Nearly finite Chacon Transformation

Transcription

1 Nearly finite hacon Transformation Élise Janvresse, Emmanuel Roy, Thierry De La Rue To cite this version: Élise Janvresse, Emmanuel Roy, Thierry De La Rue Nearly finite hacon Transformation 2018 <hal v2> HAL I: hal Submitte on 19 Jan 2018 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés

2 NEARLY FINITE HAON TRANSFORMATION ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Abstract We construct an infinite measure preserving version of hacon transformation, an prove that it has a property similar to Minimal Self- Joinings in finite measure: its artesian powers have as few invariant Raon measures as possible Keywors: hacon infinite measure preserving transformation, rank-one transformation, joinings MS classification: 37A40, 37A05 1 Introuction 11 Motivations The purpose of this work is to continue the stuy, starte in [7] an [3], of what the Minimal Self-Joinings MSJ property coul be in the setting of infinite-measure preserving transformations We want here to construct an infinite measure preserving transformation whose artesian powers have as few invariant measures as possible As in the aforementione papers, we restrict ourselves to Raon measures giving finite mass to compact sets, since in general there are excessively many infinite invariant measures for a given transformation think of the sum of Dirac masses along an orbit A first attempt in this irection was to consier the so-calle infinite hacon transformation introuce in [2] Inee, the construction of this infinite measure preserving rank-one transformation is strongly inspire by the classical finite measure preserving hacon transformation, which enjoys the MSJ property [4] The ientification of invariant measures for artesian powers of the infinite hacon transformation was the object of our previous work [7] In aition to the proucts of graph measures arising from powers of the transformation see the beginning of Section 33 for etails, we foun in the case of infinite hacon some kin of unexpecte invariant measures, the so-calle weir measures These weir measures have marginals which are singular with respect to the original invariant measure, but it is shown in [3, Example 54] that an appropriate convex combination of weir measures can have absolutely continuous marginals We propose here another rank-one transformation, which we call the nearly finite hacon transformation, hereafter enote by T Although it preserves an infinite measure µ, its construction is esigne to mimic as much as possible the behaviour of the classical hacon transformation, so that the phenomenon of weir measures isappears Our main result, Theorem 310, is the following: there exists a µ- conull set X such that, for each 1, the ergoic T -invariant Raon measures on X are the prouct measure µ an proucts of graph measures arising from powers of T orollary 311 then ientifies all T -invariant Raon measures whose Research partially supporte by French research group GeoSto NRS-GDR3477 1

3 2 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE marginals are absolutely continuous with respect to µ as sums of countably many ergoic components which are of the form given in the theorem Beyon the question of the MSJ property in the infinite measure worl, the example presente in this paper is also of crucial importance in the stuy of Poisson suspensions A Poisson suspension is a finite measure preserving ynamical system constructe from an infinite measure preserving system: a state of the space is a realization of a Poisson point process whose intensity is the infinite invariant measure, an each ranom point evolves accoring to the ynamics of the infinite measure preserving transformation we refer to [10] for a complete presentation of Poisson suspensions Although of ifferent nature, Poisson suspensions share surprising properties with another category of finite measure preserving ynamical systems of probabilistic origin: Gaussian ynamical systems, which are constructe from finite measures on the circle A beautiful theory has been eveloppe in [9], concerning a special class of Gaussian systems calle GAGs a French acronym for Gaussian systems with Gaussian self-joinings The keystone for the construction of a GAG system is a striking theorem ue to Foiaş an Strătilă [5]: if a measure supporte on a Kronecker subset of the circle appears as the spectral measure of some ergoic stationary process, then this process is Gaussian The Poisson counterpart of GAG, calle PaP Poisson suspension with Poisson self-joinings is presente in [8], where the construction of a PaP example relies on a theorem à la Foiaş-Strătilă see [8, Theorem 34] Roughly speaking, accoring to this theorem, if some ergoic point process evolves uner a ynamics irecte by an infinite measure preserving transformation with special properties, then this point process is Poissonian The special properties neee here are precisely those given by orollary 311 Therefore, systems enjoying those properties play in the theory of Poisson suspensions the same role as measures supporte on Kronecker subset in the setting of Gaussian systems For some applications in the stuy of Poisson suspensions eveloppe in [8], we also nee an aitional property which is the existence of a measurable law of large numbers Proposition 83 shows that the nearly finite hacon transformation satisfies a stronger property calle rational ergoicity 12 Roamap of the paper Section 2 is evote to the construction of the nearly finite hacon transformation, an to first elementary results For peagogical reasons, we start in Section 21 by efining the nearly finite hacon transformation with the cutting-an-stacking metho on R + equippe with the Lebesgue measure, as it is easier to visualize the structure of the Rokhlin towers in this setting Most steps of the construction are ientical to construction of the classical hacon transformation There is just a fast increasing sequence n l of integers such that each n l -th step of the construction iffers from classical hacon, which ensures that the invariant measure has infinite mass Then we turn in Section 22 to a more convenient but isomorphic moel for our purposes, which is a transformation T on a set X of sequences on a countable alphabet In Section 23, we escribe basic properties of a typical point with respect to the invariant measure µ, an efine the conull set X referre to in Theorem 310 Section 3 contains the main results concerning Raon measures on X which are T -invariant Section 31 first states some basic facts about Raon measures on X We give a criterion for such a measure to be T -invariant Lemma 32 We also efine a notion of convergence of Raon measures Definition 33, which

