂ to the whole space XB, we can set µ̂(XB \ X̂B) = 0. Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 5 S. Bezuglyi, O. Karpel, and J. Kwiatkowski Specifically, take a finite path e ∈ E(v0, w) from the top vertex v0 to a vertex w ∈ Wn that belongs to the subdiagram B. Let [e] denote the cylinder subset of XB determined by e. For any finite path s ∈ E(v0, w) from the diagram B with the same range w, we set µ̂([s]) = µ([e]). In such a way, the measure µ̂ is defined on the σ-algebra of Borel subsets of X̂B generated by all clopen sets of the form [z] where a finite path z has the range in a vertex from B. Clearly, the restriction of µ̂ on XB coincides with µ. We note that the value µ̂(X̂B) can be either finite or infinite depending on the structure of B and B (see below Theorems 2.1 and 2.2). Furthermore, the support of µ̂ is, by the definition, the set X̂B. Set X̂ (n) B = {x = (xi) ∈ X̂B : r(xi) ∈ Wi, ∀i ≥ n}. (1.1) Then X̂ (n) B ⊂ X̂ (n+1) B and µ̂(X̂B) = lim n→∞ µ̂(X̂(n) B ) = lim n→∞ ∑ w∈Wn h(n) w p(n) w . (1.2) 2. Characterization of Subdiagrams Supporting a Measure Given a Bratteli diagram B, we consider the incidence matrix Fn = (f (n) v,w), v ∈ Vn+1, w ∈ Vn and set An = F T n , the transpose of Fn. Together with the sequence of incidence matrices (Fn), we consider the sequence of stochastic matrices (Qn) whose entries are q(n) v,w = f (n) v,w h (n) w h (n+1) v , v ∈ Vn+1, w ∈ Vn. The following result was obtained in [3, Proposition 6.1] for Bratteli diagrams of finite rank. We note here that this result remains true for arbitrary Bratteli diagrams, the proof is the same as in [3]. Theorem 2.1. Let B be a Bratteli diagram with incidence stochastic matrices {Qn = (q(n) v,w)} and let B be a proper vertex subdiagram of B defined by a sequence of subsets (Wn) where Wn ⊂ Vn. (1) Let µ be a probability invariant measure on the path space XB such that the extension µ̂ of µ on X̂B is finite. Then ∞∑ n=1 ∑ w∈Wn+1 ∑ v∈W ′ n q(n) w,vµ(X(n+1) w ) < ∞. (2.1) 6 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures (2) If ∞∑ n=1 ∑ w∈Wn+1 ∑ v∈W ′ n q(n) w,v < ∞, (2.2) then any probability invariant measure µ defined on the path space XB of the subdiagram B extends to a finite measure µ̂ on X̂B. The example below shows that in general case the sufficient condition (2.2) is not necessary and the necessary condition (2.1) is not sufficient. E x a m p l e. (1) First, we give an example of an infinite measure µ̂ on a Bratteli diagram B such that µ̂ is an extension of a probability measure µ from a subdiagram B(W ) and condition (2.1) is satisfied. Let B be a stationary Bratteli diagram with the incidence matrix F =   3 0 0 1 2 0 0 1 3   . Suppose the sequence (Wn) is stationary and formed by the second and third vertices of each level. Then (W ′ n) is formed by the first vertex. Since q3,1 = 0, we have ∞∑ n=1 ∑ v∈Wn+1 ∑ w∈W ′ n q(n) v,wµ(X(n+1) v ) = ∞∑ n=1 q2,1µ(X(n+1) 2 ). Compute q2,1 = h (n) 1 h (n+1) 2 = 3n−1 2n + ∑n−1 k=0 2k3n−1−k = 3n−1 2n + (3n − 2n) = 1 3 . It is easy to see that µ(X(n+1) 2 ) = 2n−1 3n . Then q2,1µ(X(n+1) 2 ) = 2n−1 3n+1 , and thus condition (2.1) is satisfied. On the other hand, we know that the extension µ̂ is an infinite measure because the Perron–Frobenius eigenvalue of the incidence matrix of B is 3, the same as for the odometer corresponding to the first vertex (see [1]). (2) For any stationary Bratteli diagram, the sufficient condition (2.2) is never satisfied. Thus, to show that (2.2) is not necessary, we can consider any stationary Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 7 S. Bezuglyi, O. Karpel, and J. Kwiatkowski diagram with the finite full measure µ̂. For instance, one can take a diagram with the incidence matrix F =   2 0 0 1 2 0 0 1 3   and µ being the measure on the subdiagram B defined as in (1). In contrast to Theorem 2.1, the following result gives a necessary and sufficient condition for the finiteness of a measure extension. Theorem 2.2. Let B, B, Qn,Wn be as in Theorem 2.1 and µ be a probability measure on the path space of the vertex subdiagram B. The measure extension µ̂(X̂B) is finite if and only if ∞∑ n=1 ∑ w∈Wn+1 µ̂(X(n+1) w ) ∑ v∈W ′ n q(n) w,v < ∞ (2.3) or ∞∑ i=1   ∑ w∈Wi+1 h(i+1) w p(i+1) w − ∑ w∈Wi h(i) w p(i) w   < ∞. (2.4) P r o o f. Indeed, let X̂ (n) B be defined as in (1.1). Then µ̂(X̂B)=limn→∞ µ̂(X̂(n) B ). Since X̂ (n) B = X̂ (1) B ∪ (X̂(2) B \ X̂ (1) B ) ∪ · · · ∪ (X̂(n) B \ X̂ (n−1) B ), we obtain µ̂(X̂(n) B ) = 1 + n−1∑ i=1   ∑ w∈Wi+1 h(i+1) w p(i+1) w − ∑ w∈Wi h(i) w p(i) w   . This relation proves (2.4). We remark that condition (2.4) is formulated by using the vertices related only to the subdiagram B. On the other hand, X̂ (n+1) B \ X̂ (n) B = {x = (xi) ∈ X̂B : r(xn) ∈ W ′ n, r(xi) ∈ Wi, i ≥ n + 1}. and therefore µ̂(X̂(n+1) B \ X̂ (n) B ) = ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p(n+1) w = ∑ w∈Wn+1 ∑ v∈W ′ n q(n) w,vh (n+1) w p(n+1) w = ∑ w∈Wn+1 µ̂(X(n+1) w ) ∑ v∈W ′ n q(n) w,v. 8 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures Thus, µ̂(X̂B) = 1 + ∞∑ n=1 ∑ w∈Wn+1 µ̂(X(n+1) w ) ∑ v∈W ′ n q(n) w,v. R e m a r k. In Theorem 2.1, we found the necessary condition (2.1) and the sufficient condition (2.2) for the extension µ̂(X̂B) to be finite. We can show that these conditions, in fact, follow from Theorem 2.2. Indeed, we have q(n) w,vµ̂(X(n+1) w ) ≥ q(n) w,vµ(X(n+1) w ). Hence, if µ̂(X̂B) < ∞, then condition (2.1) holds by Theorem 2.2. To obtain that (2.2) is a sufficient condition for the extension µ̂(X̂B) to be finite, it suffices to show that there exists M > 0 such that f (n) w,vh (n) v p(n+1) w ≤ Mq(n) w,v = M f (n) w,vh (n) v h (n+1) w . We need to show that p(n+1) w h(n+1) w = p(n+1) w h (n+1) w h (n+1) w h (n+1) w ≤ M for some M > 0. Note that in the proof of Proposition 6.1 in [3] (see also the proof of Proposition 2.3 below), it was shown that there exists M such that h (n+1) w h (n+1) w ≤ M (the proof in [3] was given for the case of Bratteli diagrams