Gabriela I. Sebe

Journal de Théorie des Nombres de Bordeaux

JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUXGABRIELA I. SEBEA Gauss-Kuzmin theorem for the Rosen fractionsJournal de Théorie des Nombres de Bordeaux, tome 14, no 2 (2002),p. 667-682<http://www.numdam.org/item?id=JTNB_2002__14_2_667_0>© Université Bordeaux 1, 2002, tous droits réservés.L’accès aux archives de la revue « Journal de Théorie des Nombresde Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’uneinfraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/667-A Gauss-Kuzmin theoremfor the Rosen fractionspar GABRIELA I. SEBERÉSUMÉ. En utilisant les extensions naturelles des transforma-tions de Rosen, nous obtenons une représentation de la chaîned’ordre infini associée à la suite des quotients incomplets des frac-tions de Rosen. Associé au comportement ergodique d’un certainsystème aléatoire homogène à liaisons complètes, ce fait nous per-met de résoudre une version du problème de Gauss-Kuzmin pourle développement en fraction de Rosen.ABSTRACT. Using the natural extensions for the Rosen maps, wegive an infinite-order-chain representation of the sequence of theincomplete quotients of the Rosen fractions. Together with theergodic behaviour of a certain homogeneous random system withcomplete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.1. IntroductionThe Rosen fractions, introduced in [11], form an infinite family whichgeneralizes the nearest integer continued fractions. They are related to theso-called Hecke groups. There is an extended literature on these (see thepapers by Rosen and Schmidt [12], Gr6chenig and Haas [2], Schmidt [13],Haas and Series [3], and Lehner [7], [8]). It is only recently (see the papersby Burton, Kraaikamp and Schmidt [1] and Nakada [10]) that the ergodicproperties of these expansions have been studied. It should be stressedthat the ergodic theorem does not yield rates of convergence for mixingproperties; for this a Gauss-Kuzmin theorem is needed.The technique developped by Iosifescu in [5] for the case of the regularcontinued fraction expansion could be used for different types of continuedfractions. The aim of this paper is to take up the Gauss-Kuzmin problemfor the Rosen fractions in this manner.This paper is organized as follows. Using the natural extensions for theRosen maps, in Section 3 we give an infinite order-chain representationof the sequence of the incomplete quotients of the Rosen fractions. InSection 5 we show that the random systems with complete connectionsManuscrit regu le 20 d6cembre 2000.668(RSCCs ) associated with the Rosen continued fraction expansion are withcontraction and their transition operators are regular w.r.t. the Banachspace of Lipschitz functions. This leads further, in Section 6, to a solutionof a version of Gauss-Kuzmin problem for the Rosen fractions.Let A = Aq equal 2 cos 7r/q for q E {3, 4, ... }. Fix some such q &#x3E; 4 andlet Iq = [-A/2, A/2). Then the map Tq : Iq -3 Iq defined byis called a Rosen map. Puttingin case 0, and = 0, = 00 in case Tq -1 (x) = 0, andusing the Rosen map Tq, one easily sees that every x E ~Iq has a uniqueexpansionwhich is called the Rosen or A-expansion (ACF) of x.Now, related to Hecke (triangle) group Gq, q &#x3E; 4, we call x a Gq-irrationalif x has a Rosen expansion of infinite length. For Gq-irrationals, thereare restrictions on the set of admissible sequences of êi and ai. Theserestrictions are determined by the orbit of A/2 [cf. [11] and [1]).2. Natural extensions for the Rosen mapsConsider (see [1]) the so-called natural extension T for any Rosen intervalmaps, which is defined as follows: for any fixed q &#x3E; 4, let A = Aq, T(x) beTq (x) andwhere we have suppressed the dependence of a = a, and e = 61 on x. Alsonotice that T is a transformation which on the first coordinate is simplythe interval map while on the second coordinate is directly related to the"past" of the first coordinate.It has been shown in ~1~, that T is a bijective transformation of a domainn in R2 except for a set of Lebesgue measure zero. We consider two cases.2.1. Even indices. Fix q = 2p, with p &#x3E; 2, and let I be the interval Iq.Putting §j = -T~ a~2, with ~o = -A/2, we construct a partition of I by669considering the intervalsFurthermore, let Kj = [0, Lj], j E {I, ... p - 1 ~ and Kp = [0, R~, whereLj, 1  j  p - 1, and R satisfy the systemLet S2 = U1=1 Jk x Kk. One has that R = 1, and that T is bijectiveon Q except for a set of Lebesgue measure zero. In [1] it is shown that Tpreserves the probability measure v (abolutely continuous with respect tothe Lebesgue measure on S2) with density ~i~£, where C is a normalizingconstant. Actually, for q even, the constant C is given by2.2. Odd indices. Fix an odd q and recycle notation as above. Let Ibe the interval Iq and Oj = -T~.