On scaling entropy sequence of dynamical system

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1 O scalig etropy sequece of dyamical system P. B. Zatitskiy arxiv: v1 [math.ds] 22 Sep 2014 November 6, 2017 Abstract We preset a series of statemets about scalig etropy, a metric ivariat of dyamical systems proposed by A. M. Vershik i the late 90 s (see [9], [7], ad [10]). We show that it is a metric ivariat ideed ad calculate it i several special cases. All measure spaces cosidered below are stadard probability spaces (Lebesgue Rokhli spaces). Probability spaces with cotiuous measures are of major iterest of this article, however all the defiitios deal with geeral Lebesgue spaces (i.e. its measures may have atoms). We omit the sigma-algebra otatio whe we talk about probability spaces. 1 Defiitio ad properties of scalig etropy sequece Recall the defiitio of admissible semimetric ad its ε-etropy (see, e.g., [12], [11]): Defiitio 1. Semimetric ρ o a stadard probability space (X, µ) is called measurable if it is measurable o (X 2,µ 2 ) as a fuctio of two variables, ad admissible if it is separable o some subset X 1 X of full measure. I this latter case we call the triple (X,µ,ρ) a admissible semimetric triple. The followig defiitio goes back to A. N. Kolmogorov. Defiitio 2. The fuctio ε H ε (X,µ,ρ) is called a epsilo etropy of a semimetric triple (X,µ,ρ) if it is defied for ε > 0 as follows. Let k N be the least atural umber, such that the space X ca be partitioed ito disjoit measurable subsets X 0,X 1,...,X k, such that µ(x 0 ) < ε ad for ay j = 1,...,k the diameter of the set X j i semimetric ρ is less tha ε. Defie the ε-etropy of the triple (X,µ,ρ) as H ε (X,µ,ρ) = log 2 k. If oe ca ot fid such a umber k, the we defie H ε (X,µ,ρ) = +. Supported by Chebyshev Laboratory (SPbSU), RF Govermet grat 11.G , by JSC Gazprom Neft, by the RFBR grats ofi_m2 ad _A, by Presidet of Russia grat for youg researchers MK The author ackowledge Sait-Petersburg State Uiversity for a research grat

2 Recall that a measurable semimetric ρ o a stadard probability space(x, µ) is admissible if ad oly if H ε (X,µ,ρ) < + for ay ε > 0 (see [12] for details). I what follows we assume T to be a ergodic automorphism of a stadard probability space (X,µ). Let ρ be a measurable semimetric o (X,µ). Clearly, the shifts T k ρ(x,y) = ρ(t k x,t k y) are also measurable semimetrics o (X,µ), moreover, for ay ε > 0 oe has H ε (X,µ,ρ) = H ε (X,µ,T k ρ). Defie fiite averages of a semimetric ρ with respect to T by the formula Tavρ = 1 T k ρ. 1 k=0 The followig geeratig property is a importat dyamical property of semimetrics: Defiitio 3. We say that a measurable semimetric ρ o (X,µ) is (two-sided) geeratig for a metric dyamical system (X,µ,T) if its shifts T k ρ,k Z, separate poits mod zero, i.e. there exists some measurable subset X X of full measure, such that for ay two differet x,y X oe ca fid k Z, such that T k ρ(x,y) > 0. Note that ay admissible metric is a geeratig semimetric for ay automorphism T because it separates poits itself. Recall the defiitio give by A. M. Vershik i [9], [10], [7]. Defiitio 4. Let (X,µ,T) be a metric dyamical system ad ρ be a admissible semimetric o (X,µ). A sequece {h } of positive umbers is called a scalig sequece for the semimetric ρ, if for ay ε > 0 oe has lim sup ad for ε > 0 small eough oe has 0 < limif H ε (X,µ,T avρ) h < +, (1) H ε (X,µ,T av ρ) h. (2) We deote by H(X,µ,T,ρ) the class of all scalig sequeces for the semimetric ρ. Remark 1. If ρ is some admissible semimetric, h = {h } H(X,µ,T,ρ), ad h = {h } is a sequece of positive umbers, the h H(X,µ,T,ρ) if ad oly if 0 < limif h h limsup h h <. The followig theorem cofirms the hypothesis of Vershik [10] about idepedece of a scalig sequece from a semimetric. It was proved i [12] (see Theorem 4 below) for automorphisms with purely poit spectrum. Theorem 1. Let ρ, ρ Adm(X, µ) be two summable admissible geeratig semimetrics o (X,µ). The H(X,µ,T,ρ) = H(X,µ,T, ρ). Theorem 1 leads to the followig defiitio. 2

