Younès Tierce

The natural extension of the random beta-transformation

. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random βtransformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [15] about the greedy transformation.

Introduction

Expansions in base β

In all the paper, we fix a real number β, 1 < β < 2. For x ∈ [0, 1), an expansion of x in base β is a sequence (x n ) n 1 of {0, 1} such that

x = +∞ n=1 x n β n .
Rényi [START_REF] Rényi | Representations for real numbers and their ergodic properties[END_REF] introduced the greedy map T β defined on [0, 1) by T β (x) = βx mod 1, which provides a particular expansion of any real number in [0, 1), given by the sequence x n := 1 [1/β,1) (T n-1 β (x)), for all n 1. This expansion is called the greedy expansion of x. In fact, almost every x ∈ [0, 1) has an infinite number of expansions in base β, and the greedy expansion is the greatest in the lexicographic order [START_REF] Erdős | Characterization of the unique expansions 1 = ∞ i=1 q -ni and related problems[END_REF].

Rényi proved that T β has a unique absolutely continuous invariant probability measure ν β , and Parry [START_REF] Parry | On the β-expansions of real numbers[END_REF] proved that its density is proportional to the function

n 0 1 β n 1 [0,T n β (1)] ,
and that the associated measure-preserving system is weakly mixing. Smorodinsky then showed in [START_REF] Smorodinsky | β-automorphisms are Bernoulli shifts[END_REF] that the natural extension of this system is a Bernoulli automorphism. This result was also obtained by Dajani, Kraaikamp and Solomyak in [START_REF] Dajani | The natural extension of the β-transformation[END_REF] through a geometric construction of this natural extension, in the form of a tower.

The greedy map can be extended to a map T g defined on the interval I β := [0, 1 β-1 ] by setting T g (x) :=

T β (x) for x ∈ [0, 1[ βx -1 for x ∈ [1, 1 β-1 ]
. We will still refer to this extended map as the greedy map. The measure ν β extended to I β by setting ν β ([1, 1 β-1 ]) = 0 is still the unique absolutely continuous invariant probability measure of T g on I β .

We can obtain the smallest expansion of any real number of the interval I β with the lazy map T [START_REF] Erdős | Characterization of the unique expansions 1 = ∞ i=1 q -ni and related problems[END_REF], defined by

T : I β → I β x → βx if x 1 β(β-1)
βx -1 otherwise . For x ∈ I β , the lazy expansion of x is given by the sequence (x n ) defined for all n 1 by x n = 1 ( 1

β(β-1) , 1 β-1 ] (T n-1 (x)).
Let s be the symmetry on I β , defined by s(x) = 1 β-1 . The pushforward measure νβ := ν β • s -1 is the unique absolutely continuous invariant measure for T , and s conjugates T g and T , hence the systems (I β , B, ν β , T g ) and (I β , B, νβ , T ) are isomorphic [START_REF] Dajani | From greedy to lazy expansions and their driving dynamics[END_REF].

In 2003, Dajani and Kraaikamp [START_REF] Dajani | Random β-expansions[END_REF] introduced the random β transformation. We set Ω := {g, } N∪{0} , the set of sequences of g and . We then define the transformation K β on Ω × I β by

K β : Ω × I β → Ω × I β (ω, x) → (σ(ω), T ω0 (x))
,

where σ is the left shift on Ω. The sequence ω ∈ Ω describes the successive transformations that will be applied to the real x ∈ I β : once T ω0 is applied to x, we shift the sequence ω to the left and then apply T ω1 , and so on. If we fix ω ∈ Ω, we obtain a particular expansion of x in base β, by setting, for any n 1:

x ω n :=

1 {π(K n-1 β (ω,x)) 1 β } if ω n-1 = g 1 {π(K n-1 β (ω,x))> 1 β(β-1) } if ω n-1 = ,
where π is the projection on I β . Any expansion of x can be obtain with a sequence ω ∈ Ω [START_REF] Dajani | Invariant densities for random βexpansions[END_REF].

In the model we consider, the sequence ω will be drawn with the product Bernoulli measure m p , where p ∈ (0, 1) is a fixed parameter. In other words, we draw each transformation independently with probability p for T g and 1 -p for T . Dajani and De Vries proved in [START_REF] Dajani | Invariant densities for random βexpansions[END_REF] that there is a unique absolutely continuous probability measure µ p on I β such that m p ⊗ µ p is K β -invariant. Kempton [START_REF] Kempton | On the invariant density of the random β-transformation[END_REF] obtained an expression of the density ρ 1/2 of µ 1/2 with the construction of a natural extension of (Ω × I β , m 1/2 ⊗ µ 1/2 , K β ), using two symmetric towers. Suzuki [START_REF] Suzuki | Invariant density functions of random β-transformations[END_REF] then generalized this expression for any p with the use of the Perron-Frobenius operators.

Roadmap of this paper

In this paper, we consider several extensions of the random β-transformation. In section 1, we construct a first geometricosymbolic extension K of the random transformation. This extension is defined on two towers (the greedy tower and the lazy tower), each tower having a "base".

By providing a simple invariant measure on this extension and projecting it on I β , we obtain the expression of the density of µ p for any p ∈ (0, 1) (formula [START_REF] Glasner | Ergodic theory via joinings[END_REF]). We then study the transformation K g induced by K on the base of the greedy tower, and define a particular partition of this base. We prove that this partition is an independent generator of the induced system (Proposition 10), thus proving that this induced system is isomorphic to a unilateral Bernoulli shift. This provides a new proof of the ergodicity of the random system and of the uniqueness of µ p . This first extension is not yet invertible. In section 2, we construct a geometrico-symbolic version of the natural extension of this first extension. With the use of the relatively independent joining above a factor, we prove that this new extension is in fact the natural extension of the random βtransformation (Theorem 21). We then prove that this extension is isomorphic to a bilateral Bernoulli shift (Theorem 26), by "unfolding" the previous partition on the two towers.

