An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms

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An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms

Mostapha Benhenda

To cite this version:

Mostapha Benhenda. An uncountable family of pairwise non-Kakutani equivalent smooth diffeomor-

phisms. 2013. �hal-00795683�

An uncountable family of pairwise non-Kakutani equivalent smooth di ff eomorphisms

Mostapha Benhenda February 27, 2013

Abstract

We construct an uncountable family of smooth ergodic zero-entropy diffeo- morphisms that are pairwise non-Kakutani equivalent, on any smooth compact connected manifold of dimension greater than two, on which there exists an ef- fective smooth circle action preserving a positive smooth volume. To that end, we first construct a smooth ergodic zero-entropy and non-Loosely Bernoulli diffeo- morphism, by suitably modifying a smooth construction by Anosov and Katok. A construction of this kind was announced by Katok in 1977 and 1980 [8, p.141], [9, p.293].

1 Introduction

An important question on the interface between smooth dynamics and abstract ergodic theory is: what ergodic properties, if any, are imposed upon a dynamical system by the fact that it should be smooth? [11, p.89] [15, p.232] Only one restriction is known, which is that the entropy must be finite, because the dimension of the manifold is finite.

The core of the problem is when the invariant measure is smooth, and the manifold is compact (otherwise, see [10], [2]). No other restriction has been found yet, but exam- ples have been provided: Brin, Feldman and Katok [5] showed that any compact man- ifold of dimension greater than one admits a smooth Bernoulli di ff eomorphism. Ka- tok [9] and Rudolph [14] gave examples of smooth non-Bernoulli K-di ff eomorphisms.

Ratner [13] showed that the Cartesian square of the horocycle flow is non-Loosely Bernoulli, thus giving an algebraic (hence analytic) example.

In this paper, we construct an uncountable family of smooth ergodic zero-entropy di ff eomorphisms that are pairwise non-Kakutani equivalent, on any smooth compact connected manifold of dimension greater than two, on which there exists an e ff ective smooth circle action preserving a positive smooth volume.

Our construction originates from an example given by Feldman [7], of an ergodic transformation of zero entropy that is non-Loosely Bernoulli (i.e. non-Kakutani equiv- alent to an irrational circle rotation). Ornstein, Rudolph and Weiss [12] extended Feld- man’s construction to obtain an uncountable family of ergodic zero-entropy transfor- mations that are pairwise non-Kakutani equivalent. Their transformations are discon- tinuous, they are obtained by "cutting and stacking". The construction given in this

Contact: mostaphabenhenda@gmail.com. I would like to thank Jean-Paul Thouvenot for communicat-

paper corresponds to a smooth version of the construction given by Ornstein, Rudolph and Weiss. To obtain it, we suitably modify a smooth construction by Anosov and Katok [1].

In particular, we also obtain a smooth ergodic non-Loosely Bernoulli di ff eomor- phism on any smooth compact connected manifold of dimension greater than two, on which there exists an e ff ective smooth circle action preserving a positive smooth volume. This allows to generalize Katok’s result on the existence of a smooth non- Bernoulli K-di ff eomorphism to every manifold of dimension greater than 4 [9, p.293].

This smooth version of Feldman’s transformation has been announced by Katok [8, p.141], [9, p.293]. However, up to now, it has not been written.

In this paper, we show the theorems:

Theorem 1.1. Let M be a smooth compact connected manifold of dimension d ¥ 2, on which there exists an e ff ective smooth circle action p S t q t P

”¹

preserving a positive smooth measure µ. There exists an ergodic non-Loosely Bernoulli di ff eomorphism T P Di ff ⁸ p M, µ q .

Theorem 1.2. Let M be a smooth compact connected manifold of dimension d ¥ 2, on which there exists an e ff ective smooth circle action p S _t q t P

”¹

preserving a positive smooth measure µ. There exists an uncountable family of ergodic di ff eomorphisms T u P Di ff ⁸ p M, µ q such that if u v, then T u and T v are not Kakutani-equivalent.

First, in sections 2 and 3, we show theorem 1.1, which gives a smooth version of Feldman’s transformation. In section 4, we show theorem 1.2, by adapting the proof of Ornstein, Rudolph and Weiss [12, pp. 84-95].

1.1 Main ideas

We recall some definitions found in [7] (we slightly modify the terminology some- times). We fix an automorphism T of p M, B, µ q . Let I be an alphabet of size N and P t c i , i P I u a finite measurable partition indexed by this alphabet. For any x P M and integer n P Ž, and for any i 0, ..., n 1, let a i P I such that T ⁱ p x q P c a

_i

. The n-trajectory of x by T with respect to P is the word of length n on the alphabet I given by: apT , n, xq a 0 ...a _{n 1} , such that , T ⁱ pxq P c a

i

(we do not mention the partition P in the notation when it is fixed once for all). The length of this word is denoted

| a p T, n, x q| . The trajectory of x (by T with respect to P) is the infinite word a ₀ a ₁ ...

where T ⁱ p x q P c _a

_i

for any i P Ž.

Let α a ₁ ...a _n and β b ₁ ...b _m be two words on the alphabet I. A match π between α and β is an injective, order-preserving partial function π : t 1, ..., n u Ñ t 1, ..., m u such that for any j in its domain of definition D p π q € t 1, ..., n u , b _{π p j q} a j . The cardinal of D p π q is denoted | D p π q| . Let R p π q π p D p π qq . π is denoted:

π : D p π q € t 1, ..., n u Ñ R p π q € t 1, ..., m u

Let D p π q t j ₁ , ..., j _|

_D

_{p π q|} u . We identify D p π q with the subword a _j

₁

...a _j

_|Dpπq|

of α, and R p π q with the subword b _{π p j}

₁

_q ...b _{π p j}

_|Dpπq|

_q of β.

The fit of π, fit p π q is given by:

fit p π q | D p π q|

1

2 p m n q

Let also

π ¹ : R p π q € t 1, ..., mu Ñ D p π q € t 1, ..., nu

such that π ¹ p π p i qq i. Since | R p π q| | D p π q| , then fit p π ¹ q fit p π q . The distance f ¯ p α, β q between α and β is:

f ¯ p α, β q 1 max t fit p π q , π : α Ñ β match u Since fit p π ¹ q fit p π q for any match π, then ¯ f p α, β q f ¯ p β, α q .

Moreover, if α, β and γ are words of equal length, ¯ f p α, γ q ¤ f ¯ p α, β q f ¯ p β, γ q . A characterisation of Loosely Bernoulliness in the case of zero entropy is given in [7, p.22]. In this paper, we rather give a definition of non-Loosely Bernoulliness (nLB) in the case of zero entropy, because we want to obtain this property.

Definition 1.3. Suppose T has no entropy. T is non-Loosely Bernoulli (nLB) if there exists ¡ 0 and a finite partition P such that, for an infinity of integers n, and for any A P B , if µ pAq ¥ 1 , there exists x, y P A, f ¯ papT, n, xq , apT, n, yqq ¥ .

