Centre de Physique Th�eorique�, CNRS Luminy, Case 907F-13288 Marseille { Cedex 9Self-similarity and �nite time intermittent e�ects inturbulent sequencesEdgardo UgaldeAbstractWe propose a method to analyze a turbulent sequence focusing on the self-similarproperties of the discrete system it generates. First we show how to approximate theself-similar exponent (� = 1=3 in the case of hydrodynamic turbulence) from the �nitetime measurements, and then we establish a criterion to characterize the deviations fromasymptotic scaling of the empirical measure generated by �nite time sequences. Thecomparison between an hydrodynamic turbulent sequence and a numerical sequence ofGaussian random variables (which generates a system with self-similarity exponent � =1=2), suggests the possibility of a self-similar asymptotic behavior for the system generatedby the �rst one. In the second part of this work, assuming the asymptotic self-similarityfor the system generated by a sequence, we study the conditions ensuring the appearanceof anomalous scaling on the structure functions, due to �nite time e�ects. In this way weestablish a relation between this anomalous scaling and the presence of intermittent largeoscillations superimposed on the self-similar sequence, giving rise to a concave deviationin the scaling of the structure functions for �nite times.Key Words: TurbulenceNumber of �gures: 4March 1996CPT-96/P.3321anonymous ftp or gopher: cpt.univ-mrs.fr�Unit�e Propre de Recherche 7061

IntroductionIn this paper we study two questions concerning the self-similarity of an ergodic discretesystem, both related to the empirical characterization of self-similarity. First we statewhat we mean by discrete system and self-similarity. This turns out to be a property ofthe ergodic measure with respect to a family of observables like the family of the velocitydi�erences in turbulence. In fact the old problem of scaling in fully developed turbulenceis the main motivation of this work, although we hope that this approach can also beapplied to other natural phenomena where the same kind of e�ects is encountered, thatis why we try to keep some generality in our discussion. Nevertheless, to illustrate allour results we use the physical example of a turbulence sequence, and a mathematicalexample: the sum of Gaussian random variables, which has well known self-similarityproperties. Our approach is su�ciently general to handle these two examples with thesame formalism.We state the problem as follow: suppose that as time goes to in�nity a given sequence gen-erates a self-similar system, then the problem is to detect at �nite time this self-similarity?We consider the case where a single self-similarity exponent completely determines thescaling properties of the system. We propose a procedure to compute this exponent froman experimental realization of the system (a single orbit, recorded during a �nite timeinterval) and a criterion to check the validity of the obtained value. This procedure isapplied to a moderated Reynolds number turbulent sequence and to a sequence of in-dependent Gaussian random variables with known asymptotic behavior. In this way wemay compare the results and validate the conclusions we obtain.In the second part we consider the scaling of the structure functions, again in a slightlygeneral framework, although it will be of interest mostly for the turbulence community.We �nd the relation between the deviations on the maximumvalue of the observables (thevelocity di�erences for the turbulence sequence), from to the exact self-similar behavior,and the anomalies in the scaling of the structure functions. In this way we show that thestructure functions measured from a self-similar system may display anomalous scaling.The deviations on the maximumvalue, when they are preferentially biased to large values,may be thought of as the superposition of an intermittent perturbation to our originalself-similar system, in a way that the perturbation is asymptotically metrically irrelevant.This would imply that the anomalous scaling of structure functions is only perceptible at1

�nite times.1 Discrete self-similar systemsWe consider dynamical systems for which the con�guration space is a space of real se-quences � � IRIN, for instance the space of realizations of a stochastic discrete processor the sequence of uctuations of a physical real �eld, experimentally recorded with aspeci�c acquisition frequency. The evolution in time is generated by the action of thetime shift on the sequences of the con�guration space. The state of the system at timet 2 IN, is then given by the sequencevt = v1;v2;v3; : : : 2 �;which evolves at time t+ 1 to the statevt+1 = �(vt) = v2;v3;v4; : : : 2 �:We suppose that our system satis�es the following \ergodic property". 1 For any mea-surable set B � �,limN!1 # f0 � t � N : f � �t(v) 2 BgN + 1 = Z� f(v) d�(v); (1:1)independently of the initial condition v in a set �0 � � of measure �(�0) = 1. In ourcase we consider a product sigma-algebra on � of copies of the usual Borel sigma-algebraon IR. In practice only the sigma-algebra of Borel sets is needed, because the observableswe consider take values in IR.The self-similarity of a discrete system is a property of the ergodic measure with respectto a family of observables. In the case of the turbulence sequence we consider the familyof velocity di�erences f�� : �! IR v 7! v� � v1 : � 2 INg.In general, the ergodic measure � is self-similar with respect to a family of observablesff� : �! IR : � 2 INg on the set T � IN, if for any two values of the parameter �; � 0 2 Tthere exists a rescaling function �=� 0 7! �(�=� 0) such that�ff� 0 2 Bg = � ff� 2 �(�=� 0) Bg ; (1:2)1About dynamical systems and ergodicity see, for instance, Halmos [1].2

for any measurable set B � �. This is equivalent to say that the asymptotic frequencyfor f� 0 to take values on the set B is exactly the same frequency for f� to take a valueon the rescaled set �(�=� 0) B, for �; � 0 2 T . In this way the observation of the systemthrough the function f� 0 is equivalent, up to a renormalization, to its observation throughf� . The equation (1.2) de�nes the \similarity property". We call \similarity range" therange T where (1.2) is valid and \scaling function" the map � which we extend to thewhole positive real axis IR+.In a self-similar system it is su�cient to know the temporal distribution of the f� values fora single value of the parameter � , to determine the temporal distribution of the f� valuesfor any other � 2 T , by applying the scale change de�ned by �. In more generality it ispossible to consider scale changes depending on the initial condition (a orbit-dependentrescaling), or rescaling factors depending not on the quotient �=� 0 but on both parameters� and � 0. In the �rst case we are dealing with a measure which is not ergodic, for whichthere are disjoint classes of orbits showing di�erent asymptotic behaviors. The formermay be compatible with an ergodic measure and the reason why we restrict our study tothe case of a scaling function depending only on the quotient �=� 0, stresses the invarianceunder changes of time unit on the underlying continuous-time system that, in the case ofa sequence of uctuations of a physical �eld, we model by a discrete-time one.Proposition 1If the temporal distribution of f� values is not concentrated at the origin, the scalingfunction follows a power law behavior, that is, �(�) = �� for some � 2 IR 2We prove of this proposition in the appendix. From now on we place ourselves in thiscase, so our scaling function will always be of the form �(�) = ��. � is the \similarityexponent".The classical example of a self-similar discrete system is a sum of independent randomvariables with the same zero-mean Gaussian distribution. In this case the states of thesystem are all the realizations of a sequence of independent zero-mean Gaussian randomvariables, and the family of observables is the family of sums fs� : � ! IR : � 2 INgsuch that s� (v) = P��1t=0 vt. The self-similarity comes from the stability of the Gaussianmeasures under convolutions and in this case the rescaling function is the power law3