The first extension

In this section, we extend the dynamics K β on two towers. Kempton [START_REF] Kempton | On the invariant density of the random β-transformation[END_REF] implemented the same type of strategy in the case p = 1 2 , but his construction does not generalize to any p. We construct a different extension, valid for any p ∈ (0, 1).

The domain of the extension: the greedy and lazy towers

As in [START_REF] Kempton | On the invariant density of the random β-transformation[END_REF], the towers are made of floors. We first define the "base" of each tower. Let E g be the base of the greedy tower G, defined by:

E g := Ω × {g} × [0, 1].
The label of the greedy base is (g), telling us that it is the first floor of the greedy tower G. Likewise, E is the base of the lazy tower L, defined by:

E := Ω × { } × s(1), 1 β -1 .
A floor in one of the towers is denoted by E e , where e is the label of the floor. This floor is of the form:

E e := Ω × {e} × I e ,
where I e is a sub-interval of I β that will be specified later. Thus, a point in one of the two towers is of the form c = (ω, e, x), where:

• The sequence ω ∈ {g, } N∪{0} describes the successive transformations T g and T to be applied to x.

• The label e is of the form e = (h, ω -n , ω -n+1 , . . . , ω -1 )

and characterizes the floor containing the point c. The letter h indicates the tower in which the floor is: if h = g, it is a floor from the greedy tower, if h = , from the lazy tower. The integer n is the level of the floor E e . Therefore, there are 2 n floors of level n in each tower, the bases being at level 0. Finally, the symbols ω -n , . . . , ω -1 indicate the applied transformations since the last passage in one of the two bases: in other words, the maps T ω-n to T ω-1 were applied to the real component of a point in E e since the base E h .

• x represents the real component of the point c, on which will be applied the transformation T ω0 . We have x ∈ I e where

I e := [0, T ω-n,ω-n+1,...,ω-1 (1 + )] if h = g I e := T ω-n,ω-n+1,...,ω-1 (s(1) -), 1 β-1 if h = ,
where the + andsigns respectively refer to the right and left limit at the considered points, and T ω-n,ω-n+1,...,ω-1 is the composition

T ω-1 • • • • • T ω-n .
For simplification, we will often note T v with v ∈ {g, } n to refer to this type of composition.

We then define the greedy tower as the disjoint union of all floors whose label starts with a g, and the lazy tower as the disjoint union of all floors whose label starts with an :

G := e ∈{g, } * E g•e , L := e ∈{g, } * E •e ,
where the point • represents the concatenation, and the set {g, } * is the set of finite sequences (including the empty sequence) of g and . We then note X := G L the disjoint union of the two towers, on which will be defined K.

Finally, given a floor of X, we call length of this floor the Lebesgue measure of the associated interval.

Applying the dynamics K to a point in a tower consists in going up a level in the tower, or going back to the base of the tower, or going to the base of the other tower, depending on certain conditions that we will detail in the next section.

We represent the two towers in Figure 2, one above the other (the lazy tower is represented upside down). The ω-component is represented vertically. Two points on a same vertical line have the same real coordinate x.

The dynamics on the two towers

We define the dynamics K on the greedy tower first. We consider a floor of G, with label e := (g, ω -n , . . . , ω -1 ). The floor E e is split into two parts, depending on ω 0 :

E e = (E e ∩ [g] 0 ) (E e ∩ [ ] 0 ),
where E e ∩ [g] 0 abusively refers to the set of points (ω, e, x) of E e such that ω 0 = g (and in the same way for ). The dynamics K is different on each of these parts. In the following, we note t the upper bound of I e .

1. If ω 0 = g and if t < 1 β (see Figure 3):

K : E e ∩ [g] 0 → E e•g
(ω, e, x) → (σ(ω), e • g, βx) .

K sends E e ∩ [g] 0 onto the floor E e•g .

2.

if ω 0 = g and if t 1 β , the left part of the floor is sent back onto the base E g (see Figure 4):

K : E e ∩ {ω 0 = g, x < 1 β } → E g (ω, e, x)
→ (σ(ω), g, βx) ;

and the right part goes up onto E e•g :

K : E e ∩ {ω 0 = g, x 1 β } → E e•g (ω, e, x)
→ (σ(ω), e • g, βx -1) . 3. if ω 0 = and if t < 1 β(β-1) (see Figure 5):

K : E e ∩ [ ] 0 → E e• (ω, e, x) → (σ(ω), e • , βx) .
The part E e ∩ [ ] 0 is sent onto the floor E e• .

4. if ω 0 = and if t 1 β(β-1) (see Figure 6), the central part of the floor is sent onto the base E :

K : E e ∩ {ω 0 = , s(1) β < x 1 β(β-1) } → E (ω, e, x)
→ (σ(ω), , βx) ;

and the left and right parts go up onto E e• :

K : E e ∩ {ω 0 = , x s(1) β } → E e• (ω, e, x)
→ (σ(ω), e • , βx) ;

K : E e ∩ {ω 0 = , x > 1 β(β-1) } → E e• (ω, e, x)
→ (σ(ω), e • ω 0 , βx -1) .

On the lazy tower, the dynamics K is defined in the same way by symmetry (see Figure 8).

We note π : X → Ω × I β the projection onto Ω × I β . By construction, we have π

• K = K β • π.

Construction of an invariant measure

The goal of this section is to define a K-invariant measure µ on X, such that the projection of this measure on I β is absolutely continuous. We denote the Lebesgue measure by λ, regardless of the interval on which we consider it. We recall that m p is the product Bernoulli measure of parameter p on Ω (we draw g with probability p and with probability 1 -p, independently at each step).