3 Defiitio 5. A sequece h = {h } of positive umbers is called a scalig etropy sequece of a metric dyamical system (X,µ,T) if h H(X,µ,T,ρ) for some (ad the for ay) summable admissible geeratig semimetric ρ Adm(X,µ). We will deote by H(X,µ,T) the class of all scalig etropy sequeces for the dyamical system H(X,µ,T). Corollary 1. The class of scalig etropy sequeces is a metric ivariat of ergodic dyamical systems. It is a iterestig questio whether a scalig etropy sequece exists for ay dyamical system (i.e. the set H(X,µ,T) is oempty). The class of scalig etropy sequeces is mootoe i the followig sese: Corollary 2. Suppose that a metric dyamical system (X 2,µ 2,T 2 ) is a factor of a system (X 1,µ 1,T 1 ), ad h 1 = {h 1 }, h2 = {h 2 } are correspodig scalig etropy sequeces. The h 2 = O(h1 ), +. 2 Examples 2.1 Relatio to Kolmogorov etropy The followig theorem clarifies the relatio betwee scalig etropy sequeces ad Kolmogorov etropy. Theorem 2. Let T be a ergodic automorphism of a stadard probability space (X, µ). The: 1) if Kolmogorov etropy h µ (T) is fiite ad positive, the the sequece h = is a scalig etropy sequece of a system (X,µ,T); 2) if h is a scalig etropy sequece of a system (X,µ,T), the h µ (T) = 0 if ad oly if h = o(), +. It is ot difficult to calculate scalig etropy sequeces of Beroulli shifts. Let (A,ν) be a stadard probability space, X = A Z be the space of two-sided sequeces, µ = ν Z be the product measure o X, ad T: X X be the left shift. Theorem 3. If the measure ν does ot have atoms, the the sequece h = is a scalig etropy sequece of the Beroulli shift (X,µ,T). If the measureν is cotiuous, the the etropy of the space(a,ν) is ifiite. I this case accordig to the Kolmogorov theorem (see [1, 2, 3]) Kolmogorov etropy h µ (T) is ifiite. Therefore the coverse of Theorem 2 first statemet does ot hold i geeral. 2.2 Automorphisms with pure poit spectrum The followig characterizatio of automorphisms with pure poit spectrum i terms of scalig etropy sequeces was proved i [12]. Theorem 4. Let T be a ergodic automorphism of a stadard probability space (X,µ). The the followig are equivalet: 3

4 1) the spectrum of the system (X,µ,T) is purely poit; 2) the set H(X,µ,T) cosists of all positive bouded separated from zero sequeces. We compare this result with aother criterio of the pure poitess of the dyamical system spectrum due to A. G. Kushireko (see [4]). Recall the defiitio of A-etropy (sequetial etropy) of a ergodic automorphism T of a stadard probability space (X,µ). Suppose that A = {a } is a icreasig sequece of atural umbers ad ξ is a measurable partitio of (X,µ) with fiite etropy H(ξ) < + (H stads for etropy of measurable partitio). Defie h A (T,ξ) = limsup H( i=1 Ta iξ) ad A-etropy h A (T) = sup ξ h A (T,ξ) of automorphism T, where supremum is take over all partitios with fiite etropy. I [4] Kushireko proved that a automorphism T has purely poit spectrum if ad oly if h A (T) = 0 for ay sequece A. We compare this result with Theorem 4 ad obtai that if h A (T) > 0, the the scalig etropy sequece icreases. The followig statemet gives a quatitative estimate of this remark. Theorem 5. If A = {a }, h A (T) > 0, ad h = {h } is the scalig etropy h sequece of a automorphism T, the limif a loglog > Substitutios of costat legth Now we calculate the scalig etropy sequeces of substitutioal dyamical systems of costat legth. We refer the reader to the moograph [8] for detailed iformatio cocerig substitutioal dyamical systems. Let ξ be a substitutio of legth q o a alphabet A (i.e. it is a map from A toa q ). We assume that for some symbolα A the first symbol of the wordξ(α) is α agai. I such a case oe ca fid a ifiite word u A N0 startig with α such that ξ(u) = u. Let T be the left shift o A N0, ad X ξ be the closure of the sequece T u, N, i the product topology. If the substitutio ξ is primitive, the the topological dyamical system (X ξ,t) is miimal ad uique ergodic with uique ivariat measure µ ξ. I such a maer we associate the metric dyamical system (X ξ,µ ξ,t) with the primitive substitutio ξ. It turs out that the scalig etropy sequece of this dyamical system ca be expressed i terms of two combiatorial parameters of the substitutio. First of them, the height h(ξ), ca be defied for example as the greatest umber k relatively prime to q such that if u = u 0 for some, the k. Secod, the colum umber c(ξ), is defied as mi{ {ξ (a) k : a A} : N,0 k < q } (ote that this defiitio differs a little from the classical oe give i [6] ad [8]). Theorem 6. Let ξ be a ijective primitive costat legth substitutio o a alphabet A, A > 1. The the sequece h = 1+(c(ξ) h(ξ))log is a scalig etropy sequece of the substitutioal dyamical system (X ξ,µ ξ,t). The idea of the proof of this theorem is to study ξ-ivariat semimetrics o a alphabet A. Theorems 6 ad 4 imply the followig corollary. Corollary 3. Let ξ be a ijective primitive costat legth substitutio o a alphabet A, A > 1. The the spectrum of the correspodig dyamical system (X ξ,µ ξ,t) is purely poit if ad oly if c(ξ) = h(ξ). 4