�(�) = �1=2. 2More important to us is the example of a one point turbulent signal, which is supposed tobe self-similar with respect to the family of velocity di�erences, f�� : � ! IR : � 2 INgsuch that ��(v) = P��1t=0 v(t). In this case the space of sequences � � IRIN includes allthe possible velocity sequencesv = v(x; t0);v(x; t0+ dt);v(x; t0+ 2dt) : : : ;recorded in a fully developed turbulence experiment. An ergodic measure � is assumedto exist, representing the statistical temporal behavior of the system. Assuming self-similarity in the so called inertial range, it is possible to deduce the self-similarity exponentfrom the statistical theory of turbulence,3 which gives � = 1=3.How do we decide whether a given experimental sequence is the �nite time realizationof a self-similar system? This question presumes a suspicion about the self-similarity ofthe sequence we analyze, when self-similarity has been deduced form a previous analysisor from a theoretical prediction as in the case of the one point turbulence. We need ofcourse some information about the self-similarity range T and, which is essential, aboutthe family of observables to consider. So we place ourselves in the situation where boththe family of observables with respect to which our system is supposed to be self-similarand a subset of the self-similarity range, are known.In the next section we propose a procedure to determine a candidate for the self-similarityexponent from an experimental sequence and then a criterion to decide whether the se-quence is a �nite time realization of a self-similar system, given the self-similarity exponentand a �nite subset of the self-similarity range for the asymptotic system.2 Empirical investigation of self-similarityLet v be a sequence in � and N 2 IN the time length during which we record the sequence.We suppose that this sequence is a realization of a system which is self-similar with respect2This example belongs to a more general class where the Gaussian distribution is replaced by any zeromean stable measure of exponent � 6= 1, giving a scaling function �(�) = �1=�. We refer the reader tothe book of Feller [2] for this generalization.3We refer to Landau and Lifshitz [3], for a deduction of this exponent, �rst due to Kolmogorov.4

to the family ff� : � ! IR : � 2 INg, in a �nite range T0. We will characterize eachobservable f� at �nite time N by \the empirical(N; � )-maximum"�N (� ) � maxn���f� � �t(v)��� ; 0 � t � No : (2:1)For any �nite subset T0 of the self-similarity range, and thanks to the ergodic propertystated in equation (1.1), the empirical (N; � )-maximum approximately follows, for a largeclass of systems, the same scaling as the measure.Proposition 2Under some technical assumptions about the asymptotic distribution of f� values andthe convergence rate of the empirical measures to the asymptotic distribution, for all �; � 0in a �nite subset of the self-similarity range T we have�N(� 0)�N(� ) ! � 0� !� ; when N !1; (2:2)2We postpone the proof, as well as the exact statement of this result, to the appendix.The equation (2.2) allows us to compute a �nite time approximation for the self-similarityexponent in the following way.Procedure 1: For the experimental sequence v 2 � and the observation time N 2 IN,1) compute �N (� ) for all � in the �nite range T0,2) approximate the data log(� ) 7! log(�N(� )) for � 2 T0, by an a�ne function log(� ) 7!�N log(� ) + bN , by linear regression. By doing so we obtain�N = P�<� 02T0 log(� 0=� ) log (�)N(� 0)=�N (� ))P�<� 02T0 (log(� 0=� ))2 : (2:3)In this way we get an approximation �N for the self-similarity exponent � 2Example 2.1Consider a pseudo-random sequence of size N = 5 � 104, where each member is chosenindependently, according to the Gaussian distribution of mean zero and variance one. We5

compute the maximum sum�N(� ) = max(�����n+��1Xt=n vt����� : 0 � n � N)in the range T = f1; 2; : : : ; 20g, and compare it with the power function � 7! �N (1)� � 1=2in �gure (1). Applying the procedure described above, we �nd an approximation �104 =0:4882 for the self-similarity exponent � = 1=2. 2Example 2.2We analyze a turbulent wind tunnel sequence 4 with R� = 180, which is supposed tobe a generic initial condition for a system self-similar on T � f20; 21; : : : ; 120g with self-similarity exponent � = 1=3. The observation time is N = 15 � 104 units of time. 5 Wecompute the maximum di�erence�N(� ) = maxfjvn+� � vnj : 0 � n � Ngin the range T0 = f21; 21 + 5; 21 + 10 : : : ; 21 + 95g, and compare it with the powerfunctions � 7! �N(21) � (�=21)1=3 in �gure (2). Applying the procedure to approximatethe self-similarity exponent we �nd �15�104 = 0:2831 26Once we obtain an approximation for the self-similarity exponent we need a criterionto decide whether the analyzed sequence may be considered a realization of a self-similarsystem. What we propose is to compare the empiricalmeasures generated by the sequence,using the exponent to renormalize these measures in the appropriate way.For all � in the self-similarity range T and C > 0 we de�ne \the C-normalized measure"�C such that for any measurable set B � IR�C(B) � �(�����f� � �t(v)C �� ����� 2 B) (2:4)which, because of the self-similarity of � expressed in equation (1.2), does not depend on� 2 T . In equation (2.4), the C-normalized measure counts the asymptotic frequency for4This turbulence sequence was obtained by the team of F. Anselmet to study the joint distribution ofvelocity and temperature [4]5For an acquisition frequency of 37.5 kHz6In the article of Anselmet, Gagne, Hop�nger and Antonia [5], there is an interesting discussion aboutthe convergence of the (vn+� � vn)-moments, for turbulent sequences.6