We first define the measures µ g and µ with respective support G and L as follows.

On the bases, we set:

µ g |Eg := m p ⊗ λ, µ |E := m p ⊗ λ.
On the floor E g,ω-n,...,ω-1 , the measure µ g is defined by:

µ g |Eg,ω -n ,...,ω -1 := 1 β n m p ([ω -n , . . . , ω -1 ] n-1 0 ) m p ⊗ λ,
where

[ω -n , . . . , ω -1 ] n-1 0
is the cylinder of Ω containing the sequences ω whose n first terms are (ω -n , . . . , ω -1 ).

Likewise on the floor E ,ω-n,...,ω-1 , the measure µ is defined by:

µ |E ,ω -n ,...,ω -1 := 1 β n m p ([ω -n , . . . , ω -1 ] n-1 0 ) m p ⊗ λ.
On their respective tower, the measure µ g and µ are preserved by K when it goes up in the tower, but we look for a measure µ on X, globally preserved by K. The following theorem describes the situation in a more abstract framework, and shows the existence of the measure µ. Theorem 1. Let (X, B) be a standard Borel space, K a transformation on X and G and L two disjoint subsets of X such that: 2. There exist two sets E g ⊂ G and E ⊂ L such that

K(G) ⊂ G ∪ E (1)
and K(L) ⊂ L ∪ E g 3. There exist two finite measures µ g and µ with respective support in G and L such that

µ g (E g ) = µ (E ) = 1
and for any measurable A of G \ E g , we have:

µ g (K -1 (A)) = µ g (A), (2)
and for any measurable A of L \ E , we have:

µ (K -1 (A)) = µ (A),
4. There exists positive constants g 0 , 0 , g 1 , 1 such that for any measurable A of X included in E g , we have

µ g (K -1 (A) ∩ G) = g 0 µ g (A) (3)
and

µ (K -1 (A) ∩ L)) = 0 µ g (A), (4)
and for any measurable A of X included in E , we have:

µ g (K -1 (A) ∩ G) = g 1 µ (A) and µ (K -1 (A) ∩ L)) = 1 µ (A).
Then there exist a unique probability measure µ on X, linear combination of µ g and µ , such that µ is K-invariant.

Proof. Let µ be a measure of the form

µ = C g µ g + C µ ,
with C g and C two positive constants. We wish to determine the values of C g and C such that µ is a K-invariant probability measure on X.

Let A be a measurable subset of X included in G \ E g . We have µ(A) = C g µ g (A). Then, by assumptions (1) and (2), we have:

µ(K -1 (A)) = C g µ g (K -1 (A)) = C g µ g (A) = µ(A).
Similarly, for any measurable A of X included in L \ E :

µ(K -1 (A)) = C µ (K -1 (A)) = C µ (A) = µ(A).
Let A be a measurable subset of X included in E g . We have, by assumptions (3) and (4):

µ(K -1 (A)) = µ(K -1 (A) ∩ G) + µ(K -1 (A) ∩ L) = C g µ g (K -1 (A) ∩ G) + C µ (K -1 (A) ∩ L) = g 0 C g µ g (A) + 0 C µ g (A) = (g 0 C g + 0 C )µ g (A). Therefore, µ(K -1 (A)) = µ(A) if and only if g 0 C g + 0 C = C g . (5) Likewise, if A is a measurable subset of X included in E , µ(K -1 (A)) = µ(A) if and only if g 1 C g + 1 C = C . (6)
We need both the equations ( 5) and ( 6) to be satisfied. In other words, we solve the system:

(g 0 -1)C g + 0 C = 0 g 1 C g + ( 1 -1)C = 0 . ( 7
)
From assumption (1), we have:

µ g (G) = µ g (K -1 (G \ E g )) + µ g (K -1 (E g ) ∩ G) + µ g (K -1 (E ) ∩ G),
and since

µ g (K -1 (G \ E g )) = µ g (G \ E g ), we get µ g (E g ) = µ g (K -1 (E g ) ∩ G) + µ g (K -1 (E ) ∩ G).
Since µ g (E g ) = 1, we deduce that

g 0 + g 1 = 1.
Similarly, 0 + 1 = 1. Therefore, the system (7) can be reduced to the equation

C g = l 0 g 1 C .
In order for µ to be a probability measure, we must also have

C g µ g (G) + C µ (L) = 1.
We finally obtain the (positive) values

C g = 0 0 µ g (G) + g 1 µ (L) and C = g 1 0 µ g (G) + g 1 µ (L)
.

With this choice of constants, the measure µ is a K-invariant probability measure on X.

Let us prove that the extension built in the previous section satisfies the assumptions of Theorem 1.

1. The greedy and lazy towers are disjoint.

2. Under the action of K, any element of G can go up in the tower G, or go back to the base E g , or be sent in the lazy base E , and the situtation is symmetric for the elements of the lazy tower.

3. The measures µ g and µ have their respective support in G and L, and

µ g (E g ) = µ (E ) = 1. Let A be a measurable subset of E g,ω-n,...,ω-1 of the form A := [u] |u|-1 0 × {(g, ω -n , . . . , ω -1 )} × [a, b],
where u is a finite sequence of elements of {g, }, and |u| denotes the length of this sequence. Let us prove that A and K -1 (A) have the same measure. We have:

µ g (K -1 (A)) = 1 β n-1 m p ([ω -n , . . . , ω -2 ] n-2 0 )m p ([ω -1 • u] |u| 0 ) 1 β (b -a) = 1 β n m p ([ω -n , . . . , ω -1 ] n-1 0 )m p ([u] |u|-1 0 )(b -a).
We then deduce that for any measurable set A of G\E g , we have