4 NEARLY FINITE HAON TRANSFORMATION 3 is specially aapte to the formulation of Hopf s ratio ergoic theorem, an give useful lemmas concerning this convergence In Section 32, we treat the easy case of totally issipative measures: Proposition 39 eliminates the possibility of a totally issipative T -invariant Raon measure supporte on X In Section 33, we state our main result Theorem 310 an establish the bases of a proof by inuction on At the en of Section 33, we fix once an for all a T -invariant Raon measure σ supporte on X, for some 2 The remainer of the paper is completely evote to proving that either σ is a graph measure arising from powers of T, or it can be ecompose as a prouct of two measures on which we can apply the inuction hypothesis In section 34, we choose once an for all a σ-typical point x X, on the orbit of which we estimate the properties of σ We also introuce in Definition 312 the central notion of n-crossings, which are finite subintervals of Z epening on the position of the orbit of the typical point x with respect to the n-th Rokhlin tower of the rank-one construction The analysis of the structure of those n-crossings constitutes the core of our proof In Section 35, we provie a criterion for σ to be a graph measure arising from powers of T, state in terms of n-crossings Proposition 317 Section 4 is evote to the proof of Proposition 41, which is a central result in the analysis of the structure of n-crossings Section 41 escribes a hierarchy of abstract subsets of Z an provies a lemma on the combinatorics of subsets in this hierarchy Lemma 42 Then Section 42 explains how to apply this lemma to the structure of n-crossings Section 43 provies a useful corollary of Proposition 41 in terms of the measure σ Section 5 is evote to the stuy of the convergence of empirical measures, which are finite sums of Dirac masses on points on the orbit of x, corresponing to finite subsets of Z We provie two criteria, Proposition 57 an Proposition 59, for a sequence of such empirical measures to converge to σ In Section 6 we present the main tool use to ecompose σ as a prouct measure We introuce the notion of twisting transformation efinition 61, which is simply a transformation of X acting as T on some coorinates, an as I on others Base on a theorem from [7], Proposition 62 shows that, if σ is invariant by such a twisting transformation, then σ ecomposes as a prouct measure to which we can apply the inuction hypothesis Then Proposition 63 provies a criterion for σ to be invariant by some twisting transformation All the preceing tools are use in Section 7, where the proof of Theorem 310 is complete If the criterion given by Proposition 317 for σ to be a graph measure fails, then for infinitely many integers n there exists some n-crossing, not too far from 0, with some special property We treat several cases accoring to the positions of these integers with respect to the sequence n l With the help of Propositions 63 an 62, we show that in all cases σ ecomposes as a prouct of two measures to which the inuction hypothesis applies Section 8, which can be rea inepenently, is evote to the proof of the rational ergoicity of the nearly finite hacon transformation 2 onstruction of the nearly finite hacon transformation 21 utting-an-stacking construction on R + As previously explaine, this transformation is esigne to mimic the classical finite measure preserving hacon transformation as much as possible, yet it must preserve an infinite measure The

5 4 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE construction will make use of two preefine increasing sequences of integers: 1 n 1 n 2 n l an l 0 := 1 l 1 l 2 l k, satisfying some growth conitions to be precise later see below conitions 1 an 2 For each l 1, there exists a unique integer k 0 such that l k l < l k+1, an we enote this integer by kl In the first step we consier the interval [0, 1, which is cut into three subintervals of equal length We take the extra interval [1, 4/3 an stack it above the mile piece Then we stack all these intervals left uner right, getting a tower of height h 1 := 4 The transformation T maps each point to the point exactly above it in the tower At this step T is yet unefine on the top of the tower After step n we have a tower of height h n, calle tower n, the levels of which are intervals of length 1/3 n These intervals are close to the left an open to the right At step n + 1, tower n is cut into three subcolumns of equal with If n / {n l : l 1}, we o as in the stanar finite measure preserving hacon transformation: we a an extra interval of length 1/3 n+1 above the mile subcolumn, an we stack the three subcolumns left uner right to get tower n + 1 of height h n+1 = 3h n + 1 If n = n l for some l, we a h n kl extra intervals above each of the three subcolumns, an a further extra interval above the secon subcolumn Then we stack the three subcolumns left uner right an get tower n + 1 of height h n+1 = 3h n + 3h n kl + 1 See Figure 1 At each step, we pick the extra intervals successively by taking the leftmost interval of esire length in the unuse part of R + Extra intervals use at step n + 1 are calle n + 1-spacers We want the Lebesgue measure of tower n to increase to infinity, which is easily satisfie provie the sequence l k grows sufficiently fast Inee, for each n 1 we have h n+1 6h n + 1 7h n, whence h n /h n+1 1/7 It follows that for each k 0 an each l k l < l k+1, Lebtower n l + 1 Lebtower n l 1 + h n l k k h Therefore it is enough for example to assume that for each k 0, k l k+1 l k 2 Uner this assumption, we get at the en a rank-one transformation efine on R + which preserves the Lebesgue measure We will also assume that for each l, 2 n l > n l 1 + 2l 22 onstruction on a set of sequences For technical reasons, it will be more convenient to consier a moel of the nearly finite hacon transformation in which the ambient space is a totally isconnecte non compact metric space, an each level of each tower is a compact clopen set onsier the countable alphabet A := { } N To each t R +, we associate the sequence ϕt = j n t n 0 AN efine by { if t / tower n, j n t := j if t is in level j of tower n 0 j < h n

6 NEARLY FINITE HAON TRANSFORMATION 5 Figure 1 onstruction of the nearly finite hacon transformation by cutting an stacking By conition 1, R + = n tower n, an for each n, tower n tower n + 1 Hence for each t R +, n 0 0 : n < n 0, j n t =, an n n 0, j n t {0,, h n 1} Moreover, each level of tower n+1 is either completely outsie tower n or completely insie a single level of tower n Let us introuce, for each n 1, the map p n : {0,, h n+1 1, } {0,, h n 1, } efine by p n :=, j {0,, h n+1 1}, p n j := if level j of tower n + 1 is completely outsie tower n, an p n j := j {0,, h n 1} if level j of tower n + 1 is completely insie level j of tower n Then the sequence j n t satisfies the following compatibility conition n 0 n 0, j n t = p n jn+1 t