jf�=C ��j to take values in B. Under the hypothesis ensuring the convergence, establishedin equation (1.1), the normalized C-measure of an interval [k=P �1=2P; k=P +1=2P [, canbe uniformly approximated on T0, by \the empirical (P; �;N)-normalized measure"�(P;�;N) : f0; 1; 2; : : : ; Pg ! [0; 1] such that;�(P;�;N)(k) � cardn0 � t � N : 2k�12P � ���f���t(v)�N (�) ��� < 2k+12P oN + 1 ; (2:5)which gives the �nite time frequency for jf�=�N (� )j to take values in one of the theseintervals. The ergodic property for � implies that �(P;�;N) must become independent of� , when N goes to in�nity. Since �N (� ) ! C �� when N ! 1 then, �(P;�;N)(k) !�C([k=P � 1=2P; k=P + 1=2P [) for some constant C > 0 independent of � 2 T0. In thisway we obtain a criterion of self-similarity, which obviously does not imply self-similaritybut nevertheless allows us to support or disregard from the experimental point of view,the self-similarity hypothesis.Self-similarity criterion: For an experimental sequence v 2 � and a �nite observationtime N 2 IN we consider the di�erence �(P;�;N) � �(P;� 0;N) � maxn����(P;�;N)(k)� �(P;� 0;N)(k)��� : 0 � k � Po (2:6)between two empirical normalized measures and, given a candidate for the self-similarityexponent �, we de�ne the deviationhN(�; � 0) � jlog (�N(� 0)=�N(� ))� � log (� 0=� )j : (2:7)For an experimental sequence which generates a self-similar system, for any � in thesimilarity range, the di�erence �(P;�;N) � �(P;� 0;N) between empiricalmeasures, convergesto an increasing function of the deviation hN (�; � 0), when N ! 1. At �nite N , thecorrelation between �(P;�;N) � �(P;� 0;N) and hN (�; � 0) may not be an increasing functions,but in any case the boundary functions� 7! maxnhN (�; � 0) : �(P;�;N) � �(P;�;N) < �o ; (2:8)� 7! minnhN(�; � 0) : �(P;�;N) � �(P;�;N) > �o ; (2:9)for �; � 0 2 T0 and � 2 IR+, do not decrease. We consider that the corresponding experi-mental sequence is a realization of a self-similar system, if both boundary functions may7

be bounded, (2.8) from below and (2.9) from above, by an strictly increasing function.The increasing of the boundaries stresses a real correlation between large deviations inthe scaling with large di�erences between the empirical normalized measures. In this waythe smaller the distance between the boundary functions is, the better the criterion issatis�ed. 2We apply this criterion to the sequences of examples (2.1) and (2.2). In the �rst case wetake � = 1=2 and T0 = f1; 2; : : : ; 20g. In the case of the turbulent sequence we test thecriterion for � = 1=3 and T0 = f21; 21 + 5; 21 + 10 : : : ; 21 + 95g. We show the results in�gures (3) and (4) where we plot the relation 7 �(P;�;N) � �(P;� 0;N) 7! hN (�; � 0); for �; � 0 2 T0;and its boundaries. We see that this criterion is satis�ed by both sequences and thatthe relative di�erence between the boundaries is of the same order in both cases. Weremark that this criterion does not depend on the partition P we use to de�ne the empir-ical normalized measures and that the technical conditions ensuring the convergence inequation (2.2) are needed for the validity of this criterion. These conditions are satis�edfor a large class of systems and it is di�cult to imagine that the turbulence sequence be-haves di�erently. On the other hand, the sequence of Gaussian random variables satis�esthese conditions. Of course in real life the accuracy of the results depends on the way wegenerate the numerical random sequence.Using this criterion of self-similarity we arrive at the following conclusion: The turbulentsequence of example (2.2) may be considered as the �nite time realization of a self-similarsystem with self-similarity exponent � � 1=3, and the sequence of Gaussian randomvariables of the example (2.1) is a �nite time realization of the underlying theoreticalmodel, with � = 1=2.It is important to realize that our conclusions are relative to some theoretical model wecontrol, because in general there is no a priori information neither about the speed ofconvergence of the empirical measures nor about the nature of the asymptotic one. Itseems that for turbulent sequences with larger Reynolds number, the time needed toobserve convergence from the criteria we stated above is much larger than 15 � 104 timeunits, for an acquisition frequency like the one used for the sequence in example (2.2).7Which is a function when � or � 0 is �xed. 8

In the theory of turbulence, since the work of Kolmogorov known as the K41 theory,8 another way to characterize self-similarity of such a system is well known and widelyused. It is based on the fact that the scaling of the measure is re ected in the moments ofthe observables. Van Atta and Park [7] deduced, from the self-similarity, the power lawbehavior of the moments. They study the self-similarity of the experimental measuresfrom direct comparison of the probability measure functions, concluding that a quasi self-similarity exists in a restricted range for � and vn+� �vn. In the next section we describehow the �nite time deviation may be the source of anomalous scaling, when the conditionsof convergence for the empirical measures are satis�ed. A discussion about the deviationscaused by temporal intermittency closes the section.3 Anomalous scaling from �nite time e�ectsIf the measurement time is large enough, the self-similarity property must already bepresent in the observations, however, deviations due to �nite time e�ects may modify theexpected behavior. We can estimate this �nite time e�ects in the case of a self-similarsystem for which the normalized measure �C satis�es a monotonicity property we statedbelow.When a deviation on the power law behavior of �N(� ) is found, for instance if �N(� 0) <�N(� )(� 0=� )� when � < � 0 then, the distance between the empirical probability measures�(P;�;N) and �(P;� 0;N) grows according to the self-similarity criterion. This di�erence im-plies that the \empirical structure functions" de�ned below, deviate from its asymptoticbehavior in the form of an \anomalous scaling" in the sense of Paladin and Vulpiani [8].The structure functions � 7! Sq(� ) account for the behavior of the f� asymptotic moments,for f� in a family of observables ff� : � 7! IR : � 2 INg. They follow a power law behavior,for a self-similar system, on the self-similarity range T . This follows immediately fromthe similarity property stated in equation (1.2). In fact, \the q-th structure function" isthe function Sq : IN! IR+ such thatSq(� ) � limN 7!1PNt=0 jf� (vt)jqN + 1 = Z� jf� (v)jq d�(v) (3:1)8All book about turbulence dedicate some pages to the K41 theory. Frisch [6] o�ers a view where theself-similarity hypothesis made in the original work are reformulated in a modern version.9