µ g (K -1 (A)) = µ g (A). Similarly, for any measurable set A of L\E , we have µ (K -1 (A)) = µ (A).
4. We set:

g 0 := µ g (K -1 (E g ) ∩ G) = +∞ n=0 1 β n+1 v∈{g, } n Tv(1 + ) 1 β m p ([v • g] n 0 ), 0 := µ (K -1 (E g ) ∩ L) = +∞ n=0 1 β n+1 w∈{g, } n Tw(s(1) -) 1 β m p ([w • g] n 0 ), g 1 := µ g (K -1 (E ) ∩ G) = +∞ n=0 1 β n+1 v∈{g, } n Tv(1 + ) 1 β(β-1) m p ([v • ] n 0 ) l 1 := µ (K -1 (E ) ∩ L) = +∞ n=0 1 β n+1 w∈{g, } n Tw(s(1) -) 1 β(β-1) m p ([w • ] n 0 ). Let A ⊂ E g of the form A = [u] |u|-1 0 × {g}×]a; b[, with 0 a b 1. Then: K -1 (A) ∩ G = n 0 v∈{g, } n Tv(1 + ) 1 β E g•v ∩ x ∈ a β ; b β ; ω ∈ [g • u] |u| 0 Therefore µ g (K -1 (A) ∩ G) = n 0 v∈{g, } n Tv(1 + ) 1 β 1 β n × m p ([v] n-1 0 ) × m p ([g • u] |u| 0 ) × 1 β (b -a) = g 0 µ g (A).
We obtain the three other formulas with a similar computation.

Therefore, by setting

C g := 0 0 µ g (G) + g 1 µ (L)
and

C := g 1 0 µ g (G) + g 1 µ (L) , the measure µ := C g µ g + C µ is a K-invariant probability measure on X.
On each floor of the two towers, the measure µ can be represented as the product of m p on Ω and a multiple of λ. Thus, we get : Proposition 2. By projecting the measure µ on Ω × I β , we obtain the measure m p ⊗ µ p , where µ p has for density

ρ p (x) := +∞ n=0 1 β n   C g ω1,...,ωn∈{g, } n m p ([ω 1 , ..., ω n ] n-1 0 )1 [0,Tω 1 ,...,ωn (1 + )] (x) +C ω1,...,ωn∈{g, } n m p ([ω 1 , ..., ω n ] n-1 0 )1 [Tω 1 ,...,ωn (s(1) -); 1 β-1 ] (x)   . ( 8
)
As a consequence, the natural projection π : X → Ω × I β is a factor map from the system (X, µ, K) to (Ω × I β , m p ⊗ µ p , K β ).

Properties of the extension

The aim of this section is to prove that the system (X, µ, K) is ergodic. For that, we firstly study the properties of K when following paths on the towers. We then study the induced transformation of K on the greedy base E g , and prove that the induced system is Bernoulli. From this, we derive the ergodicity of the system (X, µ, K), which implies the ergodicity of the initial random system and the uniqueness of µ p (as an absolutely continuous probability measure on

I β such that m p ⊗ µ p is K β -invariant).
To study the behaviour of the dynamics along paths in the towers, we first define a class of functions that will be useful in the following: Definition 3. Let n ∈ N. We define the class of functions F n as the set of functions f satisfying, up to a set of zero measure:

• There exists a finite number of disjoint intervals I 1 , . . . , I r and a subinterval J of I β such that f :

I 1 • • • I r → J.
• The function f is nondecreasing on its definition domain.

• On each interval I i , f is a linear map with slope β n .

• f ( In particular, the intervals (I i ) are in the same order as the intervals (f (I i )), and we have

r i=1 I i ) = r i=1 f (I i ) = J.
λ(I 1 • • • I r ) = 1 β n λ(J). ( 9
) Lemma 4. Let n 1 , n 2 ∈ N and f ∈ F n1 , g ∈ F n2 such that the definition domain D f of f is included in the image of g. Then f • g |g -1 (D f ) ∈ F n1+n2 .
Proof. We note g :

I 1 • • • I r1 → J and f : J 1 • • • J r2 → L and h = f • g |g -1 (D f ) . Then g -1 (D f )
is the disjoint union of the intervals g -1 (J i ) ∩ I j (when this set is not empty). Therefore, h is defined on i,j g -1 (J i ) ∩ I j with values in L. As the composition of nondecreasing functions, h is nondecreasing on its definition domain. Moreover, h is the composition of two linear maps of slope β n1 and β n2 on each non-empty g -1 (J i ) ∩ I j , so it is a linear map of slope β n1+n2 . Finally, for all i ∈ {1, . . . , r 2 }, we have

j g(g -1 (J i ) ∩ I j ) = J i thus i,j h(g -1 (J i ) ∩ I j ) = L.
We note E := {g, } * \ {∅} the set of non-empty finite sequences of {g, }, in other words the set of labels. For e ∈ E, we note e(0) the first term of e and e(-1) its last term.

We define the oriented graph G on E describing the set of admissible sequences of labels. A label e ∈ E can be followed by the label e ∈ E (we then note e → e ) if µ(K -1 (E e ) ∩ E e ) > 0. In other words, we have e → e if the map K sends a subset of positive measure of the floor E e onto the floor E e .