7 6 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE In particular, j n t completely etermines j m t for each 0 m n observe that p n satisfies the following property: We also 3 If p n j {0,, h n 2}, then j {0,, h n+1 2} an p n j+1 = p n j+1 Now we can efine our space X, to which belongs ϕt for each t R + : X := { j n n 0 A N : n 0, j n = p n j n+1 an n 0, j n0 } X inherits its topology from the prouct topology of A N In particular it is a totally isconnecte metrizable space, but it is not compact in fact X is not close in A N, as the infinite sequence,, is in X \ X For each n 0 an each x X, we enote by j n x the n-th coorinate of x For each j {0,, h n 1}, we efine the subset of X L j n := {x X : j n x = j} Then L j n is compact an clopen in X Moreover the family of sets L j n form a basis of the topology on X To efine the transformation T on X, we nee the following easy lemma Lemma 21 For each x = j n n 0 X, there exists n such that, for each n n, j n {0,, h n 2} Proof Remember that at each step n l + 1, some spacers are ae on the last subcolumn of tower n l Hence, j +1 = h +1 1 implies j = Now take l large enough so that j Then j +1 < h +1 1, an 3 shows by an immeiate inuction that j m < h m 1 for each m n l + 1 We efine the measurable transformation T : X X as follows: for x = j n n 0 X, we consier the smallest integer n satisfying the property given in Lemma 21 Then we set T x := j n n 0, where j n := j n + 1 if n n, an the finite sequence j 1, j 2,, j n 1 is etermine by the value of j n an the compatibility conitions j n = p n j n+1, 1 n < n Note that T is one-to-one, an T X = X \ {0}, where 0 := 0, 0, For each n 1 an each 0 j < h n 1, T L j n = L j+1 n, hence L 0 n,, L hn 1 n is a Rokhlin tower for T By construction, the family of Rokhlin towers we get in this way has the same structure as the family of Rokhlin towers we constructe by cutting-an-stacking on R + From now on, tower n will rather esignate the Rokhlin tower L 0 n,, L hn 1 n to the construction on R + is the following elementary fact Remark 22 If L jn n n n such that L jn+1 n+1 is always inclue in Ljn n is always a singleton in X The main avantage that we get compare is a sequence of levels in the successive Rokhlin towers, equivalently, j n = p n j n+1, then n Ljn n Note that such an intersection can be empty in the construction on R + Let µ be the pushforwar of the Lebesgue measure on R + by ϕ Then µ is an infinite, σ-finite an T -invariant measure on X, an it satisfies n 0, j {0,, h n 1}, µl j n = 3 n

8 NEARLY FINITE HAON TRANSFORMATION 7 Aitional notations For each n 0, we enote by n the subset of X forme by the union of all the levels of tower n Note that for each n 0, n n+1, an that X = n 0 n For x n, note that j n x inicates the level of tower n to which x belongs We also efine a function t n on n, taking values in {1, 2, 3}, which inicates for each point whether it belongs to the first, the secon, or the thir subcolumn of tower n We thus have for x n an n / {n l : l 1} j n x if t n x = 1, 4 j n+1 x = j n x + h n if t n x = 2, j n x + 2h n + 1 if t n x = 3 In the case where n = n l for some l 1, we have to replace h n by h + h kl in the above formula In particular, we always have 5 j n+1 x j n x onsier two integers 0 n < m By construction, tower n is subivise into 3 m n subcolumns which appear as bues of h n consecutive levels in tower m: we call them occurrences of tower n insie tower m These occurrences are naturally orere, from bottom to top of tower m For a point x in tower n, the precise occurrence of tower n insie tower m to which x belongs is etermine by the sequence t n x, t n+1 x,, t m 1 x For example, x belongs to the last occurrence of tower n insie tower m if an oy if t n x = t n+1 x = = t m 1 x = 3 Remark 23 Observe that for each l 2 an each n l n n l 1, there is 0 or 1 spacer between two consecutive occurrences of tower n insie tower n l 23 Behaviour of µ-typical points Lemma 24 There exists a µ-conull subset X of X satisfying: for each x X, there exists an integer lx such that, for all l lx, for each n l 1 n n l l, x n but x is neither in the first hunre nor in the last hunre occurrences of tower n insie tower n l Proof If we consier x as a ranom point chosen accoring to the normalize µ-measure on n, then the ranom variables t n x, t n+1 x,, t m 1 x are ii an uniformly istribute in {1, 2, 3} Hence the probability that x belongs to some specifie occurrence of tower n insie tower m is 1/3 m n Since the series 1/3 l converges, by Borel antelli there exists a subset X n of full µ-measure insie n such that, for each x X n, there is oy a finite number of integers l such that x belongs to the first hunre or to the last hunre occurrences of tower n l l insie tower n l Setting X := X \ n \ X n, n we get a conull subset of X, an for each x X, there exists an integer lx such that, for all l lx, x 1 l, but x is neither in the first hunre nor in the last hunre occurrence of tower n l l insie tower n l If n l 1 n n l l, the first respectively last hunre occurrences of tower n insie tower n l are inclue in the first respectively last hunre occurrences of tower n l l insie tower n l, an this conclues the proof

9 8 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Remark 25 In particular, for each x X, if n > n lx, then x oes not belong to the first level of tower n An since 0 is in the first level of tower n for each n, we have 0 / X Remark 26 As n l 1 + l < n l l by 2, we may also assume that for each x X an each l lx, x is neither in the first hunre nor in the last hunre occurrences of tower n l 1 insie tower n l l 3 Invariant Raon measures for artesian powers of the nearly finite hacon transformation We fix a natural integer 1, an we stuy the action of the artesian power T on X Recall that a measure σ on X is a Raon measure if it is finite on each compact subset of X equivalently, if σ n < for each n In particular, a Raon measure on X is σ-finite but the converse is not true Our purpose is to escribe all Raon measures on X which are T -invariant an whose marginals are absolutely continuous with respect to µ 31 Basic facts about Raon measures on X We call n-box a subset of n which is a artesian prouct L j1 n L j n, where each L ji n is a level of the Rokhlin tower n A box is a subset which is an n-box for some n 0 The family of all boxes form a basis of compact clopen sets of the topology of X We consier the following ring of subsets of X R := {B X : n 0, B is a finite union of n-boxes} Proposition 31 Any finitely aitive functional σ : R R + can be extene to a unique measure on the Borel σ-algebra BX, which is Raon Proof Using Theorems F p 39 an A p 54 aratheoory s extension theorem in [6], we oy have to prove that, if R k k 1 is a ecreasing sequence in R such that lim k σr k > 0, then k R k But this is obvious since, uner this assumption, each R k is a compact nonempty set In particular, to efine a Raon measure σ on X, we oy have to efine σb for each box B, with the compatibility conition that, if B is an n-box for some n 0, then σb = B B σb, where the sum ranges over the 3 n + 1-boxes which are containe in B We call n-iagonal a Rokhlin tower for T which is of the form B, T B,, T r 1 B, where each T j B is an n-box, an which is maximal in the following sense: B has one projection which is the bottom level L 0 n of tower n, T r 1 B has one projection which is the top level L hn 1 n of tower n, an the projections of each T j B, 1 j r 2 are neither the bottom level nor the top level of tower n See Figure 2 Lemma 32 Set X 0 := {x = x 1,, x X : i, x i = 0} Let σ be a Raon measure on X Then σ is T -invariant if an oy if the following two conitions hol: 1 σ X 0 = 0 2 for each n, all the n-boxes lying on an n-iagonal always have the same measure