and because of the similarity property, for all �; � 0 2 T (Sq(� 0)=Sq(� )) = (� 0=� )� q. Thesame was deduced by Van Atta and Park [7], from an hypothesis stronger than ourself-similarity property but equivalent to it in the case of measures which are absolutelycontinuous with respect to the Lebesgue measure. In terms of the normalized measure�C introduced in equation (2.3), we can write the structure functions as,Sq(� ) = (C ��)q � ZIR+ xqd�C(x): (3:2)In the self-similarity range, the integral in the last equation is independent of � and then,for any �; � 0 2 T , the self-similarity exponent � satis�eslog (Sq(� )) = (� q) log(� ) + log�Cq � ZIR+ xqd�C(x)� : (3:3)If the observation time is large enough, one would be tempted to use this property todetermine the self-similarity exponent of the system as the slope of the a�ne functionlog(� ) 7! log (Sq(� )) de�ned on T . To this end we need to de�ne the �nite time-�niteprecision version of the structure functions.For a �nite observation timeN 2 IN and a �nite precision P 2 IN we de�ne \the empiricalstructure function" S(q;N) : IN! IR+, such thatS(q;N)(� ) � �(� )P !q PXk=0�(P;�;N)(k) kq; (3:4)computed from the empirical (P; �;N)-normalized measure de�ned in equation (2.4),which is the �nite time-�nite precision version of a C-normalized measure �C for �N (� ) �C ��.For N large enough, the \empirical structure function" follows an approximate power lawinside T , allowing us to approximate q � by the slope of the best a�ne approximationto the empirical function log(� ) 7! log(S(q;N)(� )), for � in a �nite range T0 � T . Thisde�nes, in the same way as Anselmet et al. determine the scaling exponents in [5], afunction of q 7! �N (q) which we call \the empirical scaling law", such that�N(q) = P�<� 02T0 log �S(q;N)(� 0)=S(q;N)(� )� log (�N(� 0)=�N(� ))P�<� 02T0 (log(� 0=� ))2 (3:5)We think of this function as the �nite time-range dependent approximation to the asymp-totic scaling law q 7! � q, which is always linear for a self-similar system. The equation10

(3.5) is obtained by linear regression of the data log(� ) 7! log(S(q;N)(� ))), for � 2 T0. Wemay already decompose this quantity into a linear and a non-linear component,�N(q) = �N q + fN (q); (3:6)where �N (q) is the �nite time approximation to the scaling exponent we de�ned in equa-tion (2.3), and fN (q) = P�<� 02T0 log (� 0=� )D� 0!� (q)P�<� 02T0 (log (� 0=� ))2 ; (3:7)with D� 0!� (q) � log0@PPk=0 �(P;� 0;N)(k) � kP �qPPk=0 �(P;�;N)(k) � kP �q 1A (3:8)the non-linear component of �N(q), which we call the \correction function" . This isthe key quantity for the study of anomalous scaling from �nite time deviations. Theinterest on this function comes from the statistical theory of turbulence, and we hopethat the conclusions we obtain can be extended to other systems where the same kind ofphenomenology appears. For some time the self-similarity property was supposed to holdin turbulence, but further experimental research and re�ned theoretical assumptions ledto a new interpretation of the experimental results. Nowadays it is generally accepted thatthe structure functions of the velocity di�erences f��(v) : � 2 INg follow an anomalousscaling law in some range of values of � , instead of the self-similarity property for the ��values implying \normal scaling" (in opposition to anomalous) for the structure functions.Let us specify what we understand by anomalous scaling.A family of observables ff� : �! IR : � 2 INg possesses anomalous scaling with respectto the measure �, in the range T � IN, if for all q 2 IN there exists a constant Cq and anon-linear function q 7! �(q) such thatZ� jf�(v)jq d�(v) = Cq � ��(q): (3:9)In fact q 7! �(q) turns out to be a concave function thanks to the H�older inequality.From the experimental point of view, the anomalous exponent �(q) is computed in thesame way as we compute �N(q), and because of the decomposition in equation (3.6) onemay think the non-linearity of the function q 7! �(q) as having a �nite time origin, alonger observation time being needed to put into evidence the self-similar nature of the11

phenomena. The convergence time will then be determined by the convergence of �N (theempirical self-similarity exponent), depending on the collective behavior of the empiricalmaxima, rather than by the convergence of the moments. 9 In the last part of thissection we prove that, under certain conditions on the asymptotic normalized measure,the correction function introduced in equation (3.7) is positive for all q, and increasing andconcave for small values of q. In this way the mechanism leading to anomalous scaling willby perceptible only for �nite time measurements, whereas the underlying ergodic measuremay still be considered as a self-similar measure. By taking this point of view, we canestablish a relation between �nite time anomalies and the temporal intermittency of thevelocity sequence. 10 We will come back to this relation in the next section.The knowledge of the relative deviation between two empirical measures allows us todetermine some properties of the correction function introduced in equation (3.7), whenthe asymptotic normalized measure satis�es the following monotonicity property.In the self-similarity range T , the normalized measures are related in such a way that�C(A) = �C0(C=C 0 A) , for any measurable set A � IR+ and C;C 0 > 0. We say that�C is \simply decreasing" if for all precision P 2 IN and for all deviation � > 0 in thenormalization constant C, there exists a single intersection point p(P; �) 2 IN and aminimal di�erence �(P; �) > 0 between the corresponding normalized measures, that is�C �h2k�12P ; 2k+12P h� < �C ��1 + �C� h2k�12P ; 2k+12P h�+ � when 0 � k < p;�C �h2k�12P ; 2k+12P h�+ � > �C ��1 + �C� h2k�12P ; 2k+12P h� when p < k � P 9=; (3:10)Notice that the crossing point p can be larger than P . In that case the second inequalitynever takes place. These inequalities mean that the normalized measures �C and �C=(1+�),when they are described up to a �nite precision P , cross at a single point and, far fromthis intersection, the distance between them is bounded from below. This is the casefor a Gaussian or exponentially decreasing measure, and for some other monotonouslydecreasing measures. It seems that the asymptotic measure corresponding to the turbulentdata of example 2.2 also satis�es this property, since its empirical normalized measuresatis�es an analogous property, described in proposition 3. In fact, if the observation timeis large enough the property of simple decreasing is inherited by the empirical normalizedmeasure.9The convergence of moments was the criterion used by Anselmet et al. in [5].10We understand temporal intermittency in the sense of Pomeau and Manville [9].12