We use the new class of functions F n to describe the dynamics along paths in the two towers: Proposition 5. Let C = (e 0 = e, e 1 , . . . , e n = e ) (n 1) be a finite sequence of labels of E corresponding to a path in the graph G. We have

E C := n k=0 K -k E k = [e 1 (-1), . . . , e n (-1)] n-1 0 × {e} × J C with: 1. J C is a finite union of disjoint intervals I 1 , . . . , I r , 2. T e1(-1),...,en(-1) |J C ∈ F n .
In particular, the intervals (I i ) are in the same order as the intervals (T e1(-1),...,en(-1)

(I i )), λ(J C ) = 1 β n λ(I e
) and the mapping K n : E C → E e is a bijection. Proof. We prove each point by induction on the length of the path C. Suppose that C = (e, e ) is an edge in the graph of transitions of E. Set ω 0 = e (-1). Without loss of generality, we can suppose that e(0) = g. Finally, set

I e = [0, t], with t ∈ [0, 1 β-1 [. • If ω 0 = g, then E C = [g] 0 × {e} × J C with -if t < 1 β , then J C = I e , -if t 1 β , either e = (g), then J C = [0, 1 β [, or e = e•g then J C = [ 1 β , t]. • If ω 0 = , then E C = [ ] 0 × {e} × J C with -if t < 1 β(β-1) , then J C = I e , -if t 1 β(β-1)
, either e = ( ) then J C =] s (1) β ; 1 β(β-1) ], or e = e • then J C = [0, s (1) β ] ] 1 β(β-1) , t]. In each case, J C is a finite union of disjoint intervals. Moreover, T ω0 :

J C → I e is a function of F 1 .
Suppose now that C = (e = e 0 , e 1 , . . . , e n+1 = e ). Set C = (e 1 , . . . , e n+1 ). The path C is a path of length n in the graph of transitions of E, such that E C satisfies the conclusions of the proposition, i.e. :

• E C = [e 2 (-1), . . . , e n+1 (-1)] n-1 0 × {e 1 } × J C where J C is a finit union of disjoint intervals. • T e2(-1),...,en+1(-1) |J C ∈ F n .
We then have

E C = E e ∩ K -1 (E C ).
The component in Ω is indeed [e 1 (-1), e 2 (-1), . . . , e n+1 (-1)] n 0 .

Set g = T e1(-1) |Je 0 ,e 1 and f = T e2(-1),...,en+1(-1) C . We have g ∈ F 1 and f ∈ F n . By setting J C := g -1 (J C ), we do have that J C is a finite union of disjoint intervals and 1), e 2 (-1), . . . , e n+1 (-1)] n 0 × {e} × J C .

E C = [e 1 (-
Furthermore, T e1(-1),e2(-1),...,en+1(-1)

|J C = f • g |J C . Applying Lemma 4, the function T e1(-1),e2(-1),...,en+1(-1) |J C is in F n+1 .
We now study the induced transformation of K on the base E g . We note K g this induced transformation. The measure µ(.|E g ) is the measure µ g , and is preserved by the induced map K g [1, Prop. 3.6.1 p.58]. In the following, we identify the base E g with Ω × [0, 1].

We consider the set C of possible paths for a first return to the base E g . In other words, C is the set of paths of the form C = (e 0 = (g), e 1 , . . . , e n = (g)) (where |C| := n is the length of the path C), with e k = (g) for 1 k n -1 such that

µ n k=0 K -k (E e k ) > 0.
We then define the set P of subsets of E g by For almost every a ∈ E g , we call P-name of a the unique sequence (P n ) n 0 of atoms of P such that for any n 0, K n g (a) ∈ P n .

P := {E C , C ∈ C }.
Proposition 7. The sequence of partitions (K -n g (P)) n 0 is independent for the measure µ g , i.e. for any integer n ∈ N, the partitions K -n g P and 0 k n-1 K -k g P are independent.

Proof. Consider a path C = (e 0 = (g), e 1 , . . . , e N = (g))

starting and ending with the label of the greedy base E g . Then this path is a unique concatenation of paths C 0 , . . . , C k ∈ C , and

E C = k i=0 K -i g E Ci = N i=0 K -i (E ei ) = [e 1 (-1), . . . , e N (-1)] N -1 0 × J C with λ(J C ) = 1 β N λ(I (g) ) = 1
β N , from Proposition 5. Then, we have

µ g (E C ) = m p ([e 1 (-1), . . . , e N (-1)] N -1 0 ) × 1 β N ,
and we have a similar equality for all j , 0 j k. Therefore, it is clear that

µ g (E C ) = k j=0 µ g (E Cj ).
From Proposition 5, each atom of P is of the form [ω 0 , . . . , ω r-1 ] × J C . Likewise, for each N 0, each atom of N j=0 K -j g P is again of this form. Definition 8. Let N ∈ N ∪ {0} and (ω, x) ∈ E g such that there exists an atom of N j=0 K -j g P containing (ω, x). We note C the associated path. Then, we define the set A N (ω, x) included in the interval [0, 1] by

A N (ω, x) := J C . Lemma 9.
For almost every (ω, x) ∈ E g , for all n ∈ N ∪ {0} and m ∈ N, there exists N n and t ∈ [0, 1] such that

A N +m (ω, x) = A N (ω, x) ∩ [0, t]. Proof. Let P 0 = [g] 0 × [0, 1 β [.
The set P 0 is an atom of the partition P, of positive measure, corresponding to the set of points returning to the base E g in one step. The process that maps a point in E g to its P-name (defined for almost every (ω, x) ∈ E g ) is a Bernoulli process. Therefore, for almost every (ω, x) ∈ E g , the orbit of (ω, x) under K g meets infinitely often the atom P 0 . Even better: for any m ∈ N, the orbit of (ω, x) meets the atom P 0 m times in a row, infinitely often.

Let n ∈ N ∪ {0}. For such a point (ω, x), let N n be an integer such that

K N g (ω, x), . . . , K N +m-1 g (ω, x) ∈ P 0 .
Observe that the atom of

N j=0 K -j g P containing (ω, x) is of the form [ω 0 , . . . , ω r-1 ] r-1 0 × A N (ω, x). If we note A N (ω, x) = I 1 • • • I m , Proposition 5 implies [0, 1] = T ω0,...,ωr-1 I 1 • • • T ω0,...,ωr-1 I m ,
where T ω0,...,ωr-1 I i is an interval for every 1 i m, and the order of the intervals (T ω0,...,ωr-1 I i ) is the same as the order of the intervals (I i ).