Our construction is inspired by the example given by Feldman [7] of a transforma- tion T ₀ that is ergodic, of zero entropy but non-Loosely Bernoulli. His transformation is not smooth (not even continuous) and is carried on r 0, 1 s . He constructs words (of length N p n q at the n ^th step) by induction. They are defined by a _0,i a _i P I for i 0, ..., N p 0 q 1, and for n ¥ 0 and i 0, ..., N p n 1 q 1:

a _{n 1,i}

a ^N _n,1 ^{p n q}

^{2pi 1q}

...a ^N _n,N ^{p n} _p ^q _n

^2pi

_q

^1q

^{N p n q}

^{2pNpn 1qpi 1qq}

T 0 is constructed so that, up to minor details, the Npnq -trajectory of points in r 0, 1 s are, with equal Lebesgue measure, given by the a n,i , i i, ..., Npnq . Under suitable assumptions, this property implies that T ₀ is nLB.

To get a construction looking like Feldman’s, but smooth, we rely on three obser- vations: first, the ¯ f -distance is quite flexible: the fit of a match is a ratio of two lengths, and therefore, the addition of unknown letters into a word does not sensibly a ff ect its fit with another word, if the total length of one of the words (or both) is taken su ffi ciently large. This property allows to approximate Feldman’s map by smooth maps.

Second, we rely on a phenomenon of "quasi-concatenation" of finite trajectories:

for example, let q n ¡ 0, let ζ tr 0, 1 sr i { q n , p i 1 q{ q n r , 0 ¤ i ¤ q n 1 u the partition of r 0, 1 s ” ¹ , S

1

qn

the rotation of angle _q ¹

n

of r 0, 1 s ” ¹ . Let w be the q n -trajectory of 0 with respect to S

¹

qn

and ζ , and let q n 1 ¡ 0 be an integer that is strictly divided by q ² _n . Let p n 1 {q n 1 1 {q n 1 {q n 1 , and σ be the circular permutation on words defined by: σ : a ₁ a ₂ ...a _p ÞÑ a ₂ ...a _p a ₁ , where a _i , i 1, ..., p, are letters of a word of length p.

The q _{n 1} -trajectory of 0 by S

1 qn

is:

w

qn 1 q2

n

p σ p w qq

^qnq2n1

p σ ^q

ⁿ

¹ p w qq

^qnq2n1

We assume that q ² _n divides q n 1 , and not simply that q n divides q n 1 , because the map σ is applied every q n 1 { q n iterations, and for convenience, we prefer not to cut a word in the middle.

If q n 1 { q n is su ffi ciently large, we can neglect the e ff ect of the circular permutation σ on this trajectory, which fit becomes close to the fit of w

qn 1

qn

. This phenomenon is

The third observation allows to concatenate different words (the second observation only allows to concatenate the same word). It consists in introducing a smooth "quasi- permutation" that allows to permute "tracks" on which rely the "trajectories" of points by our transformation T . By using quasi-permutations on separated tracks, we can obtain di ff erent trajectories, and thus obtain nLB. This method is possible because the manifold M has dimension greater than two.

This technique of taking "di ff erent tracks" is a novelty with respect to the origi- nal Anosov-Katok method [1], which does not use dimension two as fully as we do. In their method, they only use one single "track". Basically, most of their construction can be carried on a circle. They need dimension two only when they take the limit in the construction. This approach complicates the coexistence of di ff erent trajectories on the same manifold: indeed, in their method, each trajectory is approximated by periodic trajectories. At step n 1, we need that the rotation S

p1

n 1 q1n 1

of the annulus acts on a hori- zontal partition like a permutation having Npn 1 q q n 1 {q ¹ _{n 1} cycles, each of length q ¹ _{n 1} . But the main problem is that the cycles are too closely intertwined (figure 1).

This does not allow the convergence of the di ff eomorphism T n 1 B _n ¹ ₁ S

p1 n 1 q1

n 1

B n 1 : the norm } B n 1 } will be of order q n 1 at least, whereas in order to get a smooth map at the limit, we need that the series °

n } B

_{n 1}

}

q

¹_{n 1}

converges (it is a consequence of a gen- eralized mean value theorem). In our construction, we put cycles vertically (figure 2), so that we do not get this problem.

However, we still rely on the core ideas of the Anosov-Katok method: we obtain the smooth di ff eomorphism T as the C ⁸ -limit of a sequence T n B _n ¹ S

^pn

qn

B n of periodic di ff eomorphisms, with B n A n A n 1 ...A 1 , A n 1 S

¹

qn

S

¹

qn

A n 1 and q n divides q n 1 . Convergence in the C ⁸ -norm is possible because T n 1 is taken very close to T n : q n 1 is taken large, so that the distance between S

pn 1

qn 1

and S

^pn

qn

is small with respect to the norm of the conjugacy } B _n 1 } , which norm is related to scale of the smallest quasi-permutation at step n 1 (each quasi-permutation has its own scale, di ff erent of others, to allow nLB).

Moreover, in order to get nLB, T n 1 is also taken very close to T n : indeed, this closeness implies that T _n ⁱ ₁ does not significantly di ff er with T _n ⁱ for i q n , so that both transformations give similar i-trajectories. However, these two maps di ff er when i ¡¡ q n (typically, when i q n 1 { q n ). For example, T n is q n -periodic but not T n 1 . This closeness allows approaching Feldman’s construction: Feldman’s maps T n and T n 1 always have the same N p n q -trajectories, these two maps exactly coincide on in- creasingly larger sets. In our construction, for most points, there are Npn 1q different kinds of q n 1 -trajectories, obtained by concatenating q n -trajectories in different ways.

Up to a circular permutation of letters, and up to other minor modifications, these q n 1 - trajectories are, for i 0, ..., N p n 1 q 1:

a _{n 1,i}

a

qn 1 Npnqqn,i qn

n,0 a

qn 1 Npnqqn,i qn

n,1 ... a

qn 1 Npnqqn,i qn

n,N p n q 1

q

n,i

(1) The parameters q n,i are suitably chosen to get nLB: an important characteristic of this choice is that q n    q n,i    q n,i 1    ...    q n 1 .

In section 2, we construct T n on r 0, 1 s ”. In section 3, we show that the limit T

is smooth, nLB and ergodic. In subsections 3.3.2 and 3.3.3, we extend the construction

to the cases of r 0, 1 s ^{d 1} ” and more general manifold M. In section 4, we gener-

Figure 1: If we take a horizontal partition of r 0, 1 s ” ¹ , the three cycles are too intertwined. This is an obstacle for the convergence of T n B n ¹ S

p1n

q1n

B n towards a smooth map.

alize the construction to an uncountable family of pairwise non-Kakutani equivalent di ff eomorphisms. Various figures illustrate the construction.

In all the paper, f denotes an explicit function of its variables. The expression of this function can vary from one estimate to the other, but we still denote all these estimating functions in the same way.

2 Construction of the transformation T _n

2.1 Smooth quasi-permutations

We introduce smooth quasi-permutations, which are the main tool for the construction of the map T .

Proposition 2.1. Let a   b and c   d be real numbers such that d c   b a.

Let a   a ¹   pa bq{2, b ¹ a b a ¹ , 0     minppa ¹ aq{2, pd cq{2q. Let

τ _u be the translation of vector pu, 0 q. There exists a smooth measure-preserving map

φ pa, b, c, d, a ¹ , q : ra, bs rc, ds and a fixed numerical function f such that (see figure

3):

Figure 2: If we take a vertical partition of r 0, 1 s ” ¹ , having to deal with three cycles is no longer an obstacle for the convergence of the periodic map T n B n ¹ S

^pn

qn

B n towards

a smooth map.