Proposition 3Let the normalized measure �C , for a self-similar system with self-similarity exponent �,be simply decreasing. Let �; � 0 2 T0, with T0 a �nite subset of the self-similarity range.For any precision P 2 IN and deviation � > 0 there exists an observation time N0 2 INsuch that, if the deviation �N(� 0)� 0� < �N(�)�� + � is found, then�(P;N;�)(k) < �(P;N;� 0)(k) when 0 � k < p;�(P;N;�)(k) > �(P;N;� 0)(k) when p < k � P 9=; (3:11)for some p � P and all N � N0.ProofBy the ergodicity of the system, for all � > 0 there exist N(�) 2 IN such that for allN � N(�) ������C(�) "2k � 12P ; 2k + 12P "!� �(P;N;�)(k)����� < �;with C(� ) = �N(� )=��. On the other hand, the simple decreasing property of �C impliesthe existence, for any precision P , of a crossing point p and a minimal distance � between�C(� 0) and �C(�), which depends on � = C(� )� C(� 0). Taking � < �=2 we obtain (3.11),for all N � N0 and some p 2 IN. We only need to insure that p < P . This is the casebecause PXk=0 �(P;N;�)(k) = PXk=0 �(P;N;� 0)(k) = 1;implying that �(P;N;�)(k) cannot be larger than �(P;N;� 0)(k) for all 0 � k � P 2Now we describe the behavior of the correction function fN (q) introduced in equation(3.7), in terms of the deviations in the empirical normalized measures. These are due todeviations from the asymptotic scaling behavior on the empirical maxima �N .From equation (3.7) it is clear that the behavior of the correction function is controlledby the di�erence D� 0!� (q) de�ned in (3.8). In the range of validity of proposition 3, thisdi�erence follows a well de�ned behavior depending on the deviations of the correspond-ing empirical maxima, obtained thanks to the properties of simply decreasing measures.Indeed, to each di�erence D� 0!� (q) corresponds a couple of maxima �N(� 0); �N(� ), whichdetermines its behavior. 13

Proposition 4Let �C be a simply decreasing measure such that for all � > 0 there exists a minimaldistance � > 0 and an associated precision P0 such that�C �� 12P ; 32P ��� �C 1 + �C!� 12P ; 32P �! > �; 8 P � P0; (3:12)where �C is the normalized measure for a self-similar system with self-similarity exponent�.Then, there exists a minimal precision Pmin � 0, such that for all precision P � Pminand all deviation � > 0 there exists an observation time N0 such that, for all N � N0, if�N(� 0)� 0� < �N (�)�� + �, then1) D� 0!� (q) � 0 for all q � 0.2) There exists a minimal precision P0 > 0 such that, if the precision P used to computeq 7! D� 0!� (q) � 0 is larger than P0, then there exists q0 > 0 such that D� 0!�(q) is aconcave function on the interval [0; q0] .ProofThe di�erence D� 0!�(q) corresponding to an observation time N and precision P 2 INcan be written asD� 0!� (q) = log0@1 + PPk=0 ��(P;N;� 0)(k)� �(P;N;�)(k)� � kP �qPPk=0 �(P;N;�)(k) � kP �q 1A :and q 7! D� 0!� (q) is positive if and only ifPXk=0 ��(P;N;� 0)(k)� �(P;N;�)(k)� kP !q � 0: (3:13)Let us remind that we use the same hypothesis as in proposition 3, so for all deviation� and precision P there exists an observation time N0 such that for all N � N0 theinequality (3.10) holds. In this casePXk=1 �(P;N;� 0)(k)� �(P;N;�)(k) = 0 � ��(P;N;� 0)(0)� �(P;N;�)(0)� > 0:14

That shows the validity of (3.13) for q = 0. Let us denote by �(P;N;� 0!�)(k) the di�erence�(P;N;� 0)(k)��(P;N;�)(k) for all k 2 IN, and suppose that at the intersection point 0 < p <P , �(P;N;� 0)(p) < �(P;N;�)(p). 11 We decompose the sum on the left hand side of (3.13) intoa positive and a negative part, 12Ppk=0 �(P;N;� 0!�)(k) � kP �q + PPk=p+1 �(P;N;� 0!�)(k) � kP �q# � # �� pP �q �Ppk=0 �(P;N;� 0!�)(k)� + � pP �q �PPk=p+1 �(P;N;� 0!�)(k)� > 0:In this way we prove part 1 of this proposition. To prove the second part, that concerns theconcave increasing, we �nd after some easy calculations that in order for (d=dq) D� 0!�(q)to be positive it is su�cient thatd=dq PXk=0�(P;N;� 0!�)(k) kP !q! > 0;and for (d=dq)2 D� 0!� (q) to be negative it is su�cient that(d=dq)q PXk=0 �(P;N;� 0!�)(k) kP !q! < 0:For each one of these derivatives we decompose the resulting sum into a positive and anegative part, and then evaluate the result at q = 0. We obtainlimq!0+ ddq PXk=0 �(P;N;� 0!�)(k) kP !q! =Ppk=1 �(P;N;� 0!�)(k) log �kp� + PP�1k=p+1 �(P;N;� 0!�)(k) log �kp�# � # �log(P )�(P;N;� 0!�)(1) + log �P�1P � andlimq!0+ d2dq2 PXk=0 �(P;N;� 0!�)(k) kP !q! =11The other two possibilities, �(P;N;� 0)(p) = �(P;N;�)(p) or �(P;N;� 0 )(p) > �(P;N;�)(p) lead to the sameresult.12The symbol # � means that the upper term is greater than the lower one in a vertical array. We useit to stress the term by term inequality. 15

Ppk=1 �(P;N;� 0!�)(k) log �kp�2 + PP�1k=p+1 �(P;N;� 0!�)(k) log �kp�2# � # �log(P )2�(P;N;� 0!)(1) + log �P�1P �2 ;At this point we need to make use of the hypothesis stated in the inequality (3.12), whichholds for a normalized measure continuous with respect to Lebesgue. It is su�cient totake a precision P greater than P0 andP � � max�exp�log(2)1=2=(� � 2�)1=2� ; exp (log(2)=(� � 2�))� ;which depends on the minimal di�erence �, and on the distance � < �=2 between em-pirical and asymptotic normalized measures. Finally, for all deviation � > 0 and P �Pmin � max(P0; P�), there exists an observation time N0 such that, if �N(� 0)� 0� < �N(�)�� + �,then (d=dq) D� 0!� (0) > 0 and (d=dq)2 D� 0!� (0) < 0. Thanks to the continuity of thederivatives of D� 0!� (q), there exists an interval [0; q0], such that (d=dq) D� 0!� (q) > 0 and(d=dq)2 D� 0!�(q) < 0 for all q 2 [0; q0] around zero 2This proposition allows us to describe the behavior of the correction function through thedi�erences D� 0 7!�(q).We can �nd an interval of values [0; q0] where, for each couple � < � 0 2 T0 suchthat �N(�)(� 0) < (� 0=� )� � �N(�)(� ), there is a positive, increasing concave contributionto the correction function, whereas this contribution is negative, decreasing convex if�N(�)(� 0) > (� 0=� )� � �N(�)(� ). The equilibrium between positive and negative deviationsdetermines the nature of the correction function and it seems impossible to have an exactcancelation between positive and negative contributions. That explains the non-linearityof the empirical scaling law at �nite time in the general case.Let us remark that the �nal shape of the correction function generally depends on theobservation time through the empirical maxima, and for a �xed observation time, onthe choice of the self-similarity range T0. The simple decreasing property de�ned bythe inequality (3.11), and the hypothesis stated in (3.12) are essential for the validity ofproposition 4. They are satis�ed by the Gaussian measure and several other measuresde�ned by decreasing densities. With respect to the turbulent sequences, the behavior ofthe empirical normalized measures support the hypothesis of an asymptotic normalizedmeasure satisfying these properties. In the following section we restrict the study tomeasures with these characteristics. 16