Since K N g (ω, x) ∈ P 0 , we have

A N +1 (ω, x) = {y ∈ A N (ω, x), T ω0,...,ωr-1 (y) ∈ [0, 1 β [}.
Then, there exists t 1 ∈ [0, 1] such that T ω0,...,ωr-1

(A N (ω, x) ∩ [0, t 1 ]) = [0, 1 β [. Hence A N +1 (ω, x) = A N (ω, x) ∩ [0, t 1 ].
Finally, since K N g (ω, x), . . . , K N +m-1 g (ω, x) ∈ P 0 , we can apply the same reasoning m times, which implies the existence of t ∈ [0, 1] such that We want to prove that for almost every (ω, x) ∈ E g , the measure µ (ω,x) equals the Dirac measure on (ω, x), which leads to the announced result. Lemma 11 implies that, for almost every (ω, x) ∈ E g , the measure µ (ω,x) is of the form δ ω ⊗ μ(ω,x) , where μ(ω,x) is a measure on [0, 1]. In the following, we identify the measures µ (ω,x) and μ(ω,x) . Therefore, it remains to prove that µ (ω,x) is the Dirac measure on x.

Let ϕ : E g → R + be a continuous function depending only on the real variable x, i.e. x) = f (x), where f is a continous function from [0, 1] into R + .

By the martingale convergence theorem, we have the almost sure convergence:

E   ϕ N j=0 K -j g P   → E[ϕ|F].
Thus, for almost every (ω, x) ∈ E g , we have the convergence

E   ϕ N j=0 K -j g P   (ω, x) → f (y)dµ (ω,x) (y).
With N fixed, and for almost every (ω, x) ∈ E g , we have:

E   ϕ N j=0 K -j g P   (ω, x) = 1 λ(A N (ω, x)) A N (ω,x)
f (y)dy.

Set µ A N (ω,x) := 1 λ(A N (ω,x)) λ |A N (ω,x) the normalized Lebesgue measure on A N (ω, x). For almost every (ω, x) ∈ E g , the sequence of measures (µ A N (ω,x) ) N converges weakly to the measure µ ω,x .

We define the set Ēg of full measure as the set of points (ω, x) ∈ E g satisfying:

• There exists a probability measure µ (ω,x) on E g such that for any continous function ϕ : E g → R + , we have:

E[ϕ|F](ω, x) = ϕdµ (ω,x) .
• For any continuous function ϕ :

E g → R + , E ϕ| n j=0 K -j g P (ω, x) tends to E[ϕ|F](ω, x).
• For any integer N , the point (ω, x) belongs to an atom of the partition N j=0 K -j g P and satisfies Lemma 9. • For any integer N and for any continous function ϕ : E g → R + such that there exists a continuous function f : R → R + such that ϕ(ω, x) = f (x) for any (ω, x) ∈ E g , we have:

E   ϕ N j=0 K -j g P   (ω, x) = f (y)dµ A N (ω,x) (y). Lemma 12. Let (ω, x) ∈ Ēg . Let [a, b] ⊂ [0, 1] such that µ (ω,x) [a, b] = 1.
Then one of these two propositions is true:

i) µ (ω,x) [a, a+b 2 ] = 1, ii) µ (ω,x) ] a+b 2 , b] = 1.
Proof. Suppose that ii) does not hold. Set η := µ (ω,x) ([a, a+b 2 ]) > 0. Let m be a large enough integer so that 1

β m < η 2 . Let ε > 0. Let f ε be the continous function defined on [0, 1] by f ε = 0 on [a, a+b 2 ], f ε = 1 on [ a+b 2 + ε, b] and f ε is a linear map on [ a+b 2 , a+b 2 + ε].
f ε dµ (ω,x) 1 -η.
Therefore, since the sequence (µ An(ω,x) ) n converges weakly to µ (ω,x) , there exists N 0 ∈ N, such that for any n N 0 ,

f ε dµ An(ω,x) 1 - η 2 .
Thus, for any n N 0 ,

µ An(ω,x) a + b 2 + ε, b 1 - η 2 1 - 1 β m .
It then implies that

µ An(ω,x) a, a + b 2 + ε > 1 β m .
Let n N 0 such that K n g (ω, x), . . . , K n+m-1 g (ω, x) ∈ P 0 . Then from Lemma 9, there exists a real number t ∈

[0, 1] such that A n+m (ω, x) = A n (ω, x) ∩ [0, t]. Moreover, λ(A n+m (ω, x)) = 1 β m λ(A n (ω, x)). It implies that A n+m (ω, x) ⊂ [a, a+b 2 + ε[, hence, for any n n + m, µ A n (ω,x) a + b 2 + ε, b = 0.
Furthermore, we have

f 2ε dµ An(ω,x) → f 2ε dµ (ω,x) .
We then get that for any ε > 0,

µ (ω,x) ([ a+b 2 + 2ε, b]) = 0, hence µ (ω,x) (] a+b 2 , b]) = 0.
By dichotomy, we deduce that for any (ω, x) ∈ E g , the measure µ (ω,x) is the Dirac measure on x, which proves Proposition 10. In particular, the induced transformation K g is isomorphic to a one-sided Bernoulli shift.

Lemma 13. There exists a floor of the greedy tower with length greater than or equal to 1 β(β-1) . In particular, it is always possible to go from the tower G to the tower L under K.

Proof. We set:

n 0 = min{n ∈ N ∪ {0}, T n (1) 1 β(β -1)
.} Such integer n 0 exists for any β > 1: if 1 < 1 β(β-1) , then T (1) = β ×1. Iterating T from 1 is just a multiplication by β, as long as the images are less than 1 β(β-1) . Thus, the length of the floor E g n 0 is greater than or equal to 1 β(β-1) . Then we have

K(E g n 0 ∩ {ω 0 = , x ∈] s(1) β , 1 β(β -1) ]}) = E .
Theorem 14. The system (X, µ, K) is ergodic.