~/2, with §o = -A/2. Putting h = Q;3, weconstruct a partition of I by considering the intervals E 11, - - - 2h+2},whereLet Kj = [0, Lj], j 2~ + 1 }, and K2h+2 = [0, R], where Lj,1  j  2h + 1, and R satisfy the systemLet q = 2h + 3, with h &#x3E; 1 and S1 = x Kj. One has that Rsatisfies the equation670and that T is bijective on f2 except for a set of Lebesgue measure zero. Inpaxticulax, A  R  1. The transformation T preserves the probabilitymeasure v with density ( 1 + ~ y) -~, where3. An infinite-order-chain representationIn this section we obtain an infinite-order-chain representation of thesequence of the incomplete quotients of the Rosen fractions. We treatsimultaneously the two subfamilies of Rosen maps, those of odd indicesand those of even one.Let us consider the setsand defineSimilarly to relations (2), for all n E N*, let us definewhich impliesWriting [~i~i,~2~2?"’ ? enzn] for the finite continued fractionfor all n E N* we have671Now, define the random variables an, n E Z = ~... , 20132, 20131,0,1,2,... }on Q bywhere TO denotes the identity map and lii (z, y) = al(x). Clearly, on f2,Since T preserves v, the doubly infinite sequence (an)nEZ is strictly sta-tionary under v.Let us put = for all n E Z and(x, y) Clearly,Projecting v, we obtain the invariant measure vx on ~- 2 , 2 ~ and vy on[0, R]. Since the invariant density h(x, y) = on St has projectionshx(t) = f h(t,y)dy and hy(t) = f h(x,t)dx on ~-2, 2~ and ~O,R), respec-tively, we have -where and denote the collection of Borel sets on [-, ] and2 22J 2 20, RJ, respectively.Now, we are able to prove the following theorem which is very importantin the sequel.Theorem 1. For any x E I, we havecontinued fraction with incomplete quotients ao, a_1, ... ).Proof. As is well known672Let us denote by In the fundamental intervalfor any arbitrarily fixed values of the ti and ai, i = 0, -1, ... , -n. Thenwe havefor some vn E In. Sincethe proof is complete.Corollary. For any i E Z* we havewhere a = [fofo, ] Proof. For i = -1 we haveHence the conditional probability above equals v-a.s. toFori=lwehaveHence the conditional probability above equals v-a.s. to673Finally, the proof for i E Z* B {-1,1} is analogous. 0Remarks. (i) The strict stationarity of (an)nEZ under v implies that theconditional probabilitydoes not depend on n and is v-a.s. equal to pi (a), with(ii) The process (lnän)nEZ is called an infinite- order- chain in the theory ofdependence with complete connections (see [6], Section 5.5).(iii) Many standard formulas for continued fractions (see [9]), hold for theRosen fractions. Lettingwhere -i and ai depend on x E I, we find thatThereforeIt follows that for any Gq-irrational number x E I and any positive integern we haveLet us put sn = E N*. Note that the equation qn = anÀqn-l +enqn-2 implies "with so = 0, i.e. sn = [an, ê2al]; clearly, Sn E [0, RJ.Motivated by Theorem 1, we shall consider the family of probabilitymeasures (Va)aE[O,R] on BI defined by their distribution functionsIn particular v° is the Lebesgue measure on I, if A = 1. For any a E [0, R]put Sô = a and674Let us consider the quadrupleBy the corollary to Theorem 1, the sequence is an W-valuedMarkov chain on which starts at so = a and has the followingtransition mechanism: from state s E W the only possible transitions arethose to states (l, i) E X B {(-1,1)}, and to ~19 or to R if (l, i) =(-1,1) and s E ~0, ~’ R 1~ or s E ~~ R 1,R~, respectively, the transitionprobability being Let B(W) be the Banach space of bounded measurable complex-valuedfunctions f on W under the supremum norm III = Clearly,the transition operator of transforms f E B(W) into the functiondefined bywhatever a E W.Note that for a E W, A = ~ (an+1, ... ) and n E N*,This follows from Theorem 1 for all irrational a E W and by continuityfor all rational a E W . In particular, it follows that the Brod6n-Borel-L6vyformula holds under va for any a E W , that isforxEl, nEN*.4. The Gauss-Kuzmin type equationLet p be an arbitrary non-atomic probability on BI and define675because TO is the identity map. Sincewe can write