10 NEARLY FINITE HAON TRANSFORMATION 9 Figure 2 An n-iagonal insie n here = 2 Proof Assume first that σ is a T -invariant Raon measure on X Recalling that 0 has no preimage by T, we see that T 1 X0 =, whence σ X0 = 0 Moreover, since n-boxes on an n-iagonal are levels of a T -Rokhlin tower, the secon conition obviously hols Reciprocally, assume that the two conitions given in the statement of the lemma hol For each n, let Ω n be the subset of n constitute of all n-boxes of the form L j1 n L j n, where for each i j i 0 Then the secon conition implies that σ an T σ coincie on Ω n for each n But Ω n = X \ X0 n 0 On the other han, we have T σx0 = σ T 1 X0 = σ = 0 With the first conition we see that σ an T σ also coincie on X0, hence they are equal Definition 33 onvergence of Raon measures on X We say that a sequence σ k of Raon measures on X converges to the nonzero Raon measure σ if, for each n large enough so that σ n > 0, we have σ k n > 0 for all large enough k, for each n-box B, σ k B σ k n σb k σn Observe that, when a sequence of Raon measures converges in the above sense, then its limit is unique up to a multiplicative positive constant Observe also that the convergence is unchange if we multiply each measure σ k by a positive real number which may vary with k Remark 34 If the sequence σ k of Raon measures on X converges to the nonzero Raon measure σ, then for n such that σ n > 0 an for each m n, for

11 10 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE each m-box B n, we also have σ k B σ k n σb k σn onsequently, the above hols also when B R is inclue in n Inee, as n is a finite union of m-boxes, we have σ k n σ k m σn k σm Then we can write, for an m-box B n, σ k B σ k n = σ kb σ k m σb σ σ k m σ k n m k σm σn = σb σn Proposition 35 Let σ k be a sequence of Raon measures on X, an assume that there exists some n 0 satisfying σ k n > 0 for all large enough k, for each n > n, the sequence σ k n/σ k n is boune Then there is a subsequence k j an a nonzero Raon measure σ on X such that σ kj converges to σ Proof Multiplying each σ k by a positive real number if necessary, we may assume that for all large enough k, σ k n = 1 Then the secon assumption ensures that for each box B, the sequence σ k B k is boune By a stanar iagonal proceure, we can fin a subsequence k j such that for each box B, σ kj B has a limit which we enote by σb Then σ efines a finitely aitive functional on the ring R of finite unions of boxes, with values in R + By Proposition 31, σ can be extene to a Raon measure on BX, which is nonzero since σn = 1 An we obviously have the convergence of σ kj to σ Proposition 36 Let σ k an γ k be two sequences of Raon measures on X, an assume there exist two nonzero Raon measures σ an γ, an integer n 1 an a real number θ > 0 such that σ k σ, k γ k k γ, k, γ k σ k n n, for all large enough k epening on n, γ k n θσ k n Then γ σ Proof Let m n n, an let B be an m-box For all large enough k, we have by assumption γ k B γ k n σ kb θσ k n But by Remark 34, we have It follows that γ k B γ k n k γb γn, an σ kb σ k n k γb γn σb θσn k σb σ n

12 NEARLY FINITE HAON TRANSFORMATION 11 The above inequality extens to each B R containe in n, an then to each B BX containe in n In particular, if B n is Borel measurable an satisfies σb = 0, then we also have γb = 0 An since X = n n, this conclues the proof Remark 37 For each l 1, the efinition of the ring R is unchange if we consier oy the finite unions of n l -boxes, for some l l Hence in Propositions 35 an 36, it is enough for the conclusions to hol that the assumptions be verifie oy when n {n l : l l} 32 Dissipative case Lemma 38 For each x = x 1,, x X, for l > max{lx i : i = 1,, }, we have #{j 0 : T j x n l +1} = Proof If l > max{lx i : i = 1,, }, we know by Lemma 24 that each coorinate x i is in +1, but is not in the last occurrence of tower n l + 1 insie tower n l+1 Moreover by Remark 25, x i is not in the first level of tower n l + 1 The next occurrence of tower n l + 1 insie tower n l+1 appears after 0 or 1 spacer by Remark 23 As the height of tower n l + 1 is h +1, T h n l +1 x i is either in the same level of tower n l + 1 as x i, or in the level immeiately below Thus T h n l +1 x i +1 But the same applies to any l l, an we get that T h n l +1 x i is either in the same level of tower n l + 1 as x i, or in the level immeiately below Since these two levels are both inclue in +1 we get that T h n l +1 x i +1 Proposition 39 There is no Raon, T -invariant an totally issipative measure for which X is a conull set In particular, there is no Raon, T -invariant an totally issipative measure whose marginals are absolutely continuous with respect to µ Proof Suppose that σ is such a measure Let W be a wanering set for σ, with σ X \ j ZT j W = 0 As X is a conull set, we may assume that W X By the previous lemma, W = n W n, where W n := { x W : #{j 0 : T j x n} = } Hence there exists some n with σw n > 0 The ergoic ecomposition of σ writes σ = σx, W j Z δ T j x so we get σ n =, which contraicts the fact that σ is Raon