Proof. The induced transformation of K on E g is Bernoulli, hence ergodic. Therefore, it suffices to prove that, up to a zero measure set,

n 1 K -n (E g ) = X.
In other words, we want to prove that almost every point in X reaches the base E g in a finite number of iterations of K.

We first prove that almost every point in X can be reached from the base E g . By construction of K, every floor of the greedy tower is the image of a part of E g by a power of K. More precisely, for any e ∈ E such that e(0) = g, we have:

E e = K |e|-1   |e|-1 k=0 K -k (E e(0),...,e(k) )   .
Similarly, every floor of the lazy tower is the image by a power of K of a part of the base E . Finally, from Lemma 13, we can pass from the greedy tower to the lazy tower.

Additionally, almost every point in the base E g returns to E g in a finite number of iterations of K (from Poincaré recurrence theorem). Note N g the set of points of E g which do not return to E g . Then N g is a zero measure set. Therefore, the set n 0 K n (N g ) is measurable, and of measure zero. Indeed, let n ∈ N ∪ {0}. Then

K n (N g ) = |C|=n K n (N g ∩ E C ).
For any path C of length n, the transformation K n is invertible on

N g ∩ E C from Proposition 5. Therefore, µ(K n (N g ∩ E C )) = 0 and µ(K n (N g )) = 0. Let a ∈ X \ n 0 K n (N g ). There exist a 0 ∈ E g and n ∈ N ∪ {0} such that K n (a 0 ) = a. Moreover, a 0 / ∈ n 0 K n (N g ) so there exists a path C a 0 ∈ C such that a 0 ∈ P C a 0 .
It implies that a will return to the greedy base in a finite number of iterations of K (following the path C a 0 ). Since the system (Ω × I β , m p ⊗ µ p , K β ) is a factor of the system (X, µ, K), we have:

Corollary 15. The system (Ω × I β , m p ⊗ µ p , K β ) is ergodic.
Corollary 16. The measure µ p is the unique absolutely continuous probability measure on I β such that m p ⊗ µ p is K β -invariant.

Proof. From Lemma 13, for any 1 < β < 2, there exists a floor of the greedy tower whose length is greater than or equal to 1 β(β-1) . Symmetrically, there exists a floor of the lazy tower whose left limit is less than or equal to 1 β . It implies that the support of µ p is the full interval I β , and that m p ⊗ µ p is equivalent to m p ⊗ λ.

Let ν be an absolutely continuous probability measure (with respect to λ, and so with respect to µ p ) such that m p ⊗ ν is K β -invariant. Then the measure m p ⊗ ν is absolutely continuous with respect to the ergodic measure m p ⊗ µ p , hence m p ⊗ ν = m p ⊗ µ p , and ν = µ p .

In [START_REF] Suzuki | Invariant density functions of random β-transformations[END_REF], Suzuki proves that the density of µ p is proportional to the function

B 0 (β, p)   1 [0,1] + ∞ n=1 1 β n v∈{g, } v m p ([v] n 1 )1 [0,Tv(1)]   + B 0 (β, 1 -p)   1 [s(1), 1 β-1 ] + ∞ n=1 1 β n w∈{g, } n m p ([w]) n 1 1 [Tw(s(1)), 1 β-1 ]   (10)
where

B 0 (β, p) = p   1 - ∞ n=1 1 β n w∈{g, } n-1 1 S (T w (s(1)))m p ([w, ] n 1 )   .
By uniqueness of µ p , we know that the density ( 8) and the expression obtained by Suzuki [START_REF] Janvresse | Dynamics of λcontinued fractions and β-shifts[END_REF] are equal. This result can be easily proved in the case where β is not a root of a polynomial of the form

X n0 -X n1 -• • • -X n k -1 with n 0 > n 1 > • • • > n k (
which is equivalent to say that 1 has no finite expansion in base β). Indeed, we prove that, in this case, 0 = B 0 (β, p): on one hand, we recall that

0 = +∞ n=0 1 β n+1 w∈{g, } n ,Tw(s(1) -) 1 β m p ([w • g] n 0 ). Since β is not a root of a polynomial of the form X n0 -X n1 -• • • -X n k -1, then T w (s(1) -) = T w (s(1)) for any w ∈ {g, } n . Moreover, T w (s(1) -) 1 β is equivalent to T w (s(1)) < 1
β . Therefore, after adjusting the index of summation, we have

0 = +∞ n=0 1 β n+1 w∈{g, } n 1 R (T w (1))m p ([w • g] n 0 ).
On the other hand, after a new reindexing in 1 , we have:

0 =1 -1 = 1 - +∞ n=0 1 β n+1 w∈{g, } n 1 S∪R (T w (1))m p ([w • ] n 0 ) =1 - +∞ n=0 1 β n+1 w∈{g, } n 1 S (T w (1))m p ([w • ] n 0 ) - 1 -p p 0 .
Thus

0 = p   1 - +∞ n=0 1 β n+1 w∈{g, } n 1 S (T w (1))m p ([w • ] n 0 )   = B 0 (β, p).
By symmetry, we also obtain that g 1 = B 0 (β, 1 -p), and the equality of the two densities follows.

In the case where 1 has a finite expansion in base β, there exists n ∈ N and a sequence w ∈ {g, } n such that T w (1) = 0, and the right limit T v (1 + ) and the value T v (1) can differ, thus complicating the identification of the two formulas. For the simple case where β = 1+ √ 5

2 , we refer to the computations in the appendix of [START_REF] Tierce | Étude ergodique de bêta-transformations aléatoires[END_REF].