4 Temporal intermittency and �nite time deviationsWe saw that �nite time deviations from self-similarity may produce an appearance ofanomalous scaling, because of �nite time non-linear corrections to the asymptotic scalinglaw. From the analysis of turbulence sequences, once the similarity range is determinedvia the third structure function, 13 the determination of the empirical scaling law, shows inall the reported cases, a well de�ned deviation. Usually the empirical scaling law �N (q) iscompared with concave, asymptotically a�ne increasing functions, coming from models ofthe distribution of energy dissipation in space as in She and Leveque [10] or Novikov [11],or from symmetry considerations as in Dubrulle and Graner [12]. It was Kolmogorov [13]who �rst proposed a non-linear concave function to model the nonlinearity of �N(q), froma model for the distribution of energy dissipation. Nevertheless this is an inconsistentmodel as noted by Mandelbrot [14]. In [15] we relate this kind of behavior of �N(q) withthe properties of the dynamical system generated by the corresponding sequence, withoutany assumption about asymptotic self-similarity.In some cases, when this concave non-linear scaling law is derived from an energy cascademodel, 14 the notion of intermittency is advocated to justify the deviation from a linearscaling law, but the notion of intermittency rests quite ambiguous in that framework.From our point of view, the relation between this deviation and intermittency really existsand can be stated in an explicit way. In fact, if we assume that the turbulent sequenceis the superposition of a sequence generating a self-similar system on the Kolmogorov'sinertial range and another sequence, which is temporally intermittent in the sense ofPomeau and Manville [9], we can show that the resulting empirical scaling law is composedof a linear part with slope �N < 1=3 and of a correction function fN (q) which is alwayspositive and increasingly concave on an interval [0; q0].To simplify our description let us consider that the intermittent sequenceI = I1; I2; : : : ; It; : : : 2 [�V; V ]INwith long sequences of zeros alternating with a single large oscillation u1; u2 : : : uk suchthat un 2 [�V; V ] for 1 � n � k, is superimposed on the self-similar sequencev = v1;v2; : : :vt; : : : 2 IRIN13Since, from the K41 theory, S3(� ) / � on the inertial range, our self-similarity range.14See Paladin and Vulpiani and references therein [8].17

which generates a system, self-similar on T with respect to the family f�� : IRIN ! IR :� 2 INg, with similarity exponent � > 0. In these circumstances, the observed sequenceis v � I 2 IRIN; such that v� It = 8<: vt if It = 0;It if It 6= 0: (4:1)When the intermittent sequence begins to oscillate, it replaces the usual self-similar se-quence. One can characterizes the intermittent sequence I by the temporal frequencyof the values It. The state It = 0 corresponds to the so called \laminar regime", whileIt; It+1; : : : ; It+k�1 = u1; u2 : : : ik represents the \chaotic regime" on the intermittent mod-els as in Wang [16] or Collet, Galves and Schmitt [17]. Let us simply remark that theempirical measures generated by I converges to the measure concentrated at zero, that islimN!1 # f0 � t � N : �t(I) 2 BgN + 1 = 8<: 0 if 0 62 B;1 if 0 2 B;for any measurable set B � IRIN, where 0 2 IRIN is the sequence formed only by zerosthat is, 0t = 0 for t 2 IN.Hypothesis about the superposition:1) We consider that the empirical measure generated by v � I converges to the samemeasure as v, that islimN!1 #f0 � t � N : �t(V) 2 BgN + 1 = limN!1 #f0 � t � N : �t(v) 2 BgN + 1 ; (4:2)for all measurable set B � IRN . This limit satis�es the self-similarity property, whereas theempirical maxima associated to it can be greatly modi�ed with respect to those associatedto v.2) The maximal di�erences associated to I have a negative deviation from the asymptoticscaling, that is, there exists � > 0 such that for all couple � < � 0 2 T0,maxfjun+� 0 � unj : 1 � n � k � � 0gmaxfjun+� � unj : 1 � n � k � �g < � 0� !� � �; for all �; � 0 2 T0: (4:3)Equation (4.2) ensures, for any precision P and for the deviation � appearing in the lastinequality, the existence of an observation time N0 such that for all N > N0 propositions3 and 4 hold for the empirical measure generated by v � I.18

Theorem 1Let us suppose that the superposition v�I of a self-similar and an intermittent sequencesatis�es the hypothesis stated above. There exists a minimal precision Pmin and for allP � Pmin there a minimal observation time N0 such that, if for all observation timeN0 � N � N0 +K;K > 0 we have�N(� ) = maxfjun+� � unj : 1 � n � k � �g; 8 � 2 T0; (4:4)where k is the length of the intermittent oscillation, and the empirical maximum �N (� ) iscomputed from v � I. Then the empirical scaling law �N(q) computed from v � I withprecision P , decomposes as �N(q) = �N (q) q + fN (q);where �N < � and q 7! fN (q) is a positive function, concave and increasing in a range[0; q0], and that for all N0 � N � N0 +K;K > 0.ProofThis result follows directly from the decomposition of �N(q) established in equation (3.6),�N (q) = �N q + fN (q) =P�<� 02T0 log � � 0� � log � �N0(� 0)�N(�) �P�<� 02T0 (log(� 0=� ))2 q + P�<� 02T0 log � � 0� �D� 0!� (q)P�<� 02T0 (log(� 0=� ))2 :The hypothesis in (4.3) and (4.4) ensures that for any couple �; � 0 2 T0 and N0 � N �N0 +K,log �N(� 0)�N(� 0)! = log maxfjun+� 0 � unj : 1 � n � k � � 0gmaxfjun+� � unj : 1 � n � k � �g ! < log � 0� !� � �! ;where the maxima �N(� ) are computed from the superposition v � I. In this way weobtain �N < �.In the proposition 4, which applies to the sequence v � I thanks to the hypothesis in(4.2), we stated the existence of a minimal precision Pmin such that for all P � P0 and alldeviations � > 0, there exists a minimal observation time N0 such that for all N � N0the di�erence functions q 7! D� 0!�(q) computed at those times are positive for all q � 0,19