13 12 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE 33 Main result An obvious example of a T -invariant Raon measure on X is the prouct measure µ Another example is what we call a graph measure arising from powers of T : this is a measure σ of the form 6 σa 1 A = αµa 1 T e2 A 2 T e A, for some integers e 2,, e an some fixe positive real number α Such a measure is concentrate on the subset { x1,, x X : x i = T ei x 1 for all i = 2,, } Theorem 310 For each 1, the infinite measure preserving ynamical system X, µ, T is conservative ergoic Moreover, if σ is a nonzero, Raon, T -invariant an ergoic measure on X, such that 7 σ X \ X = 0, then there exists a partition of {1,, } into r subsets D 1,, D r, such that σ = σ D1 σ Dr, where σ Dj is a graph measure on X Dj arising from powers of T orollary 311 If σ is a nonzero, Raon, T -invariant measure on X, whose marginals are absolutely continuous with respect to µ, then σ ecomposes as a sum of countably many ergoic components, which are all of the form escribe in Theorem 310 To prove Theorem 310 in the case = 1, we even o not nee assumption 7 as we can show that µ is, up to a multiplicative constant, the oy T -invariant, Raon measure on X the proof is the same as for the hacon infinite transformation, see Proposition 24 in [7] We also note that, if we have prove the secon part of the theorem for some 2, then the first one follows immeiately Inee, if µ were not ergoic, then almost all its ergoic components woul satisfy 7, hence woul be a prouct of graph measures ifferent from µ But this woul mean that for µ -almost all x X, there exist at least two coorinates of x lying on the same T -orbit, which of course is absur Hence µ is ergoic, an by Proposition 39, it is conservative The remainer of this paper is evote to the proof by inuction of the secon part of Theorem 310 So we now assume that for some 2, the statement is true up to 1 We consier a nonzero, Raon, T -invariant an ergoic measure σ on X, satisfying 7 We will show that either σ is a graph measure arising from powers of T, or it can be ecompose into a prouct of two measures σ 1 σ 2, σ i being a T i -invariant Raon measure on X i for some 1 i <, = In this latter case we can apply the inuction hypothesis to each σ i, which yiels the announce result 34 hoice of a σ-typical point By Proposition 39, the system X, σ, T is conservative ergoic By Hopf s ergoic theorem, if B X with 0 < σ <, we have for σ-almost every point x = x 1,, x X j J 8 1 BT j x j J 1 T j x σb J σ,

14 NEARLY FINITE HAON TRANSFORMATION 13 where the sums in the above expression range over a set J of consecutive integers containing 0 We say that x X is typical if, for all n large enough so that σ n > 0, Property 8 hols whenever B is an n-box an is n We know that σ-almost every x X is typical Therefore, there exists a point x = x 1,, x such that 9 For each j Z, T j x is typical Since there are oy countably many boxes, we may also assume that 10 For each box B, x B = σb > 0 Moreover, by 7, we can further assume that 11 i = 1,,, x i X From now on, we fix a point x = x 1,, x satisfying the above assumptions 9, 10 an 11 We will erive properties of σ from the observations mae on the orbit of this point x By an interval, we mean in this paper a finite set of consecutive integers We will nee the following key notion in our argument Definition 312 We call n-crossing a maximal interval J Z with the following properties: T j x n for each j J, for each 1 i, j t n T j x i is constant on J An n-crossing is sai to be synchronize if t n T j x 1 = = t n T j x for each j in this n-crossing Note that an n-crossing has at most h n elements If j is the smallest respectively the largest element of an n-crossing, then there exists 1 i such that T j x i is in the first respectively the last level of tower n Observe also that when j runs over an n-crossing, T j x successively passes through each n-box of some n-iagonal 35 haracterizations of graph measures arising from powers of T Lemma 313 The following assertions are equivalent: i σ is a graph measure arising from powers of T ; ii e 2,, e Z: x i = T ei x 1 for each i = 2,, ; iii n : n n, t n x 1 = = t n x ; iv j, n : n n, t n T j x 1 = = t n T j x Proof Let us first prove that i = ii If σ is a graph measure arising from powers of T, then there exist a positive real number α an integers e 2,, e such that for all measurable subsets A 1,, A of X, 6 hols Observe that, if l is large enough so that h kl > max{ e 2,, e }, then for each i = 2,, an each j, j {0,, h 1}, L j n l T ei L j n l = { L j n l if j = j + e i, otherwise It follows that the oy n l -boxes that may be charge by σ are of the form L j1 n l L j1+e2 n l L j1+e n l for some j 1 By assumption 10, it follows that for each

15 14 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE i = 2,,, j x i = j x 1 + e i Since this is true for all large enough l, this in turn implies that for each i = 2,,, x i = T ei x 1 onversely, if ii hols, the same argument shows that if l is large enough so that h kl > max{ e 2,, e }, then the oy n l -boxes that can contain x are the n l -boxes of the form L j1 n l Ln j1+e2 l L j1+e n l for some j 1 Note that the n l -boxes of this form constitute an n l -iagonal, which we enote by D But ii is also vali for each T j x, j Z hence the argument also applies to each T j x Thus, if B is an n l -box which is not on D, then T j x / B for each j Z Now, remembering that x is typical for σ, we have for each n l -box B = L j1 σb σ n l = lim k k j k 1 BT j x k j k 1 T j x n l L j The above limit is 0 if B is not on D Moreover, note that each time the orbit of x passes through n l, T j x successively passes through each n l -box on D Hence if B is on D, the limit is equal to the inverse of the number of n l boxes on D In particular the limit is proportional to µ L j1 n l T e2 L j1 n l T e L j The coefficient of proportionality epens a priori on l, but since each n l -box is a union of isjoint n l+1 -boxes, we see that in fact this coefficient oes not epen on l Finally, this gives 6 in the case of an n l -box for each large enough l, an this is enough to conclue that 6 hols for each measurable set of the form A 1 A We have so far prove the equivalence of i an ii Now let us turn to the proof of ii = iii Since x 1 X, we have j n x 1 an h n j n x 1 as n If ii hols, we then have j n x i = j n x 1 + e i for each i = 1,, an each n large enough so that min{j n x 1, h n j n x 1 } > max{ e 2,, e } But then for such an n we also have t n x i = t n x 1 for each i = 1,, The implication iii = iv is obvious Assume now that iv hols with j = 0 ie that, in fact, iii hols For i = 2,,, we then have by an easy inuction that j n x i j n x 1 = j n x i j n x 1 for each n n Setting e i := j n x i j n x 1 for i = 2,,, we get that x i = T ei x 1 an we have ii Now if iv hols with some j Z, we get ii for T j x, which is clearly equivalent to ii for x Thus we have prove that iv = ii an this conclues the proof of the lemma For the remainer of the paper, we also fix a real number 0 < η < 1, small enough so that η < In particular we will nee the inequality 1 η2 > 1/2 Definition 314 For each n, let I n := { h n /2,, h n /2 + h n 1} be the interval of length h n an centere at 0 For each n 0, we call substantial n- crossing any n-crossing whose intersection with I n countains at least ηh n elements Lemma 315 If n = n l 1 for some large enough l, then substantial n-crossings cover a proportion at least 1 + 2η of I n In particular, there exists at least one substantial n-crossing Moreover, if all substantial n-crossings are synchronize, then each substantial n-crossing is of size at least 1 + 2ηh n, an there are at most two of them Proof Let us start by consiering the case of an integer n which is of the form n = n l 1 for some l > max i lx i We also assume that l is large enough so that 1 3 kl 1 < η 2