φ _{|r a}

1

,b

¹

sr c ,d s id φ _{|r a ,a}

1

sr c ,d s τ _b

1

a

φ _{|r b}

1

,b sr c ,d s τ a b

¹

} φ } n ¤ f p n, a, b, c, d, a ¹ , q

Remark 2.2. Assumptions given in this proposition are not the most general in order to define a quasi-permutation. However, they will be easier to write the construction, allowing to avoid writing some unessential technical details.

Remark 2.3. The estimating function f p n, a, b, c, d, a ¹ , q could be explicitly deter- mined, but we do not need its expression in this paper.

The norm of the C ⁿ -norm of φ is denoted with } φ } n . The set where }Dφ } n 1, which includes r a ¹ , b ¹ s r c , d s Y r a , a ¹ s r c , d s Y r b ¹ , b s r c , d s Y r a, b s r c, d s pr a { 2, b { 2 s r c { 2, d { 2 sq is called safe zone. The set where } φ } n 1 is called turbulence zone.

The basic phenomenon that we use is the following: let S t be the rotation flow on

, 0   a   b   1, 0   c   d   1, x p u, v q . We have: S t p u, v q p u, v t q .

Let A :  ý such that A _{|r a,b sr c,d s} φ p a, b, c, d, a ¹ , q , where φ p a, b, c, d, a ¹ , q is the quasi-permutation defined in proposition 2.1, and such that A id elsewhere. Let R p x q A ¹ S t A p x q . We have:

1. If u R r a, b s , then R p x q S t p x q .

2. If u P ra , a ¹ s, v ¤ c and c ¤ v t ¤ d , then Rpxq τ _b

1

a S t pxq.

3. If u P rb ¹ , b s , v ¤ c and c ¤ v t ¤ d , then Rpxq τ _{a b}

1

S t pxq . 4. If u P r a ¹ , b ¹ s , v ¤ c and c ¤ v t ¤ d , then R p x q S _t p x q .

Observe also that, since R ² p x q A ¹ S _t AA ¹ S _t A p x q A ¹ S _2t A p x q , then if we take x inside a safe zone, and if some iterate of x by R falls inside the turbulence zone, then if we iterate enough by R, we get back into a safe zone, as if we went there directly, i.e. as if we never crossed turbulences. So even if we lose the trajectory of x into a turbulence zone, we recover it after su ffi cient iterations. Therefore, if turbulence zones are su ffi ciently thin, we can control most of the trajectory of x.

On the other hand, if x belongs to a turbulence zone, we cannot control its trajectory by R.

Proof of proposition 2.1. We recall the following proposition, which is found in [6, 4, 3] in a slightly modified version:

Proposition 2.4 ([6]). For any η ¡ 0, there exists a smooth measure-preserving map

φ p η q : r 0, 1 s ² ý such that:

Figure 3: A quasi-permutation: it permutes the two lateral boxes, and keeps the central one fixed.

where Rp π, p1{2, 1{2qq denotes the rotation of angle π and center p1{2, 1{2q, φ p η q _|r 0,1 s

²

pr η { 2,1 η { 2 s

²

q id

} φ p η q} n ¤ f p n, η q

Remark 2.5. In [6, 4, 3], the angle of the rotation is π { 2, whereas the angle here is π.

Let p ¥ 1 and

C _p : r 0, 1 s 0, ¹ _p

Ñ r 0, 1 s r 0, 1 s px, yq ÞÑ px, pyq Let φ p η, pq C p ¹ φ p η qC p . The map φ p η, pq : r0, 1s

0, ¹ _p

ý is smooth and measure-preserving. By the Faa-di-Bruno formula,

} φ p η, p q} n ¤ f p n, η, p q

By composing φ p η, p q with translations and homotheties, there exists φ ¹ p a, b, c, d, q : r a, b s r c, d s ý such that:

φ ¹ _{|r a,b sr c,d spr a { 2,b { 2 sr c { 2,d { 2 sq} id φ ¹ _{|r a { 2,b { 2 sr c { 2,d { 2 s} R

π,

a b 2 , c d

2

} φ ¹ } n ¤ f pn, a, b, c, d, q there exists φ ² p a, b, c, d, a ¹ , q : r a, b s r c, d s ý such that

φ ² _{|r a}

₁

_,b

₁

_{sr c ,d s} R

π, a b

2 , c d 2

and φ ² id on r a, b s r c, d s pr a ¹ , b ¹ s r c { 2, d { 2 sq

} φ ² } n ¤ f p n, a, b, c, d, q

there exists φ ³ pa, b, c, d, a ¹ , q : ra, bs rc, ds ý such that φ ³ _{|r a ,a}

₁

_{sr c ,d s} R

π,

a a ¹ 2 , c d

2

and φ ³ id on r a, b s r c, d s pr a { 2, a ¹ s r c { 2, d { 2 sq } φ ³ } n ¤ f p n, a, b, c, d, q

and there exists φ ⁴ p a, b, c, d, a ¹ , q : r a, b s r c, d s ý such that φ ⁴ _{|r b}

₁

_{,b sr c ,d s} R

π,

b b ¹ 2 , c d

2

and φ ⁴ id on r a, b s r c, d s pr b ¹ , b s r c { 2, d { 2 sq } φ ⁴ } n ¤ f p n, a, b, c, d, q

We let φ φ ⁴ φ ³ φ ² φ ¹ . We have:

φ _{|r a,b sr c,d spr a { 2,b { 2 sr c { 2,d { 2 sq} id φ _{|r a}

1

,b

¹

sr c ,d s id

φ _{|r a ,a}

1

sr c ,d s τ b

¹

a

φ _{|r b}

1

,b sr c ,d s τ a b

¹

} φ } n ¤ f p n, a, b, c, d, a ¹ , q

2.2 Definition of T n

Figures 4, 5 and 6 illustrate the definition. We define T n in the case M r 0, 1 s ”. In subsections 3.3.2 and 3.3.3, we extend the construction to the general case.

For n ¥ 0, let Npnq ¥ 2 be a sequence of integers. Additional assumptions on Npnq appear in corollary 3.7 of the next section, to get nLB of the limit T . Let N n

± n

k 0 N p k q , N ₁ 1. Let strictly positive integers q _n , q _n,0 , ..., q _{n,N p n 1 q 1} such that q _n divides q _n,0 , for i 0, ..., N p n 1 q 2, q _n,i divides q _{n,i 1} , and N p n q q _n q _{n,N p n 1 q 1} divides q _n 1 . Additional assumptions on the q _n,i appear in corollary 3.7, in order to get nLB of the limit T .

