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Publication de l'umr 6085 laboratoire de mathématiques raphaël salem

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Ajouté le : 21 juillet 2011
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CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE UNIVERSITE de ROUEN PUBLICATION de l’UMR 6085  LABORATOIRE DE MATHEMATIQUES RAPHAEL SALEM   ANEXTENSIONWHICHISRELATIVELYTWOFOLDMIXING BUTNOTTHREEFOLDMIXING Thierry de la Rue   Document 2004-07 Universite de Rouen UFR des sciences Mathematiques, Site Colbert, UMR 6085 F 76821 MONT SAINT AIGNAN Cedex Tel: (33)(0) 235 14 71 00 Fax: (33)(0) 232 10 37 94 An extension which is relatively twofold mixing but not threefold mixing Thierryde la Rue July 21, 2004 Abstract We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold. 1 Factors, extensions and relative mixing 1.1 Factors, extensions and Rokhlin cocycle We are interested in dynamical systems (X,A, ,T ), where T is an ergodic automorphism of the Lebesgue space (X,A,). We will often designate such a system by simply the symbol T. A 1factor of T is a sub--algebraH ofA such thatH =T H . The canonical example of a system with factor is given by the skew product, constructed from a dynamical system (X ,A , ,T ) (called the base of the skew product) and a measurableH H H H map x 7 → S from X to the group of automorphisms of some Lebesgue space (Y,B,) (suchx H a map is called a Rokhlin cocycle). The transformation is de ned on the product space ( X H Y,A B, ) byH H ˜T(x,y) = (T x,S y).H x ˜In this context, the sub--algebraA {Y,∅} is clearly a factor of T.H SincetheworkofAbramovandRokhlin[1],thiskindofconstructionisknowntobethegeneral model for a system with factor: IfH is a factor of T, then there exists an isomorphism ϕ between ˜T and a skew product T constructed as above, with ϕ(H ) =A {Y,∅}. In such a situation,H we say that T is an extension of T .H 1.2 Mixing relative to a factor To understand precisely the way a factor is embedded in the dynamical system, one is led to study thebehaviourofthesystem relative to the factor; tothisend, relativepropertiesarede nedwhich aregeneralizationsofabsolutepropertiesofdynamicalsystems. Forexample, onecande neweak- mixing relative to a factor (see e.g. [2]), or the property of being a K-system relative to a factor [4]. We are interested in this work in the property of being mixing relative to a factor. De nition 1.1. LetH be a factor of the system (X,A, ,T ). T is said H -relatively mixing if  probak k∀A,B ∈A,  A∩T B|H (A|H )(T B|H→) 0. (1) k→+∞ As for the absolute property of mixing, it is possible to de ne mixing relative to a factor of any order n 2. The property described by (1) corresponds to relative mixing of order 2 (twofold relative mixing); for relative mixing of order 3 (threefold relative mixing), (1) should be replaced by 2 N N 3 ∀A,B,C ∈A,  probaj k j k A∩T B∩T C|H (A|H )(T B|H )(T C|H→) 0. (2) j,k j→+∞ Whether (absolute) twofold mixing implies threefold mixing is a well-known open problem in ergodic theory. The main goal of this work is to show that as far as relative mixing is concerned, twofold does not necessarily imply threefold. Theorem 1.1. We can construct a dynamical system (X,A, ,T ) with a factorH such that T isH -relatively twofold mixing but notH -relatively threefold mixing. 2 An extension which is relatively twofold mixing but not relatively threefold mixing. 2.1 The base The dynamical system announced in Theorem 1.1 is constructed as a skew product, whose base (X ,A , ,T ) is obtained as follows: Take X := [0,1[ equipped with the Lebesgue measureH H H H H  on the Borel -algebraA . The transformation T can be viewed as a triadic version ofH H H the Von Neumann-Kakutani transformation; we describe now its construction by the cutting and stacking method (see Figure 1). We begin by splitting X into three subintervals of length 1/3; we set B := [0,1/3[. TheH 1 2transformation T translates B onto T B := [1/3,2/3[, and translates T B onto T B :=H 1 H 1 H 1 H 1 2[2/3,1[. At this rst step, T is not yet de ned on T B . In general, after the n-th step ofH H 1 nn 3 1the construction, X has been split into 3 intervals of same length: B ,T B ,...,T B .H n H n H n nTheseintervalsformaso-called Rokhlin tower withbaseB andheight3 . Suchatowerisusuallyn represented by putting the intervals one on top the other, the transformation T mapping eachH n 3 1point to the one exactly above. At this step, the transformation is not yet de ned on T B .H n Step n+1 starts by chopping the base B into three subintervals of the same length, the rst ofn which is denoted by B . The n-th Rokhlin tower is thus split into three columns, which aren+1 n3 1 stacked together to get the n+1st tower. This amounts to mapping T B onto the secondH n+1 n23 1 piece of B by a translation, and T B onto the third piece of B . T is now de nedn H n+1 n H n+13 1 everywhere except on T B .H n+1 The iteration of this construction for all n  1 de nes T everywhere on X . The transfor-H H mation obtained in this way preserves Lebesgue measure, and it is well known that the dynamical system is ergodic. 2.2 The extension In order to construct the extension of T , we will now de ne a Rokhlin cocycle x7 →S from XH x H intothegroupofautomorphismsof(Y,B,), whereY :={ 1,1} ,B istheBorel-algebraofY, and  is the probability measure on Y which makes the coordinates independent and identically distributed, with (y = 1) =(y = 1) = 1/2 for each k  0.k k j If y = (y ) ∈ Y and 0  i  j, we denote by y| the nite word y y y . For eachk k∈ i i+1 ji nn  0, we call n-block a word of length 2 on the alphabet { 1,1}. Le rst n-block of y is thus n2 1y| . If w = y ...y n and w = z ...z n are two n-blocks, we denote by w w the1 0 2 1 2 0 2 1 1 20 (n+1)-block obtained by the concatenation of w and w , and w .w the n-block de ned by1 2 1 2 the termwise product of w and w :1 2 n n n nw w := y ...y z ...z , and w .w := (y z )...(y z ).1 2 0 2 1 0 2 1 1 2 0 0 2 1 2 1 Foreachn 1,wenowde