16 NEARLY FINITE HAON TRANSFORMATION 15 We set n := n l 1 kl 1, an we observe that the above assumption ensures that h n < η h n We know by Lemma 24 that x n l 1 = n, an that the interval { 100h l,, 100h l} is containe in a single n l -crossing A fortiori, I n is containe in a single n l - crossing Therefore, if a coorinate T j x i reaches the top of tower n an comes back to n on the interval I n, then the two passages in n are separate by at most h n + 1 Moreover, this can happen at most once on the interval I n for each i It follows that the set of integers j I n such that T j x / n is constitute of at most pieces, an its carinality is boune above by ηh n by 12 Then there exist at most + 1 n-crossings intersecting I n, an they cover a proportion at least 1 η of I n Now the proportion of I n covere by n-crossings which are not substantial is less than + 1η, hence the proportion of I n covere by substantial n-crossings is at least 1 + 2η This proves the first part of the lemma Let us assume now that all substantial n-crossings are synchronize If we have oy one substantial n-crossing, then this n-crossing is of size at least 1 +2ηh n, an we have for j in this n-crossing 13 j n T j x i1 j n T j x i2 + 2ηh n If we have at least two substantial n-crossings, note that between two of them, there is at least one coorinate passing through the top of tower n, an for which t n has increase by 1 mo 3 Since the t n T j x i, i = 1,, are suppose to be equal on each substantial n-crossing, we euce that each coorinate passes through the top of tower n between two substantial n-crossings As this happens at most once for each coorinate on I n, we see that there are at most two substantial n-crossings Finally, from the first part of the lemma it follows that two consecutive substantial n-crossings are separate by at most + 2ηh n points We euce that, on any substantial n-crossing, 13 hols, hence each substantial n-crossing is of size at least 1 + 2ηh n Remark 316 The preceing lemma extens easily to the case when n l 1 n n l l Inee, when n l n n l l the proof is even simpler, as two successive passages in n are now separate by at most one Proposition 317 The measure σ is a graph measure arising from powers of T if an oy if for each large enough n, all substantial n-crossings are synchronize Proof First assume that σ is a graph measure arising from powers of T Then by Lemma 313, we know that there exists e 2,, e Z such that x i = T ei x 1 for each i = 2,, Take n large enough so that max{ e 2,, e } < ηh n Let J be a substantial n-crossing In particular the size of J is at least ηh n Hence there exists j J such that ηh n j n T j x 1 1 ηh n We euce that j n T j x i = j n T j x 1 + e i for each i = 2,, But we also have ηh n j n+1 T j x 1 1 ηh n an this ensures that j n+1 T j x i = j n+1 T j x 1 +e i By 4, the equality j n+1 T j x i j n+1 T j x 1 = j n T j x i j n T j x 1 implies t n T j x i = t n T j x 1 Finally, as j t n T j x i is constant on the n-crossing J, we see that J is synchronize

17 16 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE onversely, assume that there exists n such that for each n n, all substantial n-crossings are synchronize Without loss of generality, we may assume that n is of the form n l 1, for some l large enough to apply Lemma 315 Then we know that there exists at least one substantial n-crossing J n, of size at least 1 + 2ηh n For j J n an for each i = 2,,, j n T j x i j n T j x 1 + 2ηh n Let us prove by inuction that for each n n, there exists a substantial n-crossing J n, of size at least 1 + 2ηh n, an containing J n We alreay know that this property is true for n Assume it is true up to n for some n n Then, the n-crossing J n extens to a unique n + 1-crossing J n+1 As J n intersects I n an is of size at most h n, J n I n+1 It follows that J n+1 I n+1 J n 1 + 2ηh n ηh n+1, which proves that J n+1 is a substantial n + 1-crossing Moreover, since the size of J n is at least 1 + 2ηh n, we have for j J n an each i = 2,, j n T j x i j n T j x 1 + 2ηh n But J n is synchronize, hence by 4, we have for j J n j n+1 T j x i j n+1 T j x 1 = j n T j x i j n T j x 1 + 2ηh n + 2ηh n+1 This equality extens to j J n+1 since the ifference is constant on an n + 1- crossing This proves that the size of J n+1 is at least 1 + 2ηh n+1 Now if we take any j J n, we have j J n for each n n, an since we assume that each substantial n-crossing is synchronize, we have t n T j x 1 = = t n T j x, ie we have iv of Lemma 313 This proves that σ is a graph measure arising from powers of T Remark 318 In the preceing proof, the inuction provies in fact a stronger inequality for the sizes of the substantial n-crossings J n : J n h n + 2ηh n 4 ombinatorics of some sets of integers The purpose of this section is to establish Proposition 41 on the combinatorics of the set of integers j such that T j x n for a given large n Proposition 41 There exist constants K 1 > 0 an K 2 > 0 such that, for any large enough integer l, an any integer 1 c h, the following hols: if I Z is an interval containe in an n l+l -crossing for some l 1, an if the length of I is at least ηh +l 1, then the proportion of integers j I such that T j x n l is at least 1 η 2l ; among all the integers j I such that T j x n l, the proportion of those belonging to an n l -crossing of size c is boune above by K 1 c + K 2 h 3 l For this we will introuce a hierarchy of more an more complex subsets of Z, prove by inuction some combinatorial results on abstract sets in this hierarchy, an finally show how to apply these results in the particular case we are intereste in