For i 0, .., N p n 1 q 1, j 0, ..., N n 1 1, j ¹ 0, ..., N n 1 1, y 0, ..., N p n q 1, x 0, ..., tp N p n q y q{ 2 u , if p j j ¹ q{ N n 1   1 then we define (by using notations of proposition 2.1):

A n 1 p i, j, j ¹ , x, y q : j

N n 1

i N n 1

x N n

, N p n q x y N n

1 N n 1

j ¹ N n 1

₁

p u, v q ÞÑ j

N n 1

i N n 1

, j ¹ q n,i N n

φ x

N _n , Npnq x y N _n

1 N _n 1

j ¹ N _n 1

, y

q _n,i N p n q , y q _n,i N p n q

1 q _n,i N _n , x

N _n 1 N _n 1

, n 1

p u, v q if p j j ¹ q{ N n 1 ¥ 1 then we define (by using notations of proposition 2.1):

A n 1 p i, j, j ¹ , x, y q : j

N _n 1

i N _n 1

Npnq x y N _n

j ¹ N _n 1

, x N _n

1 N _n 1

j ¹

q n,i N n

y

q n,i Npnq , y q n,i Npnq

1 q n,i N n

ý

pu, vq ÞÑ j

N n 1

i N n 1

, j ¹ q n,i N n

φ

N p n q x y N n

, x N n

1 N n 1

j ¹ N n 1

, y

q n,i N n

, y

q n,i N p n q 1 q n,i N n

, N p n q x y N n

1 N n 1

, _{n 1}

pu, vq We distinguish the cases p j j ¹ q{ N _{n 1}   1 and p j j ¹ q{ N _{n 1} ¥ 1, because if

p j j ¹ q{ N _{n 1} ¥ 1, then _N ^j

n1

i N

n 1

N p n q x y N

n

1 N

n 1

j

¹

N

_n1

¡ 1.

We briefly explain the di ff erent roles played by the indices i, j, j ¹ , x, y: the index i is used to label the N p n 1 q di ff erent q n 1 -trajectories. The parameters x, y serve to concatenate the N p n q di ff erent q n -trajectories in the right order and everywhere. The parameters j, j ¹ serve to connect the di ff erent ergodic components of T n properly, to get ergodicity of the limit transformation T .

We extend A n 1 p i, j, j ¹ , x, y q to E p i q

N

_n1

¤ 1 j 0

N

_n1

¤ j 1 j

¹

0

N p ¤ n q 1 y 0

tp N p n ¤ q y q{ 2 u x 0

j N n 1

i N n 1

x N n

, N p n q x y N n

1 N n 1

j ¹ N n 1

j ¹ q n,i N n

y

q _n,i N p n q , y q _n,i N p n q

1 q n,i N n

¤

N

_n1

¤ 1 j 0

N

_n1

¤ 1 j

¹

N

_n1

j

N p ¤ n q 1 y 0

tp N p n ¤ q y q{ 2 u x 0

j N n 1

i N n 1

N p n q x y N n

j ¹ N n 1

, x N n

1 N n 1

j ¹ q n,i N n

y

q _n,i N p n q , y q _n,i N p n q

1 q n,i N n

by identity. Then, on each Epiq, i 0, .., Npn 1q 1, we define:

A n 1 piq ^N _j

ⁿ¹

₀ ^{1 N} _j

1ⁿ¹

0 ^{1 N} _y ^{p n} ₀ ^{q 1 tp} _x ^N ₀ ^{p n q y q{ 2 u} A n 1 pi, j, j ¹ , x, yq

The order in which we compose the maps A n 1 p i, j, j ¹ , x, y q a ff ects the definition of

A n 1 p i q , because their support are not disjoint: their turbulence zones have intersections

on sets of small measure. However, this order does not matter for the properties of A n 1 piq that we seek.

We extend A n 1 piq to Epiq p 0, l{q n,i q , l 0, ..., q n,i 1 by 1 {q n,i -equivariance, i.e.:

A _{n 1} p i qp u, v l { q _n,i q A _{n 1} p i qp u, v q p 0, l { q _n,i q The parameter i was sorted out because the q _n,i depend on i.

Finally, on r0, 1s ”, we define: A n 1

N

_n1

1

i 0 A n 1 piq.

Again, the order of composition of the maps A n 1 piq matters for the definition, but not for the properties that we seek.

Let B _n A _n ... A ₀ and T _n B _n ¹ S

^pn

qn

B _n . This defines T _n . In corollary 3.7, we add assumptions on q _n , N p n q and q _n,i to obtain that T _n converges towards a smooth, nLB and ergodic transformation T . T will have zero entropy as the limit of maps conjugated to rotations.

3 Properties of the transformation T

3.1 Convergence of T n towards a smooth map T

Showing the convergence of T n towards a smooth map T is classical (see e.g. [1, 6]). By construction, there exists f precgce p n, q n , N n 1 , q _{n,N p n} 1 q 1 , n 1 q such that } B n 1 } n 1 ¤ f precgce p n, q n , N n 1 , q _{n,N p} n 1 q 1 , n 1 q . By the Cauchy criterion, it su ffi ces to show that

°

n ¥ 0 d n p T n 1 , T n q converges. We combine the fact that A n 1 commutes with S

¹

qn

, the estimation of B _{n 1} and the fact that p _{n 1} { q _{n 1} p _n { q _n 1 { q _{n 1} . We recall the lemma [6, p.1812]:

Lemma 3.1. Let k P Ž. There is a constant C p k, d q such that, for any h P Di ffp M q , α 1 , α 2 P ’, we have:

d k p hS α

₁

h ¹ , hS α

₂

h ¹ q ¤ C p k, d q} h } ^k _k ¹ ₁ | α 1 α 2 | Since T _n B _n ¹ S

^pn

qn

B _n B _n ¹ ₁ S

^pn

qn

B _n 1 (because A _n 1 commutes with S

1 qn

), and since, for n ¥ 2, } φ _n } n 1 ¤ q ^R n

¹

^{p n q} for a sequence R 1 pnq independent of q n (because q n ¥ 2 for n ¥ 2), we obtain, for a fixed sequence f cgce pn, q n , N n 1 , q n,N p n 1 q 1 , _{n 1} q :

d _n p T _{n 1} , T _n q d _n p B _n ¹ ₁ S

pn 1 qn 1

B _{n 1} , B _n ¹ ₁ S

^pn

qn

B _{n 1} q

¤ C p k, d q} B n 1 } ⁿ _n ¹ ₁ p n 1

q _{n 1} p n

q _n

¤ f cgce p n, q n , N n 1 , q _{n,N p n 1 q 1} , n 1 q p n 1

q _{n 1} p n

q _n

For a su ffi ciently increasing sequence q n , this last estimate guarantees the conver-

gence of T n in the smooth topology.

Figure 4: Domains of the di ff erent trajectories for the first three iterations of the con-

struction, with N p k q 2, k 0, ..., 3. Some quasi-permutations are represented in

dotted lines.

Figure 5: Representation of A 1 and partial representation of A 2 (dashed lines), with

N p k q 2, k 0, ..., 2. The rectangle p i, j q is quasi-permuted with the rectangle p j, i q

of the same height. In particular, A l (l 1, 2) is the identity on p i, i q .

Figure 6: Partial representation of A 1 , with Np 0 q 4 and Np 1 q 2. The rectan-

gle pi, jq is quasi-permuted with the rectangle p j, iq of the same height. Some quasi-

permutations are represented in dotted lines.

3.2 T is non-Loosely Bernoulli

To get nLB, the idea is that the two words aaaabbbb and abababab are far from each other in the ¯ f -distance.

We fix the partition: P ! c _j

j N p 0 q , _N ^j _{p 0} ¹ _q

” ¹ , j 0, ..., N p 0 q 1 )

The aim of this subsection is to show the following proposition, which is slightly stronger than the nLB property:

Proposition 3.2. For any ¡ 0, there exists T P Di ff ⁸ p  q such that for any A P B such that µ p A q ¡ 2, there exists n ₀ ¥ 0 such that for any n ¥ n ₀ , there exists x, y P A such that f ¯ p a p T, q _n , x q , a p T, q _n , y qq ¥ 1 3.

First, we show that it su ffi ces to consider q n -trajectories by T n , instead of consider- ing q n -trajectories by T .