of finite rank, but it is easy to see that the same proof works for a general case). Since p (n+1) w h (n+1) w < 1, the same constant M can be used to prove (2.2). Here is another sufficient condition for µ̂(X̂B) to be finite. Proposition 2.3. Let B be a vertex subdiagram of B with a probability mea- sure µ on XB. If we suppose that I = ∞∑ n=1 max w∈Wn+1   ∑ v∈W ′ n q(n) w,v   < ∞, then µ̂(X̂B) is finite. P r o o f. It suffices to show that I < ∞ implies S < ∞ where S = ∞∑ n=1 ∑ w∈Wn+1 µ̂(X(n+1) w ) ∑ v∈W ′ n q(n) w,v Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 9 S. Bezuglyi, O. Karpel, and J. Kwiatkowski is defined in (2.3). Notice that S = ∞∑ n=1 ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p(n+1) w . We have ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p(n+1) w = ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p (n+1) w h (n+1) w h (n+1) w = ∑ w∈Wn+1 µ(X(n+1) w ) ∑ v∈W ′ n f (n) w,v h (n) v h (n+1) w ≤ max w∈Wn+1   ∑ v∈W ′ n f (n) w,v h (n) v h (n+1) w   ∑ w∈Wn+1 µ(X(n+1) w ). Since µ is a probability measure on XB, we obtain that ∑ w∈Wn+1 µ(X(n+1) w ) = 1. We show that there exists M > 0 such that h (n) w h (n) w < M for all w ∈ Wn and all sufficiently large n. Indeed, set Mn = max w∈Wn h (n) w h (n) w . Fix any v ∈ Wn+1. Then h (n+1) w h (n+1) w = 1 h (n+1) w   ∑ v∈Wn f (n) w,vh (n) v + ∑ v∈W ′ n f (n) w,vh (n) v   ≤ Mn h (n+1) w ∑ v∈Wn f (n) w,vh (n) v + 1 h (n+1) w ∑ v∈W ′ n f (n) w,vh (n) v = Mn + h (n+1) w h (n+1) w ∑ v∈W ′ n f (n) w,v h (n) v h (n+1) w = Mn + h (n+1) w h (n+1) w ∑ v∈W ′ n q(n) w,v ≤ Mn + h (n+1) w h (n+1) w εn, 10 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures where εn = max w∈Wn+1   ∑ v∈W ′ n q(n) w,v   . Since I < ∞, the value of εn tends to zero as n tends to ∞. From the above inequalities we obtain h (n+1) w h (n+1) w (1− εn) ≤ Mn and Mn+1 ≤ Mn 1− εn . Finally, Mn ≤ M1∏∞ k=1(1− εn) = M. Since I < ∞, we get that M is well-defined. Thus, ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p(n+1) w < M max w∈Wn+1   ∑ v∈W ′ n f (n) w,v h (n) v h (n+1) w   = M max w∈Wn+1   ∑ v∈W ′ n q(n) w,v   . E x a m p l e. In this example, we consider a class of Bratteli diagrams B whose incidence matrices Fn of size |Vn+1| × |Vn| can be written as follows: Fn =   an 1 . . . 1 1 1 . . . 1 ... ... . . . ... 1 1 . . . 1   , n ≥ 1, where an ≥ 2 for n ≥ 1. Consider the subdiagram B defined by the leftmost vertex at each level such that Wn consists of a single vertex and Fn = (an). Let µ be the unique probability invariant measure on XB. Then p(n) w = 1 a0 · · · an−1 for w ∈ Wn. Recall that, by Theorem 2.2, µ̂(X̂B) is finite if and only if the series S = ∞∑ n=1 ∑ w∈Wn+1 ∑ v∈W ′ n f (n) w,vh (n) v p(n+1) w Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 11 S. Bezuglyi, O. Karpel, and J. Kwiatkowski converges. Thus, ( µ̂(X̂B) < ∞ ) ⇐⇒  S = ∞∑ n=1 1 a0 · · · an ∑ v∈W ′ n h(n) v < ∞   . For any v ∈ W ′ n, the height h (n) v is independent of v; denote it by h (n) 0 . Hence, S = ∞∑ n=1 |Vn| − 1 a0 · · · an h (n) 0 . We formulate below some conditions for the convergence and divergence of the series S. (i) We observe that h (n) 0 does not depend on an−1 and an. Hence, for any values of the parameters {ai}n−2 i=1 and {|Vi|}n i=1 we can choose an−1an large enough to guarantee that the series S converges. (ii) Suppose for simplicity that a0 = 1. Then we have h(1) = (1, . . . , 1)T and h (n+1) 0 = h (n) 0 (|Vn| − 1) + h (n) 1 . Since an > 1, we obtain h (n+1) 0 ≥ h (n) 0 |Vn|. Thus, for every n, the inequality h (n) 0 ≥ |V1| · · · |Vn−1| holds. Therefore, S ≥ ∞∑ n=1 |V1| · · · |Vn−1|(|Vn| − 1) a0 · · · an . (2.5) If ∞∑ n=1 |V1| · · · |Vn−1|(|Vn| − 1) a0 · · · an = ∞, then µ̂(X̂B) = ∞. In other words, we can see that if the number of vertices in (W ′ n) is sufficiently large, then the extension of the measure µ is infinite. (iii) Denote bn = |Vn| − 1 a0 · · · an h (n) 0 , 12 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures then bn+1 bn = (|Vn+1| − 1)(|Vn| − 1 + h (n) 1 ) an+1(|Vn| − 1) . If (|Vn| − 1)a−1 n ≥ 1 for all sufficiently large n, then the series S diverges and the measure extension is infinite. (iv) It is obvious that h (n) 0 ≥ a0 . . . an−2. Hence, S ≥ ∞∑ n=1 |Vn| − 1 an−1an . Thus, ( ∞∑ n=1 (|Vn| − 1)(an−1an)−1 = ∞ ) =⇒ ( µ̂(X̂B) = ∞ ) . Denote S1 = ∞∑ n=1 |V1| . . . |Vn−1|(|Vn| − 1) a0 . . . an and S2 = ∞∑ n=1 |Vn| − 1 an−1an . We have seen that S1 = ∞ implies that µ̂ is infinite and S2 = ∞ implies the same. Let us compare the series S1 and S2 to find out whether their convergence implies the finiteness of the measure µ̂. Suppose |Vn| = n2 and an = n2. Then S2 = ∞∑ n=1 ( 1 n(n− 1) + 1 n2(n− 1) ) converges, but S1 still diverges (the general term is n2−1 n2 ) and the measure µ̂ is infinite. Thus, S2 cannot provide us with a necessary and sufficient condition for the finiteness of µ̂. (v) On the other hand, it follows from h (n+1) 1 = h (n) 0 (|Vn| − 1) + anh (n) 1 ≤ h (n) 1 (an + |Vn| − 1) that h (n) 0 ≤ h (n) 1 ≤ (an + |Vn| − 1) . . . (an−1 + |Vn−1| − 1) for every n. Then we have S ≤ ∞∑ n=1 (a1 + |V1| − 1) . . . (an−1 + |Vn−1| − 1)(|Vn| − 1) a0 . . . an = ∞∑ n=1 |Vn| − 1 an ( 1 + |V1| − 1 a1 ) . . . ( 1 + |Vn−1| − 1 an−1 ) . Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 13 S. Bezuglyi, O. Karpel, and J. Kwiatkowski Hence µ̂(X̂B) < ∞ if the sequence (|Vn| − 1)a−1 n tends to zero fast enough such that the above series converges. Therefore, if (|Vn| − 1)a−1 n ≥ 1 for all sufficiently large n, then the mea- sure extension is infinite; if the measure extension is finite, then the sequence (|Vn| − 1)a−1 n tends to zero fast enough. To simplify the formulation of the next statement, we assume that fw,v > 0 for every w ∈ Wn+1, v ∈ W ′ n and n > 0, i.e., for every w ∈ Wn+1 there is an edge to some vertex from W ′ n. This assumption is not restrictive since one can use the telescoping procedure to ensure the positivity of all entries of F . Corollary 2.4. Let B, B, Qn,Wn be as in Theorem 2.1 