18 NEARLY FINITE HAON TRANSFORMATION 17 Figure 3 A set F of orer 2 insie an interval I 41 A hierarchy of subsets of Z This part of the argument is completely abstract an inepenent of the rest of the paper, but we keep the notations an integer, 2 an η a positive real number between 0 an 1 We set K 1 := η 1 η We fix two sequences of positive integers c l l 1 an s l l 1, satisfying 14 l 1, an 15 l 1, s l < 1 η c l η + 1 η, c l c l+1 < η K 1 Let F Z, an let I Z be an interval We call piece of F I any maximal interval inclue in F I, an we call hole of F I any maximal interval inclue in I \ F Thus, I is the isjoint union of the pieces an the holes of F I, which alternate We say that F is of orer 1 insie the interval I if each hole of F I is of size s 1, two consecutive holes of F I are always separate by a piece of size at least c 1 Recursively, we say that F is of orer l 2 insie the interval I if there exists a subset F Z such that F F, each hole of F I is of size s l, two consecutive holes of F I are always separate by a piece of size at least c l, for each piece I of F I, F is of orer l 1 insie I See Figure 3 Note that, if F is of orer l insie the interval I, then F is of orer l insie each subinterval J I Lemma 42 Let F 1,, F be subsets of Z, an let I Z be an interval Assume that for some l 1, F i is of orer l insie I for each i = 1,,, an that the size of I is at least ηc l Set F := i=1 F i Then

19 18 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE 16 the ensity of F insie I satisfies F I I 1 η 2l, for a given integer c, 1 c < h 1, the proportion of integers in F I lying in pieces of F I with size c is boune above by c K 1 + c 1 1 c 1 c 2 1 η c l 1 1 c l 1 η 2l Proof Let us first establish the result for l = 1 We assume that each F i is of orer 1 insie I, an that I ηc 1 For each i = 1,,, let k i be the number of holes of F i I Then by efinition of orer 1, there are at least k i 1 pieces of F i I with size at least c 1, whence c 1 k i 1 I, an k i I /c Since each hole of F i I has size s 1, we euce that the carinality of I \ F i is boune by s 1 I /c This yiels by the inequality 1 1 I η c 1 an 14: I \ F s 1 I /c s 1 I /c 1 η I η We thus get F I / I 1 η 1 η 2, which is the first point Moreover, the number k of holes of F I satisfies k k k I /c 1 +, whence the number m of pieces of F I satisfies m I /c I /c I /c 1 η It follows that the number r of points of F I lying in a piece of size c satisfies r mc I c η c 1 As we alreay know that F I 1 η I, we get by efinition of K η r F I 1 η c c 1 = K 1 c c 1, which establishes the secon point for l = 1 Now we assume by inuction that the result is true up to l 1 for some l 2 an we consier a family F i 1 i of subsets of Z, which are of orer l insie an interval I satisfying I ηc l By efinition of orer l, for each i there exists a subset F i Z satisfying F i F i, each hole of F i I is of size s l, two consecutive holes of F i I are always separate by a piece of size at least c l, for each piece I of F i I, F i is of orer l 1 insie I Since I ηc l, the argument eveloppe for orer 1 applies for F := i=1 F i with c l 1, c l, s l in place of c, c 1, s 1 We thus get 17 F I 1 η I,

20 NEARLY FINITE HAON TRANSFORMATION 19 an enoting by r the number of points of F I lying in pieces of F I of size < c l 1, we have using also r F I K c l 1 1 < η c l Let G stan for the union of all pieces of F I of size c l 1 The above inequality can be rewritten as 19 G F > 1 η I Let J be an arbitrary piece of G Since for each i, F i is of orer l 1 insie J, an by efinition of G, J c l 1 ηc l 1, the inuction hypothesis gives F J J 1 η 2l 2 Summing over all pieces of G we get, using also 19 an F I F G 1 η 2l 2 G 1 η 2l I, which is the first point at orer l Moreover, if r J enotes the number of points of F J lying in pieces of F J of size smaller than c, then r J F J K 1 c + c 1 1 c 1 c 2 1 η c l 2 1 c l 1 1 η 2l 2 Now let us enote by r the number of points of F I lying in pieces of F I of size smaller than c The contribution to r of points in G is J r J where the sum ranges over all pieces J of G, an by the previous inequality, it satisfies J r J K 1 c c 1 + c 1 c η c l 2 c l η 2l 2 K 1 c c 1 + c 1 c η c l 2 c l η 2l 2 F G F I The contribution to r of points in F \ G is clearly at most F \ G, which can be boune above as follows F I \ G F I \ G because F F = r by efinition of G an r K 1 c l 1 c l F I by 18 c l 1 K 1 I c l K 1 c l 1 c l F I 1 η 2l by 20 Summing the two contributions an using the above inequalities, we get c r K 1 + c 1 1 c 1 c 2 1 η c l 2 1 c l 1 1 η 2l 2 + c l 1 1 c l 1 η 2l F I, which is the secon point at orer l