Lemma 3.3. For any ¡ 0, there exists a numerical map f nlb p , n, q n , N n 1 , q _{n,N p n 1 q 1} , n 1 q ¥ f cgce p n, q n , N n 1 , q _{n,N p n 1 q 1} , n 1 q , there exists E nlb €  such that µ p E nlb q ¥ 1 ,

and such that for any n P Ž, if q n 1 ¥ f nlb p , n, q n , N n 1 , q _{n,N p n 1 q 1} , n 1 q , then for any x P E nlb the q n -trajectory of x by T n is the same as the q n -trajectory of x by T . Proof. Let f nlb p , n, q n , N n 1 , q n,N p n 1 q 1 , n 1 q ¥ f cgce p n, q n , N n 1 , q n,N p n 1 q 1 , n 1 q ¥ 2 ⁿ q n such that, if q n 1 ¥ f nlb p , n, q n , N n 1 , q n,N p n 1 q 1 , _{n 1} q, then for any m P Ž,

¸

n ¥ m

f _cgce p n, q _n , N _n 1 , q _{n,N p n 1 q 1} , _n 1 q q n 1

¤

q ² _m N p 0 q 2 ^{m 1}

Remark that as far as q n 1 ¥ 2 ⁿ q n , f nlb does not depend on the q p , p ¥ n 1. Since dpT, T m q ¤ ¸

n ¥ m

f cgce pn, q n , N n 1 , q n,N p n 1 q 1 , _{n 1} q q n 1

then

d p T, T m q ¤ q ² _m N p 0 q 2 ^{m 1} Therefore, for 0 ¤ i ¤ q n 1,

d 0 pT ⁱ , T _m ⁱ q ¤

q m N p 0 q 2 ^{m 1} (2)

Moreover, for any F, G continuous and measure-preserving transformations, and A P B,

µ p F p A q X G p A qq ¥ µ p A q µ

F p A q f G ¯ p A q

¥ µ p A q d 0 p F, G q (3) Now, let

E _nlb £

n ¥ 0 q £

_n

1

i 0

^{N p} ¤ ^{0 q 1}

j 0

T _n ⁱ c _j X T ⁱ c _j

Since, by (3),

µ T _n ⁱ c j X T ⁱ c j

¥ µ p c j q d p T _n ⁱ , T ⁱ q

and since T ⁱ c j X T ⁱ c j

¹

H if j j ¹ , then µ

^{N p} ¤ ^{0 q 1}

j 0

T _n ⁱ c j X T ⁱ c j

^{N p} ¸ ^{0 q 1}

j 0

µ T _n ⁱ c j X T ⁱ c j

¥

N p ¸ 0 q 1 j 0

µ pc j q dpT n ⁱ , T ⁱ q ¥ 1 q n 2 ^{n 1}

Moreover, for any A, B P B, A , B ¥ 0 such that µ p A q ¥ 1 A and µ p B q ¥ 1 B , we have:

µ p A X B q µ p A q µ p B q µ p A Y B q ¥ 1 p A B q (4) Therefore,

µ p E nlb q ¥ 1 ¸

n ¥ 0 q ¸

n

1

i 0

q n 2 ^{n 1} 1

Finally, if x P E nlb , then for any n P Ž, for any i 0, ..., q n 1, there exists j 0, ..., N p 0 q 1 such that x P T _n ⁱ c j X T ⁱ c j . Therefore, T _n ⁱ x P c j and T ⁱ x P c j . Therefore, x has the same q n -trajectory by T n and by T .

Second, we show that the trajectory by T _n of most points is well approximated by the "theoretical" trajectories a _n,i , defined in (1).

Let η ₀ 0, and for any n ¥ 1, let η _n 2

n ¸ 1 k 0

_k 1 N _k 1 max

0 ¤ i ¤ N p k 1 q 1

q _k,i 1

min _{0 ¤ i ¤ N p k 1 q 1} q k,i max

0 ¤ i ¤ N p k 1 q 1

N p k q ² q ² _k q _k,i q k 1

and

E safe p n q

£ n k 0

safe p A k q We have the lemma:

Lemma 3.4. We have a partition Esafe p n q t c 0 p n q , ..., c _{N p n q 1} p n qu such that for any i 0, ..., Npnq 1,

µ p c i p n qq 1 N p n q

¤ µ p turb p B n qq and for any x P c i p n q ,

f ¯ p a p T n , q n , x q , a n,i q ¤ η n

Proof. The proof is by induction on n. If n 0, E safe p 0 q . Moreover, a p T 0 , q 0 , x q i x , where i x P I is such that x P c i

_x

. Therefore, apT 0 , q 0 , xq a 0,i

_x

, and ¯ f papT 0 , q 0 , xq , a 0,i

_x

q ¤ η ₀ 0.

Suppose the lemma holds at step n, and let x P E safe pn 1 q . Since safe pA n q is stable by A _n , then E safe p n 1 q € safe p B _{n 1} q , and therefore, x P safe p B _{n 1} q . By construc- tion, up to a circular permutation, x has N p n 1 q possible types of q _{n 1} -trajectories (i.e.

N p n 1 q if we neglect turbulences, otherwise there are N _{n 1} possible q _{n 1} -trajectories),

depending on which "track" it stands. We denote these (non-connected) elements of this partition c 0 pn 1 q , ..., c N p n 1 q 1 pn 1 q . We have:

µ p c i p n 1 qq 1 N p n 1 q

¤ µ pturb p B n 1 qq

We show the second estimate. By construction, up to a circular permutation, any y P E safe p n q € safe p B _n q has N p n q possible types of q _n -trajectories by T _n . We denote them a _n,i, e ff , with i 0, ..., N p n q 1. Labels i of a _n,i, e ff are chosen such that, by induction assumption, ¯ f p a _n,i, e ff , a _n,i q ¤ η _n . a _n,i, e ff is the "e ff ective" trajectory: it corresponds to an "ideal" trajectory a _n,i perturbed by turbulences coming from B _n . These turbulences depend on the point y, and for better precision, we could write a _n,i, e ff p y q .

First, we neglect turb p A _{n 1} q (we suppose it infinitely thin). By construction, the q _{n 1} -trajectory of x, a _{n 1,i,} neg (for 0 ¤ i ¤ N p n 1 q 1) is of the form:

a _{n 1,i,} negl σ ^u

σ ^l

⁰

p a _n,0, e ff q

_Npnq^{qn 1}

qn,i qn

...

σ ^l

^Npnq1

p a _{n,N p n q 1,} e ff q

_Npnq^{qn 1}

qn,i qn

^qn,i

qn

...

σ ^q

ⁿ

^{1 l}

⁰

p a _n,0, e ff q

Npnqqn,i qn^{qn 1}

...

σ ^q

ⁿ

^{1 l}

^Npnq1

p a _{n,N p n q 1,} e ff q

Npnqqn,i qn^{qn 1}

^qn,i

qn

for some integers u, l 0 , ..., l N p n q 1 . In particular, at y fixed, there are only N p n q possible words a _n,i, e ff p y q , i 0, ..., Npnq1 that compose the q n 1 -trajectory of y in the formula above. Turbulences coming from B n are the same in all these words. This fact is important for the construction of the uncountable family of pairwise non-Kakutani equivalent di ff eomorphisms.