and concave increasing in a range [0; q0]. Since fN(q) is a positive linear combination ofsuch di�erences, the linear part of �N also has these properties 2In this way we construct a \model sequence" for which the asymptotic behavior agreeswith the self-similar predictions, but for which the �nite time deviations give to �N(q)\anomalous" characteristics.Given a precision P > Pmin and for observation times N0 � N � N0 +K, the empiricalscaling law decomposes as a sum�N(q) = �N q + fN (q);where the empirical scaling exponent �N is smaller than the asymptotic one, and thenon-linear correction function fN(q) is always positive and concave increasing in a range[0; q0]. In this range, the empirical scaling law is a non-linear, concave increasing function,bounded from below by a linear increasing function q 7! �N q. This is exactly the samebehavior as predicted by the energy cascade models cited above, with the di�erence thatin this case it is valid only for �nite times and the concavity is ensured only on a �niterange of q.The next step on this investigation about the possible �nite time origin for the anomalousscaling, is to confront this simpli�ed model, of superposition of self-similarity and inter-mittency, with the experiment. In practice, if the turbulent sequence can be decomposedas the superposition of a self-similar and an intermittent component, the characteristicsof the intermittent one may be not so trivial as we have considered. We took the oscilla-tions on the intermittent sequence as being are always of same same type, while in a realexperimental they may not be. In this way that the behavior of �N(� ) may be dominatedsome times by the intermittent oscillation and at some other times by the self-similarsequence itself, depending on the observation time and/or the values of � we consider.5 ConclusionsWe have discussed two questions related to the �nite time behavior of a sequence thatgenerates a self-similar system. The �rst one concerns the way to approach the self-similar20

exponent associated to the system, from the analysis of the sequences observed duringa �nite time. After this approximation procedure we established a criterion to decidewhether the analyzed sequence may be considered to be a �nite time subsequence of agenerating sequence for a self-similar system. This criterion which regard the propertiesof the empirical measure generated by the sequence at a �xed time, the observationtime, does not imply that the sequence really generates a self-similar measure, simplybecause it is impossible to give su�cient conditions to ensure it, when we regard only �nitetime realizations. Nevertheless it is essential to relate the �nite time realization to someasymptotic known behavior, otherwise is impossible to develop consistently our analysis.The main motivation of this work is the investigation of self-similarity on the systemgenerated by a turbulent sequence, implied by the postulates of Kolmovorov's theory ofturbulence and by the Taylor hypothesis [18]. This prediction was already considerated onthe work of Van Atta and Park [7], unfortunately their analysis is mostly focused on thesearch for non self-similar e�ects, and some interesting observations deserving a furtherinvestigation were not developed. Using the criterion of self-similarity on both a Gaussiansequence and a turbulent sequence, we arrive at the conclusion that the same degree ofself-similarity must be assigned to both sequences, although only in the �rst case we maybe sure that the system is. Then the observations of Van Atta and Park about the lackof self-similarity mainly for extreme events, leading to anomalous scaling in the structurefunctions may be thought of as a �nite time e�ect. That leads to the formulation of thesecond question. We suppose that a given sequence is really a �nite time realization ofa self-similar system, and we look for the conditions ensuring anomalous scaling at �nitetime, taking into account the particular kind of deviation on the scaling law found on theanalysis of turbulent data. In fact, the empirical scaling laws reported on the literature,despite the precise value for the exponents, have all in common a concave increasing shape,which is related to \intermittency" on the energy cascade models (Novikov [11], Frisch [19],She and Leveque [10] and others). To introduce this particular kind of deviations, givena �nite time realization which is compatible with the self-similar asymptotic behavior,we suppose that an intermittent e�ect, with no incidence on the measure convergence,but modifying the behavior of the empirical empirical maxima, is superimposed on theself-similar sequence. In this way the scaling law obtained from such a sequence possessesthe same type of deviation as the one predicted by some of the energy cascade models.It remains to develop an analysis capable of putting into evidence this kind of temporal21

intermittent e�ects and allowing a decomposition of the turbulent signal into a real self-similar sequence superimposed on an intermittent one. It is also interesting to apply thiskind of reasoning to other systems where anomalous scaling may be related to �nite timee�ects.Acknowledgments This work was partially supported by the Universit�e dela Mediterran�ee by an ATER contract. We thank all the members of the DynamicalSystems team at CPT-Marseille for their interesting comments and suggestions at eachlevel of the realization of this work. We also thank to F. Anselmet for giving us access tothe experimental turbulent sequence we used in example 2.2.

22

Bibliography1. Halmos, P. R., 1956, Lectures on Ergodic Theory, Chelsea Publishing Company, NewYork, N. Y.2. Feller W., 1966, An introduction to probability theory Vol. II, . Wiley, New York.3. Landau, L. and Lifshitz E. M., 1959, Fluid Mechanics, Pergamon Press.4. Ould-Rouis, M., Anselmet, F., Legal P. and Vaienti S., 1995, Physica D 85, 405-424.5. Anselmet, F., Gagne, Y., Hop�nger, E. J. and Antonia, R. A., 1984, J. Fluid Mech.140, 63-89.6. Frisch, U., 1995, Turbulence, Cambridge University Press.7. Paladin, G. and Vulpiani, A., 1987, Physics Reports 156 4, 147-225.8. Van Atta, C. W. and Park, J., 1972, in Statistical Models and Turbulence, (Ed. M.Rosenblatt and C. W. Van Atta) Lecture Notes in Physics Vol. 12, 402-426, Springer-Verlag.9. Pomeau Y. and Manville, P., 1980, Commun. Math. Phys. 74, 189-197.10. She, L. S. and Leveque, E., 1994, Phys. Rev. Lett. 72 3, 336-339.11. Novikov, E. A., 1970, Prikl. Math. Mech. 35 2, 266-277.12. Dubrulle, B. and Graner, F., 1995, Scale invariance and scaling exponents in fullydeveloped turbulence, submitted to Journal de Physique France.13. Kolmogorov, A. N., 1962, J. Fluid Mech. 13, 82-85.14. Mandelbrot, B. B., 1972, in Statistical Models and Turbulence, (Ed. M. Rosenblattand C. W. Van Atta) Lecture Notes in Physics Vol. 12, 333-351, Springer-Verlag.15. Ugalde E. and Lima R., 1995, On a discrete dynamical model for local turbulence,Preprint, to be published in Phisica D.16. Wang, X. J., 1989, Phys. Rev. Lett. 40 11, 6647-6661.23

17. Collet, P, Galves, A. and Schmitt, B., 1992, Ann. Inst. Henri Poincar�e 57 3, 319-33118. Taylor, I. G., 1935, Proc. R. Soc. Lond. A 151, 421-478.19. Frisch, U. and Sulem, P. and Nelkin J., 1978, J. Fluid Mech. 87, 719-736.