the Gauss-Kuzmin type equation asAssuming that for some m E N the derivative F§£ exists everywhere inI and is bounded, it is easy to see by induction that exists and isbounded for all n E N*. Differentiating the Gauss-Kuzmin type equationwe arrive atWe consider two cases.4.1. Even indices. Let h(x, y) = 1+xy~ be the invariant density onwith p &#x3E; 2 (see Subsection 2.1). For} and Kj = [0, Lj] we obtainand, forThus we havewhereFurther, writeto get676for n &#x3E; m. Clearly, the above equation reduces towith V being the linear operator defined on B(I) bywhereNote that V is the transition operator of the homogeneous Markov chaindefined asand yo = 0. Clearly, (yn )n&#x3E;o satisfies the recursion equationand yn E I for all indices n. Let us consider the quadruplewhere v : I x X - I is given by v (x, (l, i)) = vii (x) and Q : I x X - [0, 1]is given by Q (x, (l, i)) = Let us prove that 1:(I,i),Cx qli(x) = 1, for all x E I. First, we show that677for all (1, i) E X and, Hx = 1 on [0, ~), we obtainSince it follows thatHence the right-hand side ofthe above equation equalsNext, to prove that we note thatHenceFor the other cases, similar proofs hold, and we leave them to the reader.Thus we have shown that the sequence is an I-valued Markovchain on which start at yo - 0 and has the following transition678mechanism: from state s E I, the only possible transitions are those tostates m, (l, i) E X B {(-1,1), (1,1)~, and to 2 or to ’ if l = ~1,i = 1 and s E ~- 2 , ~ - A) or s E ( ~ - a, 2 ) , respectively, the transitionprobability being qls(s).4.2. Odd indices. We have the same goal as in the previous subsection.Let q = 2h + 3, with h &#x3E; 1, and recycle notation as above. Since h(x, y) =c is the invariant density on fl = Kj (see Subsectionj 2.2), it follows that for t E E { 1, ... , 2 h + 1} and Kj = [0, while for t E J2h+2 = [0, 2 ~ and K2h+2 = [0, RI, we getThuswhereFurther, write f n (x) - EC7Fn(x), x E I, to get m,where V is the linear operator defined on bywith qli and vlz, (1, i) E X defined as in the previous subsection. As in theeven case, we may prove that qii(x) = 1 on I. Also, it appears thatV is the transition operator of the homogeneous Markov chain on-5. Characteristic properties of the transition operators U and VHere we restrict our attention to the transition operators U and V in-troduced in Sections 3 and 4. In this section we deal with the properties ofU and V on function spaces different from B(W) and B (I ) .In connection with the operators U and V, we note the following prop-erties. First, if we define679andthen we have U’U’f = U°° f for all f E B(W) and V’V’f = V°° ffor all f E B(I) and n E N*. Second, let L(W) (respectively L(I)) bethe Banach space of all bounded complex-valued Lipschitz functions on W(respectively I ) under the usual norm IIIIIL = I f + s ( f ), whereThen U (respectively V) sends boundedly L(W) (respectively L(I)) intoitself. Moreover, we have the following results.Theorem 2. The random system with complete connections (RSCC)associated with the aCF-expansion is with contraction and its transitionoperator U is regular w.r.t L(W).Proof. We have for all (l, i) E XHence the requirements of the definition of an RSCC with contractionare met with k = 1 (see Definition 3.1.15 in [6]). By Theorem 3.1.16in [6], it follows that the Markov chain (s:)n&#x3E;O associated with the RSCC{(W,W), (X, X), u, P} is a Doeblin-Fortet chain. Hence by Definition 3.2.1in [6], the Markov chain (sn)n&#x3E;o is compact, because its state space is acompact metric space (W, d) _ (~0, R~, d), with d(x, y) _ ~x - y I, dx, y E Wand its transition operator is a Doeblin-Fortet operator.To prove the regularity of U w.r.t. L(W), let us define recursively wn+1 =(w" + 2)-1, n E N, with wo = w. A criterion of regularity is expressedin Theorem 3.2.13 in [6], in terms of the supports En(w) of the n-steptransition probability functions P~(w, ~), n E N* where, with the usualnotation,Clearly Therefore, Lemma 3.2.14 in [6] and an induc-tion argument lead to the conclusion that wn E E N*. Butlimn,,, wn = V2- - 1 for any w E W . Consequently680Now, the regularity of U w.r.t. L(W) follows from Theorem 3.2.13 in [6]and the proof is complete. 