For all i 0, ..., N p n 1 q 1, let also:

a _{n 1,i,} seme ff

a

qn 1 Npnqqn,i qn

n,0, e ff a

qn 1 Npnqqn,i qn

n,1, e ff ... a

qn 1 Npnqqn,i qn

n,N p n q 1, e ff q

_n,i

(the index "semeff" is for "semi-effective": a _{n 1,i,} seme ff is halfway between the

"e ff ective" trajectory a _{n 1,i,} e ff and the "ideal" trajectory a n 1,i ). Moreover, for any integer N ¥ 2, integer k, and word a, σ ^k p a ^N q a ¹ a ^{N 2} a ² , where a ¹ and a ² are words such that | a ¹ | | a ² | | a | . Therefore,

f ¯ p a ^N , σ ^k p a ^N qq ¤ 1 p N 2 q| a | N | a | 2

N (5)

Therefore, f ¯

a n 1,i, neg , σ ^u

a _{n 1,i,} semeff ¤ 2N p n q ² q ² _n q _n,i q n 1

(6) Now, we take into account turb p A n 1 q . The q n 1 -trajectory of x crosses turbu- lences from quasi-permutations making up its own trajectory, but also from quasi- permutations making up other trajectories (see figure 4). Therefore, it crosses at most

2N n 1 max _{0 ¤ i ¤ N p n 1 q 1} q n,i turbulence zones (the factor 2 is because we cross one

f apT ¯ n 1 , q n 1 , xq , a n 1,i, neg

¤ 2 _{n 1} N n 1 max

0 ¤ i ¤ N p n 1 q 1 q n,i (7) To conclude the proof, we also need the lemma:

Lemma 3.5. Let a, b, a ¹ , b ¹ words such that | a | | b | | a ¹ | | b ¹ | . We have:

f ¯ p ab, a ¹ b ¹ q ¤ 1 2

f ¯ p a, a ¹ q f ¯ p b, b ¹ q

By applying lemma 3.5 and the induction assumption, for any integer u, we get:

f ¯ p σ ^u pa _{n 1,i,} seme ff q , σ ^u pa n 1,i qq f ¯ pa _{n 1,i,} seme ff , a n 1,i q ¤ max

0 ¤ j ¤ N p n q 1

f ¯ pa _n,j, e ff , a n, j q ¤ η _n (8) Moreover, by estimation (5),

f ¯ p σ ^u p a _{n 1,i} q , a _{n 1,i} q ¤ 2

q n,i (9)

Therefore, by combining estimates (6), (7), (8), (9), we get:

f ¯ p a p T _{n 1} , q _{n 1} , x q , a _{n 1,i} q ¤ f a ¯ p T _{n 1} , q _{n 1} , x q , a _{n 1,i,} neg f ¯

a _{n 1,i,} neg , σ ^u

a _{n 1,i,} seme ff f ¯ p σ ^u p a _{n 1,i,} seme ff q , σ ^u p a _{n 1,i} qq f ¯ p σ ^u p a n 1,i q , a n 1,i q ¤ 2 n 1 N n 1 max

0 ¤ i ¤ N p n 1 q 1 q n,i 2N p n q ² q ² _n q _n,i q n 1

η n max

0 ¤ i ¤ N p n 1 q 1

2 q _n,i Therefore,

f ¯ pa p T n 1 , q n 1 , x q , a n 1,i q ¤ η n 1

Proof of lemma 3.5. Let π _a : a Ñ a ¹ and π _b : b Ñ b ¹ two matches. Let π : ab Ñ a ¹ b ¹ defined by π _{| a} π _a and π _{| b} π _b . π is a match because π _a and π _b are matches (it is an order-preserving, injective function). Moreover,

fit p π a q fit p π b q | D p π _a q|

1

2 p| a | | a ¹ |q

| D p π _b q|

1

2 p| b | | b ¹ |q 2 | D p π _a q| | D p π _b q|

1

2 p| a | | a ¹ | | b | | b ¹ |q 2fit p π q Moreover, fit p π q ¤ 1 f ¯ p ab, a ¹ b ¹ q . By taking the maximum on possible fits of π a

and π b in the previous equality, we get: 1 f ¯ p a, a ¹ q 1 f ¯ p b, b ¹ q ¤ 2 1 f ¯ p ab, a ¹ b ¹ q . Hence lemma 3.5.

To get nLB, it remains to give a lower bound on ¯ f pa n 1,i , a n 1,j q , when i j. Our method is analogous to [7, p. 34].

Let

u n max

! fitp π q{ π : a ^r _n,i Ñ a ^s _n,j match, r, s P Ž, 0 ¤ i   j ¤ Npnq 1 )

For i 0, ..., N p n 1 q 1, let r n 1,i _{N p} ^q _n

ⁿ

_{q q}

_n,i¹

_q

_n

, and for j 0, ..., N p n 1 q 1, j ¡ i, let λ _n,i,j q _n,j { q _n,i . Note that since j ¡ i, λ _n,i,j is a positive integer.

We show the slightly stronger lemma:

Lemma 3.6. We have:

u _{n 1} ¤

u _n 2

Npnq 1 max

0 ¤ i   j ¤ N p n 1 q 1

2N p n q

λ _{n,i, j} 1 max

0 ¤ i   j ¤ N p n 1 q 1

2λ n,i,j

r n 1,i

Corollary 3.7. If N p n q ¥ 2 ^{n 3} { , and if for any 0 ¤ i   j ¤ N p n 1 q 1, λ _n,i,j ¥ 2 ^{n 5} N p n q , and r _{n 1,i} ¥ 2 ^{n 5} λ _n,i,j , then for any r, s ¡ 0,

f ¯ p a ^r _n,i , a ^s _n,j q ¥ 1 Proof of lemma 3.6. We denote λ λ n,i,j . We have:

a ^r _{n 1,i}

a ^r _n,0

^{n 1,i}

...a ^r _n,N

^{n 1,i}

_{p n q 1} ^q

^n,i

^r

a _n ^s _1,j

a

rn 1,i

n,0

λ

...a _n,N

^rnλ1,i

p n q 1

q

n,i

sλ

For l 0, ..., Npnq 1, let α _n,l a

rn 1,i

n,l

λ

. We can write:

a ^r _{n 1,i}

α ^λ _n,0 ...α ^λ _{n,N p n q 1} ^q

^n,i

^r α ¯ ^λ _n,0 ... α ¯ ^λ _{n,N p n q q}

n,i

r 1

Let

π : D p π q € a ^r _{n 1,i} Ñ R p π q € a _n ^s _1,j

be a match. For l 0, ..., Npnqq n,i r 1, let D p π q l D p π q X α ¯ ^λ _n,l (i.e. D p π q l is the part of the word D p π q that is included in the subword ¯ α ^λ _n,l of a ^r _{n 1,i} ). We have:

D p π q D p π q 0 ...D p π q N p n q q

_n,i

r 1

Let R p π q l π p D p π q l q . We can write:

a _n ^s _1,j a n 1,j,0 ...a _{n 1, j,N p n q q}

_n,i

_{r 1} such that R p π q l € a n 1,j,l , for l 0, ..., N p n q q n,i r 1.

Let

π _l : D p π q l € α ¯ ^λ _n,l Ñ R p π q l € a n 1,j,l

be a match, with π _l π _|

_D

_{p π q}

_l

. We have: D p π _l q D p π q l . a n 1,j,l is of the form:

a n 1,j,l α ˜ l p α n,0 ...α _{n,N p n q 1} q ^t

^l

p α l

with t _l ¥ 0, and such that max p| α ˜ _l | , |p α _l |q ¤ N p n q| α _n,0 | .