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0 2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

Figure 1. The circles represent the function � 7! �N(� ) on the range � 2 f; 1; 2; : : : ; 20g,computed from the random sequence of example 2.1 (N � 104). The continuous is thepower law function � 7! �N(1)� � 1=2 for � 2 [1; 20].25

20 30 40 50 60 70 80 90 100 110 1202.5

3

3.5

4

4.5

5

5.5

Figure 2. The circles represent the function � 7! �N(� ) on the range � 2 f21; 21+5; 21+10 : : : ; 21 + 95g, computed from the turbulent sequence of example 2.2 (N � 105). Thecontinuous line is the power law function � 7! �N(31) � (�=31)1=3 for � 2 [21; 116].26

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10−3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

possible differences

po

ssib

le d

evia

tio

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Figure 3. From the Gaussian sequence of the example 2.1 we plot the possible scalingdeviations as a function of the possible di�erences on the empirical normalized measures, �(P;�;N) � �(P;�;N) ! hN (�; � 0) � ���log ��N(� 0)=�N (� ) (�=� 0)1=2����with �; � 0 2 f1; 2; : : : ; 20g and �(P;�;N) � �(P;�;N) as de�ned in equation (2.6). One circleat the point (�; h) represents a couple �; � 0 for which the di�erence between the empiricalnormalized measures is � and the deviation on the scaling is h. The continuous lines arethe upper and lower boundary functions de�ned in equation (2.8) and (2.9) respectively.27

0 1 2 3 4 5 6 7

x 10−3

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Figure 4. From the turbulent sequence of example 2.2 we plot the possible scalingdeviations as a function of the possible di�erences on the empirical normalized measures, �(P;�;N) � �(P;�;N) ! hN (�; � 0) � ���log ��N(� 0)=�N (� ) (�=� 0)1=3����with �; � 0 2 f21; 21+5; 21+10 : : : ; 21+95g and �(P;�;N) � �(P;�;N) as de�ned in equation(2.6). One circle at the point (�; h) represents a couple �; � 0 for which the di�erencebetween the empirical normalized measures is � and the deviation on the scaling is h.The continuous lines are the boundary functions de�ned in equations (2.8) and (2.9)respectively. 28

AppendixOne of the consequences of the self-similarity property, introduced in equation (1.2) isthe fact that under certain circumstances, the scaling functions � 0=� 7! �(� 0=� ) follow apower law.Proposition 1Let (�; �; �) be a self-similar system with respect to ff� : � 7! IR : � 2 INg, withself-similarity range T and scaling function � 0=� 7! �(� 0=� ). If for all � 2 T , � � f (�1)� isnot the atomic measure at 0, then �(�) = �� for some � 2 IR.ProofLet �; � 0, et � 00 2 T , by similarity we have:� ff� 00 2 Bg = � ff� 2 �(�=� 00) Bg ;� ff� 00 2 Bg = � ff� 0 2 �(� 0=� 00) Bg ;� ff� 0 2 �(� 0=� 00)�Bg = � ff� 2 �(�=� 0)�(� 0=� 00) Bg ; andthen, � ff� 2 �(� 0=� 00)�(�=� 0) Bg = � ff� 2 �(�=� 00) Bg :for any Borel set B � IR, �; � 0 and � 00 2 T .Let us take B = B�(0)�(�=� 00) with B�(0) = fx 2 IR : jxj � �g, then � f�� 2 B�(0)g =� f�� 2 `B�(0)g, with ` = �(� 0=� 00)�(�=� 0)�(�=� 00) :If ` 6= 1, � ff� 2 B�(0)g = � ff� 2 B�`n(0)g ;8� > 0;for all � > 0; n 2 IN, which implies that � � f�1� is the measure concentrated at 0. Thisbeing impossible by hypothesis,�(�=� 00) = �(� 0=� 00)�(�=� 0);8�; � 0; � 00 2 T29

then, �(�) = �� for some � 2 IR 2Proposition 2Let (�; �; �) be a self-similar system with respect to ff� : � ! IR : � 2 INg, withself-similarity range T and exponent �.Thanks to the ergodic property stated in equation (1.1), for all � > 0 there exists anobservation time N(�) 2 IN such that�����# f0 � t � N : jf� � �t(v)j 2 BgN + 1 � �fjf� j 2 Bg����� < �; (H:1)for all � 2 T0; N � N(�) and any measurable set B � �, T0 being a �nite subset of theself-similarity range T . Let us suppose that this convergence is such that,lim�!0 � N(�) = 0: (H:2)Under these hypothesis, for all �; � 0 in T0,�N(� 0)�N(� ) ! � 0� !� ; when N !1:ProofFor � > 0 and all � 2 IN we de�ne��(� ) � inf nR 2 IR+ : � fjf� (v)j � Rg > �o :Thanks to the self-similarity property,� fjf� (v)j � Rg = � fjf�(v)j � (� 0=� )� �Rg ; 8�; � 0 2 T;which implies that ��(� 0)��(� ) = � 0� !q : (T:1)The ergodic property stated in (H.1) implies that,if #n0 � t � N(�) : jf� � �t(v)j > Ro = 0 then �fjf� j � Rg < �;30

and from here we deduce that ��(� ) � �N(�)(� ), for all � 2 T0.On the other hand, if �fjf� j � Rg < � for some R > 0, then#n0 � t � N(�) : jf� � �t(v)j 2 Bo < 2�(N(�) + 1):The hypothesis (H.2) implies the existence of �c > 0 such that for all � � �c, 2�(N(�)+1) <1, which implies # f0 � t � N(�) : jf� � �t(v)j 2 Bg = 0.In this way we �nally get �N(�)(� ) = ��, for all � < �c or equivalently, for all N � N(�c).Replacing �� by �N(�) in (T.1) the proposition is proved 2

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