5 Note that the similar criterio was first obtaied i [5] ad [6], but it was formulated a little bit differet there. The differece is as follows. If the height h(ξ) is bigger tha oe, the c(ξ) is defied via some modified substitutio. The result of Corollary 3 coicides with their criterio for the case h(ξ) = 1: the spectrum is pure poit if ad oly if c(ξ) = 1. But if h(ξ) > 1, the criterio 3 (ulike the Kamae-Dekkig criterio) does ot require to cosider modified substitutios. The author thaks F. V. Petrov ad A. M. Vershik for a lot of discussios, advises ad help. Refereces [1] A. N. Kolmogorov, New Metric Ivariat of Trasitive Dyamical Systems ad Edomorphisms of Lebesgue Spaces, Doklady of Russia Academy of Scieces, 119, N5, , 1958 (I Russia). [2] A. N. Kolmogorov, Etropy per uit time as a metric ivariat of automorphism, Doklady of Russia Academy of Scieces, 124, , 1959 (I Russia). [3] Ya. G. Siai, O the Notio of Etropy of a Dyamical System, Doklady of Russia Academy of Scieces, 124, , 1959 (I Russia). [4] A. G. Kushireko, Metric ivariats of etropy type, UMN, 22:5(137), 57-65, 1967 (I Russia); Traslatio: Russia Math. Surveys, 22(5), 53 61, [5] Kamae, T. A topological ivariat of substitutio miimal sets., J. Math. Soc. Japa, 24 (1972), [6] Dekkig, F. M. The spectrum of dyamical systems arisig from substitutios of costat legth., Z. Wahrscheilichkeitstheorie ud Verw. Gebiete, 41 (1977/78), o. 3, [7] A. M. Vershik, A. D. Gorbul skii, Scaled etropy of filtratios of σ-fields, TVP, 52(3), , 2007 (I Russia); Traslatio: Theory Probab. Appl., 52:3, , [8] Queffélec, M. Substitutio dyamical systems spectral aalysis. Secod editio., Lecture Notes i Mathematics, Spriger-Verlag, Berli, xvi+351 pp. ISBN: [9] A. Vershik, Dyamics of metrics i measure spaces ad their asymptotic ivariats, Markov Processes ad Related Fields, 16:1, , [10] A. M. Vershik, Scalig etropy ad automorphisms with pure poit spectrum, Algebra i aaliz, 23:1, , 2011 (I Russia); Traslatio: St. Petersburg Math. J., 23:1, 75 91, [11] P. B. Zatitskiy, F. V. Petrov, Correctio of metrics, Zap. Nauch. Semi. POMI, 390, , 2011 (I Russia); Traslatio: J. Math. Sci., New York 181, No. 6, (2012). 5

6 [12] A. M. Vershik, P. B. Zatitskiy, F. V. Petrov, Geometry ad dyamics of admissible metrics i measure spaces, Cetral Europea Joural of Mathematics, 11 (3), , Chebyshev Laboratory, St. Petersburg State Uiversity, , Russia, Sait Petersburg, 14th Lie, 29b; PDMI RAS, , Russia, St. Petersburg, Fotaka, paxa239@yadex.ru 6

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