Moreover, we have fit p π _l q fit p π _l ¹ q because | D p π _l q| | R p π _l q| . We have:

π _l ¹ : R p π l q € α ˜ l p α n,0 ...α _{n,N p n q 1} q ^t

^l

p α l Ñ D p π l q € α ¯ ^λ _n,l Let

π ˜ _l ¹ : R p π _l q € p α _n,0 ...α _{n,N p n q 1} q ^t

^l

² Ñ D p π _l q € α ¯ ^λ _n,l

π ¹ π ¹

Like previously, we can write: p α _n,0 ...α _{n,N p n q 1} q ^t

^l

² α ¯ _n,0 ... α ¯ _{n, p t}

_l

_{2 q N p n q 1} . Moreover, let R p π _l q p R p π _l q X α ¯ _n,p , for p 0, ..., Npnqpt l 2 q 1. We have:

R p π l q R p π l q 0 ...R p π l q N p n qp t

l

2 q 1

Let D p π l q p π _l ¹ r R p π l q p s . We can also write:

α ¯ ^λ _n,l α n,l,0 ...α _{n,l, p t}

_l

_{2 q N p n q 1}

for l 0, ..., Npnqq n,i r 1, with α _n,l,p such that D p π _l q p € α _n,l,p . Moreover, since

| D p π _l q p | | R p π _l q p | ¤ | α ¯ _n,p | | α _n,0 |

we can choose α _n,l,p such that, if l p mod N p n q , | α _n,l,p | ¤ | α _n,0 | . Let

π l,p : R p π l q p € α ¯ n,p Ñ D p π l q p € α n,l,p

α n,l,p is of the form α n,l,p aa ˜ ^u _n,l p a. with max p| a ˜ | , |p a |q ¤ | a n,l | q n . We have:

D p π l,p q R p π l q p

We have:

π l,p : D p π l,p q € α ¯ n,p a

rn 1,i

n,p

λ

Ñ R p π l,p q € aa ˜ ^u _n,l p a Let

π ˜ l,p : D p π l,p q € a

rn 1,i

n,p

λ

Ñ R p π l,p q € a ^u _n,l ²

Let 0 ¤ p ¹   N p n q such that p ¹ p mod N p n q and 0 ¤ l ¹   N p n q such that l ¹ l mod N p n q .

If p ¹   l ¹ , then fit p π ˜ l,p q ¤ u n , by induction hypothesis.

If p ¹ ¡ l ¹ , then we can apply the induction hypothesis to p π ˜ l,p q ¹ , and therefore, fit p π ˜ l,p q fit p π ˜ _l,p ¹ q ¤ u n .

If p ¹ l ¹ , then fit p π ˜ l,p q ¤ 1 (i.e. we cannot say anything).

Now, let us relate fits of ˜ π _l,p , π _l,p , π ˜ _l ¹ , π _l and π. First, we relate fits of π _l,p and ˜ π _l,p . We have:

fit p π ˜ _l,p q 2 | D p π l,p q|

r

_{n 1,i}

λ u 2

| a n,l | and on the other hand:

fitp π _l,p q 2 | D p π _l,p q|

r

_{n 1,i}

λ | a _n,l | u | a _n,l | | a ˜ | |p a | ¤ 2 | D p π _l,p q|

p ^r

ⁿ

_λ

^1,i

u q| a _n,l | Therefore,

fit p π _l,p q ¤

r

n 1,i

λ u 2

r

_{n 1,i}

λ u fit p π ˜ _l,p q ¤

1 2

r

_{n 1,i}

λ u

fit p π ˜ _l,p q

Since u ¥ 0, we get:

fit p π _l,p q ¤

1 2λ

r n 1,i

fit p π ˜ _l,p q (10)

We relate fits of π _l,p and ˜ π _l ¹ . We have:

fit p π _l,p q 2 | D p π _l,p q|

| α ¯ n,p | | α n,l,p | Therefore,

N p n qp ¸ t

l

2 q 1 p 0

| D p π l,p q| 1 2

N p n qp ¸ t

l

2 q 1 p 0

fit p π l,p q r| α ¯ n,p | | α n,l,p |s

If p l mod N p n q , then by estimation (10), fit p π _l,p q ¤ 1 _r ^2λ

n 1,i

u _n . If p l

mod Npnq, we still have fitp π _l,p q ¤ 1 (all fits are smaller or equal to one). Therefore, we get:

N p n qp ¸ t

_l

2 q 1 p 0

| D p π _l,p q| ¤

1 2λ

r n 1,i

u n

1 2

N p n qp ¸ t

_l

2 q 1 p 0,p l mod N p n q

| α ¯ _n,p | | α _n,l,p |

1 2

N p n qp ¸ t

_l

2 q 1 p 0,p l mod N p n q

| α ¯ n,p | | α n,l,p | On the other hand,

fit p π ˜ _l ¹ q 2 | R p π l q|

p t l 2 q N p n q| α n,0 | | α ¯ ^λ _n,l | 2 ° N p n qp t

l

2 q 1

p 0 | D p π _l,p q|

° N p n qp t

_l

2 q 1

p 0 | α ¯ _n,p | | α _n,l,p | Therefore,

fitp˜ π _l ¹ q ¤

1 2λ

r n 1,i

u n

° N p n qp t

_l

2 q 1

p 0,p l mod N p n q | α ¯ _n,p | | α _n,l,p |

° N p n qp t

l

2 q 1

p 0 | α ¯ _n,p | | α _n,l,p |

° N p n qp t

_l

2 q 1

p 0,p l mod N p n q | α ¯ _n,p | | α _n,l,p |

° N p n qp t

l

2 q 1

p 0 | α ¯ _n,p | | α _n,l,p | fit p π ˜ _l ¹ q ¤

1 2λ

r n 1,i

u n

° N p n qp t

l

2 q 1

p 0,p l mod N p n q | α ¯ _n,p | | α _n,l,p |

° N p n qp t

_l

2 q 1

p 0 | α ¯ n,p | | α n,l,p |

Moreover, | α ¯ n,p | | α n,0 | , and when p l mod N p n q , by construction, | α n,l,p | ¤

| α _n,0 |. We also have:

N p n qp ¸ t

_l

2 q 1 p 0,p l mod N p n q

1 ¤ t l 2 Therefore,

fit p π ˜ _l ¹ q ¤

1 2λ

r n 1,i

u _n 2 p t l 2 q| α n,0 |

Npnqpt l 2 q| α _n,0 | λ | α _n,0 | ¤

1 2λ

r n 1,i

u _n 2 Npnq

(11)

π ¹ π

fit p π ˜ _l ¹ q 2 | R p π _l q|

p t l 2 q N p n q| α n,0 | | α ¯ ^λ _n,l | 2 | R p π _l q|

pp t l 2 q N p n q λ q| α n,0 |

On the other hand, since a _{n 1,j,l} is of the form a _{n 1,j,l} α ˜ _l p α _n,0 ...α _{n,N p n q 1} q ^t

^l

α p _l , we get:

fit p π _l q fit p π _l ¹ q 2 | R p π l q|

p N p n q t λ q| α n,0 | | α ˜ l | |p α l | ¤ 2 | R p π l q|

p N p n q t l λ q| α n,0 |

fit p π _l q ¤ p t l 2 q N p n q λ

N p n q t l λ fit p π ˜ _l ¹ q ¤

1 2N p n q N p n q t l λ

fit p π ˜ _l ¹ q ¤

1 2N p n q λ

fit p π ˜ _l ¹ q Finally,

fit p π q

° rq

n,i

1

l 0 2 | D p π l q|

° rq

n,i

1

l 0 | α ^λ _n,l | | a n 1, j,l | ¤

° rq

n,i

1

l 0 max _{0 ¤ rq}

_n,i

₁ p fit p π _l qq | α ^λ _n,l | | a _{n 1, j,l} |

° rq

n,i

1

l 0 | α ^λ _n,l | | a n 1,j,l |

¤ max

0 ¤ l ¤ rq

n,i

1 p fit p π l qq By taking the max on all possible fitp π q, we get:

u n 1 ¤

u n

2

N p n q 1 2N p n q

λ 1 2λ

r n 1,i

By taking the max on all possible λ, we get the conclusion.