0Theorem 3. The random system with complete connections (RSCC)is with contraction and its translation operator V is regular w.r.t. L(I).Proof. The proof parallels that one of Theorem 2. We have for all (l, i) E XWe deduce that the Markov chain associated with the RSCC(X, X), v, Q} is a Doeblin-Fortet chain. Moreover, the Markovchain is compact and its transition operator is a Doeblin-Fortetoperator.To prove the regularity of V w.r.t. L(I) we proceed as in the precedingproof. Here the transition probability function isA similar argument leads to the regularity of V. 0Remark. Theorem l’ in [4] shows that there exist positive constantsKl, K2, ,~  1 and 0  1 such that6. A Gauss-Kuzmin type problemThe results obtained allow to a solution of a Gauss-Kuzmin type problem.The solution presented here is based on the ergodic behaviour of the RSCCassociated with the transition operator V.It should be mentioned that it is only very recently that there has beenany investigation of the metrical properties of the Rosen fractions (Nakada(in ~10~) started investigations of metrical properties of the Rosen fractions,but only with even q’s) . Thus, we may emphasize that our solution obtainedin an elementary manner is surprisingly simple and could be used in similarcontexts.For any arbitrarily given n E N*, take681(see Section 3). By equation (4), we haveBy elementary computations we obtainand If:1  2, 8(1:)   3 for all possible values of sn.Now, we note that on account of the last remark in Section 5, we actuallyproved the following Gauss-Kuzmin type theorem.Theorem 4 (Solution of the Gauss-Kuzmin problem). For all a E W thereexist two positive constants K and 0  1 such thatthere p denotes the invariant measure vx for T on x3I.Remark. Theorem 4 reduces for a = 0 and A = 1 to a version of Gauss-Kuzmin type theorem for the nearest integer continued fraction, where p°represents the Lebesgue measure on [-1/2, 1/2]  2 ’It should be emphasized that, to our knowledge, Theorem 4 is the firstGauss-Kuzmin result proved for the Rosen fractions.Acknowledgements. I would like to thank Marius Iosifescu for manystimulating discussions and Cor Kraaikamp for useful information concern-ing the present subject. I also wish to thank an anonymous referee for anumber of suggestions and comments which allowed to improve the contentand presentation of the paper.References[1] R. M. BURTON, C. KRAAIKAMP, T. A. SCHMIDT, Natural extensions for the Rosen fractions.Trans. Amer. Math. Soc. 352 (2000), 1277-1298.[2] K. GRÖCHENIG, A. HAAS, Backward continued fractions and their invariant measures.Canad. Math. Bull. 39 (1996), 186-198.[3] A. HAAS, C. SERIES, The Hurwitz constant and Diophantine approximation on Heckegroups. J. London Math. Soc. 34 (1986), 219-234.[4] M. IOSIFESCU, A basic tool in mathematical chaos theory: Doeblin and Fortet’s ergodic the-orem and Ionescu Tulcea-Marinescu’s generalization. In: Doeblin and Modern Probability(Blaubeuren, 1991), 111-124. Contemp. Math. 149, Amer. Math. Soc. Providence, RI, 1993.[5] M. IOSIFESCU, On the Gauss-Kuzmin-Lévy theorem, III. Rev. Roumaine Math. Pures Appl.42 (1997) , 71-88.[6] M. IOSIFESCU S. GRIGORESCU, Dependence with complete connections and its applications.Cambridge Univ. Press, Cambrigde, 1990.682[7] J. LEHNER, Diophantine approximation on Hecke groups, Glasgow Math. J. 27 (1985), 117-127.[8] J. LEHNER, Lagrange’s theorem for Hecke triangle groups. In: A tribute to Emil Grosswald:number theory and related analysis, 477-480, Contemp. Math. 143, Amer. Math. Soc.,Providence, RI, 1993.[9] H. NAKADA, Metrical theory for a class of continued fraction transformations and theirnatural extensions. Tokyo J. Math. 7 (1981), 399-426.[10] H. NAKADA, Continued fractions, geodesic flows and Ford circles. In: Algorithms, fractals,and dynamics (Okayama/Kyoto, 1992), 179-191, Plenum, New York, 1995.[11] D. ROSEN, A class of continued fractions associated with certain properly discontinuousgroups. Duke Math. J. 21 (1954), 549-563.[12] D. ROSEN, T. A. SCHMIDT, Hecke groups and continued fractions. Bull. Austral. Math. Soc.46 (1992), 459-474.[13] T. A. SCHMIDT, Remarks on the Rosen 03BB-continued fractions. In: Number theory with anemphasis on the Markoff spectrum (Provo, UT, 1991), 227-238, Lecture Notes in Pure andAppl. Math., 147, Dekker, New York, 1993.Gabriela I. SEBEUniversity Politehnica of BucharestDepartment of Mathematics ISplaiul Independentei 31377206 BucharestRomaniaE-mail : gisebeQmathem. pub . ro

A Gauss-Kuzmin theorem for the Rosen fractions

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A Gauss-Kuzmin theorem for the Rosen fractions

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