and µ be a probability measure on the path space of the vertex subdiagram B. Let the measure extension µ̂(X̂B) be finite. Then ∞∑ n=1 min w∈Wn+1 max v∈W ′ n q(n) w,v < ∞. In particular, ∞∑ n=1 min w∈Wn+1 max v∈W ′ n h (n) v h (n+1) w < ∞. (2.6) P r o o f. By Theorem 2.2, we have µ̂(X̂B) = 1 + ∞∑ n=1 ∑ w∈Wn+1 µ̂(X(n+1) w ) ∑ v∈W ′ n q(n) w,v ≥ 1 + ∞∑ n=1 ∑ w∈Wn+1 µ̂(X(n+1) w ) max v∈W ′ n q(n) w,v ≥ 1 + ∞∑ n=1 min w∈Wn+1 max v∈W ′ n q(n) w,v ∑ w∈Wn+1 µ̂(X(n+1) w ). Since ∑ w∈Wn+1 µ̂(X(n+1) w ) → µ̂(X̂B) > 0, there is a constant C > 0 such that ∑ w∈Wn+1 µ̂(X(n+1) w ) > C for all n. Hence we obtain ∞∑ n=1 min w∈Wn+1 max v∈W ′ n q(n) w,v < ∞. Since fw,v > 0 for every w ∈ Wn+1, v ∈ W ′ n and n > 0, then there follows relation (2.6). 14 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures R e m a r k. Let B be a stationary Bratteli diagram. If B is simple, then there is a unique ergodic invariant measure ν on XB. Suppose that λ is the Perron– Frobenius eigenvalue for the incidence matrix of B. Then all the heights h (n) v of B grow as λn. Thus, for any choice of Wn ⊂ Vn, for any v ∈ W ′ n and w ∈ Wn+1, the ratio h (n) v h (n+1) w will tend to 1 λ as n tends to infinity. Hence, by Corollary 2.4, there is no proper subdiagram B such that ν could be the extension of an invariant ergodic measure from XB. In the case of a non-simple stationary diagram B, the minimal support of an ergodic invariant measure is some simple stationary subdiagram B(W ) whose incidence matrix F has the Perron–Frobenius eigenvalue λ. Then, for every w ∈ Wn, the height h (n) w grows again as λ n, but for every v ∈ W ′ n, the height h (n) v grows as δn, where δ < λ (see [1]). We recall that for a finite rank Bratteli diagram the support of any probabil- ity measure µ is determined by a vertex subdiagram B(W ),W = (Wn), whose vertices v satisfy the condition: there exists some δ > 0 such that µ(X(n) v ) > δ for all sufficiently large n and all v ∈ Wn (see [3]). In particular, a Bratteli dia- gram B is of exact finite rank if the condition µ(X(n) v ) > δ holds for all vertices v ∈ Vn. Clearly, the above result cannot be true for general Bratteli diagrams. Nevertheless, we can find another characterization for vertices that belong to the support of a probability measure by studying how the measure of towers X (n) v changes when v is in the subdiagram and when it is not in the subdiagram. R e m a r k. Let µ̂ be the extension of the measure µ defined on an exact finite rank subdiagram B of a Bratteli diagram B. Suppose that µ̂(X̂B) < ∞. Then we have max v∈W ′ n µ̂(X(n) v ) ≤ ∑ v∈W ′ n µ̂(X(n) v ) = µ̂(X̂B)− ∑ w∈Wn µ̂(X(n) w ) → 0 as n →∞. Since the measure of any tower X (n) w is bounded from zero, it follows that lim n→∞ max v∈W ′ n µ̂(X(n) v ) min w∈Wn µ(X(n) w ) = 0, (2.7) and therefore lim n→∞ max v∈W ′ n µ̂(X(n) v ) min w∈Wn µ̂(X(n) w ) = 0. (2.8) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 15 S. Bezuglyi, O. Karpel, and J. Kwiatkowski It is very