Proof of corollary 3.7. By induction on n, we show:

u n ¤

1 1 2 ⁿ

If n 0, u 0 0, so the estimate holds. Suppose the estimate holds at rank n. By lemma 3.6,

u n 1 ¤

u n

2

Npnq 1 max

0 ¤ i   j ¤ N p n 1 q 1

2N p n q

λ _{n,i, j} 1 max

0 ¤ i   j ¤ N p n 1 q 1

2λ n,i,j

r _{n 1,i}

Moreover, for any i   j,

1 2λ _{n,i, j} r n 1,i

¤ 1 1 2 ^{n 4} and

1 2N p n q λ n,i,j

¤ 1 1

2 ^{n 4}

Therefore,

1 2λ _n,i,j

r n 1,i 1 2 λ n,i,j

¤

1 1

2 ^{n 4} 2

1 2 2 ^{n 4}

1

p 2 ^{n 4} q ² ¤ 1 1 2 ^{n 2} Moreover, by induction assumption,

u n

2 N p n q ¤

1 1

2 ⁿ 1 2 ^{n 2}

By combining these two estimates, we get:

u n 1 ¤

1 1

2 ^{n 1}

Hence the estimate at step n 1.

Proof of proposition 3.2. Let _n ¹ ₁ 4 _{n 1} N _n ² ₁ q n,N p n 1 q 1 . Each quasi-permutation constituting A _{n 1} has a Lebesgue density of at most 4 _{n 1} . Moreover, there is less than N ² _{n 1} q _{n,N p n 1 q 1} quasi-permutations in A _{n 1} . Therefore,

µ p turb p A n 1 qq ¤ _n ¹ ₁ By applying estimation (4), we get:

µ p E safe q ¥ 1 ¸

n ¥ 0

_n ¹ ₁

There exists f turb p , n, N n 1 , q _{n,N p n 1 q 1} q such that if n 1 ¤ f turb p , n, N n 1 , q _{n,N p n 1 q 1} q , then

µ p E safe q ¥ 1 Therefore,

µ p E safe X E nlb q ¥ 1 2

There also exists f dist p , n, N n 1 , q n,N p n 1 q 1 q such that if _{n 1} ¤ f dist p , n, N n 1 , q n,N p n 1 q 1 q, then η _n ¤ . We take for _{n 1} a function of , n, N n 1 , q n,N p n 1 q 1 such that

_{n 1} ¤ min p f _dist p , n, N _{n 1} , q _{n,N p n 1 q 1} q , f _turb p , n, N _{n 1} , q _{n,N p n 1 q 1} qq Let A P B such that µ p A q ¡ 2. Then µ A X p E safe X E nlb q

¡ 0.

Since N p n q Ñ n Ñ 8 8 then µ p max 0 ¤ i ¤ N p n q 1 c i p n qq Ñ n Ñ 8 0. Therefore, for any n su ffi ciently large, and by applying lemmas 3.3, 3.4 and corollary 3.7, there exists x, y P A X p E safe X E nlb q and i j such that

f ¯ papT , q n , xq , a n,i q ¤ η _n f ¯ papT, q n , yq , a n,j q ¤ η _n Therefore,

f ¯ p a p T, q _n , x q , a p T, q _n , y qq ¥ 1 3

3.3 Ergodicity

3.3.1 The case M r0, 1s ” Let

P n

"

i N _n , i 1

N _n

j

q _n , j q _n

, 0 ¤ i ¤ N n 1, 0 ¤ j ¤ q n 1

*

Since P n is a partition generating the Lebesgue sigma-algebra, it is su ffi cient to show that T is ergodic with respect to B p P n q .

Lemma 3.8. Let q ¡ 0 that divides q m and η tri{q, pi 1q{qr , 0 ¥ i ¥ q 1u. R

^pm

qm

is ergodic with respect to p B p η q , λ q, where λ denotes the Lebesgue measure on ”, and for any A, B P B p η q,

1 q _m

q

_m

¸ 1 l 0

λ R

lpm

qm

p A q X B λ p A q λ p B q Proof. Let A P B p η _n q , λ p A q ¡ 0 that is R

^pm

qm

-invariant. Then there is 0 ¤ i 0 ¤ q _n 1 such that r i 0 { q _n , p i 0 1 q{ q _n r€ A. Therefore, ” Y 0 ¤ i ¤ q

m

1 R

ipm

qm

pr i 0 { q _n , p i 0

1 q{ q n rq € A, and R

^pm

qm

is B p η n q -ergodic. By q m -periodicity and the ergodic theorem, for any integer L ¡ 0,

1 q m

q

m

¸ 1 i 0

λ pR

^ipm

qm

pAq X Bq 1 Lq m

Lq ¸

m

1 i 0

λ pR

^ipm

qm

pAq X Bq Ñ L Ñ 8 λ pAq λ pBq Now, we define the finite algebra B p ζ n q that contains the elements of A n 1 p P n q , modulo small turbulences (see figure 7). Let

C tl pl i

¹

,j q , 0 ¤ i ¹ ¤ N n 1 1, 0 ¤ j ¤ q n,i

q n,0 , 0 ¤ i ¤ Npn 1q1, i i ¹ mod Npn 1qu For l P C, let

C p l q

N p n ¤ 1 q 1 i 0

¤

0 ¤ i

¹

¤ N

_{n 1}

,i

¹

i mod N p n 1 q

i ¹ N n 1

, i ¹ 1 N n 1

Y

qn,i qn,0

1 j 0

l i,j

N n q n,i , l i,j 1 N n q n,i

Lemma 3.9. Let ζ n t C p l q , l P C u (ζ n recovers r 0, 1 s ”, but it is not a partition).

For any m ¡ n, S

^pm

qm

is ergodic with respect to B p ζ n q , and for any A, B P B p ζ n q , 1

q _m

q ¸

_m

1 l 0

µ S

lpm

qm

p A q X B µ p A q µ p B q Proof. For i ¹ 0, ..., N _{n 1} 1, let

P i

¹

"

i ¹ N n 1

, i ¹ 1 N n 1

j

N n q n,i , j 1 N n q n,i

, 0 ¤ j ¤ N n q n,i 1

*

Figure 7: An element of ζ _n with N n 3, q n,0 1, q n,1 3q _n,0 , q n,2 2q _n,1 . S

^pm

qm

is

ergodic with respect to B p ζ _n q .