21 20 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE 42 Application to the structure of n-crossings We want now to apply the preceing lemma in orer to obtain some statistical results on long range of successive n-crossings We fix some integer l, large enough to satisfy some conitions to be precise later, an we set k := kl We efine the sequences c l l 1 an s l l 1 as follows c 1 := h, s 1 := h k + 1, in general, c l := h +l 1, an s l := h +l 1 kl+l Using the fact that we always have h n /h n+1 < 1/3, we observe that for each l 1, with k := kl + l 1, s l = h n l+l 1 k + 1 < 2 h n l+l 1 k < 2 c l h +l 1 h +l 1 3 k 2 3 k Hence 14 is satisfie if l is large enough The fact that 15 hols if l is large enough follows from the following easy consequence of 2: h < h n l+1 l + 1 < 1 h +1 h +1 3l+1 l 0 We can therefore assume that l is large enough so that both 14 an 15 hol We want to apply Lemma 42 to the subsets F i i = 1,, efine by F i := { j Z : T j x i } Let I Z be an interval, n 1 an i {1,, } We say that x i climbs into tower n along I if for each j I, T j x i n, an there is no j I such that j + 1 I, T j x i L hn 1 n an T j+1 x i L 0 n Note that I is inclue in an n-crossing if an oy if each coorinate x i climbs into tower n along I Lemma 43 For each interval I Z an each i {1,, }, if x i climbs into tower n l+l along I, then F i is of orer l insie I Proof By construction of the Nearly Finite hacon Transformation, two successive occurrences of tower n l insie tower n l+1 are separate either by h k or by h k + 1 spacers Hence, if x i climbs into tower n l+1 along I, F i is of orer 1 insie I This proves the lemma in the case l = 1 Assume that the statement of the lemma is true up to l 1 for some l 2 We consier F i := { j Z : T j x i +l 1 } We clearly have F i F i Two successive occurrences of tower n l+l 1 insie tower n l+l are separate either by h +l 1 k or by h +l 1 k + 1 spacers, where k is etermine by l k l + l 1 < l k+1 Hence, if x i climbs into tower n l+l along I, each hole of F i I is of size h +l 1 k + 1 = s l, an two consecutive holes of F i I are separate by a piece of F i I of size h n l+l 1 = c l Moreover, along each piece of F i I, x i climbs into tower n l+l 1 Therefore the property for l 1 ensures that F i is of orer l 1 insie each piece of F i I It follows that F i is of orer l insie I, an the lemma is prove by inuction

22 NEARLY FINITE HAON TRANSFORMATION 21 Proof of Proposition 41 With the subsets F i efine as above, we see that F := } F i = {j Z : T j x 1 i Observe that the pieces of F are precisely the n l -crossings Assume that the interval I Z is inclue in an n l+l -crossing for some l 1 remember that this is equivalent to: each coorinate x i climbs into tower n l+l along I Then, putting together Lemma 42 an Lemma 43, an provie that the length of I be at least ηc l = ηh +l 1, we get: the proportion of j I such that T j x n l is at least 1 η 2l, for each 1 c h, the proportion of j F I belonging to an n l -crossing of size c is boune above by c 21 K 1 + h n l 1 h h +1 1 η h n l+l 2 1 h +l 1 1 η 2l Let us estimate the general term of the above sum, using the inequality h /h +1 < 1/3 l+1, an the assumption 1 η 2 > 1/2 h +l 2 h +l η 2l < 1 3 l+l η 2l = 1 3 l 1 1 < 1 3 l 1 31 η 2 l l 2 3 c It follows that there exist a constant K 2 such that 21 K 1 + K 2 h 3 l 43 Measure of the ege of n For each n 0, we say that an n-box L j1 n L j n is on the ege of n if there exists i {1,, } such that j i = 0 or j i = h n 1 We enote by n the union of all such n-boxes As a first application of Proposition 41, we have the following result orollary 44 δn := σ n σ n n 0 sketch of proof This is a irect consequence of the following facts: Since x is typical for σ, the quotient δn can be estimate by the ratio j I 1 n T j x j I 1 n T j x for a large interval I containing 0 The subset of j Z such that T j x n is partitione into n-crossings, an in each n-crossing J there are exactly two integers j the minimum an the maximum of J such that T j x n By Proposition 41, most n-crossings are large if n is large

23 22 ÉLISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE 5 onvergence of sequences of empirical measures For each finite subset J Z, we enote by γ J the empirical measure γ J := j I δ T j x The valiity of Property 8 whenever B is an n-box an is n remember that x has been chosen as a typical point means that, if J n is a sequence of intervals containing 0, with J n, then we have the convergence γ J n σ n n Our purpose in this section is to exten this convergence to the case where the intervals J n o not necessarily contain 0, but are not too far from 0 We will also treat the case where the subsets J n are no longer intervals, but union of intervals with a sufficiently regular structure We fix a real number ε > 0, small enough so that 1 ε 2 > 1 η Then we consier an integer c 1, large enough so that c 1 c > 1 ε In Sections 51 an 52, we consier a fixe integer l, large enough so that the result of Proposition 41 hols We can also assume that 22 K 1 c h + K 2 3 l < ε We are going to estimate the behaviour of empirical measures with respect to n l - boxes The following lemmas are evote to the control of γ I = 1 T j x for particular intervals I j I 51 onsecutive n-intervals For n 1, we call n-interval any interval I = {j, j + 1,, j + h n 1} of length h n an such that j is a multiple of h n The secon conition is completely artificial, it is oy useful to efine canonically a cutting of any interval into intervals of length h n Lemma 51 Let p 1 be the smallest integer such that 3 p1 > an p 1 > There exists a constant 0 < θ 1 < 1 epening oy on η an for which the following hols Let l > l + 1, an let n be such that n l 1 kl 1 + p 1 n < n l Whenever I 1 an I 2 are two consecutive n-intervals, both containe in the same n l -crossing, we have θ 1 γ I1 < γi2 < 1 θ 1 γ I1 Proof We ivie the proof into two cases ase 1: n l n n l Set j 1 := min I 1 an j 2 := min I 2 = j 1 + h n Proposition 41 applies to I 1, an this ensures that, among the γ I1 integers j such that T j1+j x n l, a proportion at least 1 ε by 22 belong to an n l -crossing of size at least c Then, among those belonging to an n l -crossing of size at least c, a proportion at least c 1 c are not the minimum of their n l -crossing By the choice of ε an c, we get the partial following result: a proportion at least 1 η of integers j {0,, h n 1} are such that, for each i = 1,,, T j1+j x i, but T j1+j x i is not in the bottom level of tower n l Let us consier such an integer j Observe that, since I 2 is in the same n l -crossing as I 1, the coorinate T j1+j x i