plausible that (2.8) is true for any uniquely ergodic Bratteli subdi- agram B. However, this is still under the question. On the other hand, we are able to prove the following result. Proposition 2.5. Let B be a Bratteli diagram with the incidence matrices Fn = {(f (n) v,w)}. Let B = B(W ) be a proper vertex subdiagram of B such that |Wn| |V \Wn| ≤ C for every n and some constant C > 0. Suppose µ̂ is a finite invariant measure on the path space XB which is obtained as the extension of a probability measure µ defined on XB. Then lim n→∞ min w∈W ′ n µ̂(X(n) v ) max w∈Wn µ(X(n) w ) = 0. (2.9) P r o o f. Let W ′ n = Vn \Wn. For every n, we have µ̂(X̂B) = ∑ v∈W ′ n µ̂(X(n) v ) + ∑ w∈Wn µ̂(X(n) w ) ≥ |W ′ n| min v∈W ′ n µ̂(X(n) v ) + ∑ w∈Wn µ̂(X(n) w ). Thus, min v∈W ′ n µ̂(X(n) v ) ≤ µ̂(XB)−∑ w∈Wn µ̂(X(n) w ) |W ′ n| . Since µ(XB) = 1, we obtain max w∈Wn µ(X(n) w ) ≥ 1 |Wn| . Hence, min w∈W ′ n µ̂(X(n) v ) max w∈Wn µ(X(n) w ) ≤ |Wn|(µ̂(XB)−∑ w∈Wn µ̂(X(n) w )) |W ′ n| . Notice that µ̂(XB)− ∑ w∈Wn µ̂(X(n) w ) → 0 as n →∞. This proves that equality (2.9) holds. 16 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 Subdiagrams of Bratteli Diagrams Supporting Finite Invariant Measures R e m a r k. Since µ̂(X(n) w ) ≥ µ(X(n) w ) for every w ∈ Wn and every n, we obtain the following simple corollary of the proved result lim n→∞ min v∈W ′ n µ̂(X(n) v ) max w∈Wn µ̂(X(n) w ) = 0. Acknowledgment. This article was finished when the first named author was a visitor of the Department of Mathematics, University of Iowa. He is thank- ful to the department for hospitality and support. References [1] S. Bezuglyi, J. Kwiatkowski, K. Medynets, and B. Solomyak, Invariant Measures on Stationary Bratteli Diagrams. — Ergodic Theory Dynam. Syst. 30 (2013), 973– 1007. [2] S. Bezuglyi and O. Karpel, Homeomorphic Measures on Stationary Bratteli Dia- grams. — J. Funct. Anal. 261 (2011), 3519–3548. [3] S. Bezuglyi, J. Kwiatkowski, K. Medynets, and B. Solomyak, Finite Rank Bratteli Diagrams: Structure of Invariant Measures. — Trans. Amer. Math. Soc. 365 (2013), 2637–267. [4] S. Bezuglyi, J. Kwiatkowski, and R. Yassawi, Perfect Orderings on Finite Rank Bratteli Diagrams. — Canad. J. Math. 66 (2014), 57–101. [5] S. Bezuglyi and R. Yassawi, Perfect Orderings on General Bratteli Diagrams. Preprint, 2013. [6] F. Durand, Combinatorics on Bratteli Diagrams and Dynamical Systems. In: Com- binatorics, Automata and Number Theory, V. Berthé and M. Rigo (eds.), Encyclo- pedia of Mathematics and its Applications 135, Cambridge University Press (2010), 338–386. [7] T. Giordano, I. Putnam, and C. Skau, Topological Orbit Equivalence and C∗- Crossed Products. — J. Reine Angew. Math. 469 (1995), 51–111. [8] R.H. Herman, I. Putnam, and C. Skau, Ordered Bratteli Diagrams, Dimension Groups, and Topological Dynamics. — Int. J. Math. 3(6) (1992), 827–864. [9] K. Medynets, Cantor Aperiodic Systems and Bratteli Diagrams. C. R., Acad. Sci. 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