The natural extension

Construction of the new extension

Any point in X which is not in a base has a unique preimage by K. However, a point in one of the two bases can come from several floors. Indeed, at each return to a base, the map K "forgets" where it comes from. This lack of information prevents K from being invertible.

We wish to construct a natural extension of K, denoted by K. In some sense, it is the smallest invertible extension of K. Therefore, to construct K, we need to extend the set X such that the information of the past floors is available. We recall the definition of a natural extension of a system (see for example [START_REF] Bruin | Natural extensions for piecewise affine maps via Hofbauer towers[END_REF]).

Definition 17. The system (Y, C, ν, F ) is a natural extension of the system (X, B, µ, T ) if there exist two sets X * ∈ B and Y * ∈ C such that µ(X * ) = ν(Y * ) = 1 and a measurable mapping π : Y * → X * such that:

1. F is a bijection of Y * , 2. µ = ν • π -1 , 3. π • F = T • π and 4. C = n 0 F n (π -1 (B)).
Moreover, all natural extensions of a system are isomorphic [START_REF] Abramovitch | Exact endomorphism of a Lebesgue space[END_REF]. We now construct the extension K of K, and then prove that it is one of its natural extensions. In the initial definition of the towers G and L, each floor has a label e. More precisely, a label e is a finite sequence of g and , where the first term of e describes the present tower and the following terms of e (when there are) describe which transformations have been applied since the base: the label describes the recent past of a point in the floor. Therefore, instead of "erasing" the label at each return to a base, we will keep track of every past label, so that we can uniquely determine the past orbit of almost any point in the two towers.

We note Z the set of left-infinite sequences generated by the graph G, i.e.

Z := {(e j ) j∈Z 0 ∈ E Z 0 : ∀j -1, e j → e j+1 }.

A sequence of labels in Z contains in particular the information of the sequence of past transformations. Given a sequence e in Z, we will note e 0 its term of index 0.

For e ∈ E, we set Z e := {e ∈ Z : e 0 = e}.

We then define the base of the two towers by

Ẽg := Ω × Z (g) × [0, 1]
and

Ẽ := Ω × Z ( ) × s(1), 1 β -1
.

If e = (g, ω -n , ..., ω -1 ), n ∈ N, we set

Ẽe := Ω × Z e × [0, T ω-n,...,ω-1 (1 + )].
If e = ( , ω -n , ..., ω -1 ), n ∈ N, we set

Ẽe := Ω × Z e × T ω-n,...,ω-1 (s(1) -), 1 β -1 .
In some sense, the measure μ of this set A is obtained by "shifting" the set A to the future, so that it can be viewed as a set in the first extension X. No information is lost by projecting it on X, allowing us to use the measure µ. The measure μ is well defined

: if A ∈ N n=0
Kn (π -1 X,X (B)), then for any integer k ∈ N, we have µ(π X,X ( K-N (A)) = µ(π X,X ( K-N-k (A))

since µ is K-invariant. For any B ∈ B, we have μ • π -1 X,X (B) = µ(B), and the measure μ is K-invariant.

Finally, by construction, K is one to one on X. It implies that ( X, μ, K) is a natural extension of (X, µ, K). Moreover, the natural extension of an ergodic system is ergodic, which provides the following result, as a direct consequence of Theorem 14:

Corollary 18. The system ( X, μ, K) is ergodic.

Natural extension of the initial system

The goal of this section is to prove that the previous extension is in fact a natural extension of the initial system (Ω × I β , m p ⊗ µ p , K β ). To do so, we first introduce a canonic way to construct a natural extension of the initial system, then prove that the two extensions are isomorphic.

We adapt the construction described in [1, p. 62], which provides a natural extension of K β . We set

X := {(ω, x) ∈ {g, } Z × I Z β : ∀k ∈ Z, T ω k x k = x k+1 }.
Let B be the σ-algebra generated by the cylinders of X. We define the measure µ on every set of the form A = [ω k , . . . , ω n+k ] n+k k × [I k , . . . , I n+k ] n+k k , with k ∈ Z, n ∈ N, and for any k i n + k, ω i ∈ {g, } and I i sub-interval of I β , by

µ(A) = m p ⊗ µ p [ω k , . . . , ω n+k ] n 0 × n i=0 T -i ω k+i (I k+i ) .
We note K the shift on X.

Proposition 19. [START_REF] Boyarsky | Laws of chaos. Invariant measures and dynamical systems in one dimension[END_REF] The dynamical system (X, µ, K) is a natural extension of the system (Ω × I β , m p ⊗ µ p , K β ).

Let (ω, e, x) ∈ X. We note ω = ω(ω, e) the sequence of {g, } Z defined by

ω(k) = ω k if k 0 and ω(k) = e k+1 (-1) if k -1.
The sequence ω is the bi-infinite sequence of applied transformations in the past, and transformations to be applied in the future. For any k ∈ Z, we note x k the real component of Kk (ω, e, x), and x := (x k ) k∈Z . We define the application φ : X → X (ω, e, x) → (ω, x) .

One easily checks that:

Lemma 20. The application φ is a factor map from ( X, μ, K) to (X, µ, K).

Theorem 21. The factor map φ is an isomorphism. Therefore, the system ( X, μ, K) is a natural extension of (Ω × I β , m p ⊗ µ p , K β ).

To prove this theorem, we use the notion of relatively independent joinings above a factor (see for example [8, pp. 126-127]).

Let P be the relatively independent self-joining of the system ( X, μ, K) above its factor X i.e. the measure on X × X such that for two measurable sets A, B ∈ X,

P(A × B) =
X μ(A|F)μ(B|F)dμ, where F = φ -1 (B(X)) is the factor σ-algebra associated to the factor (X, µ, K). We have the following properties:

• P is a self-joining so its marginals on the first and second coordinate are μ, and P is K × K-invariant;

• the measure P is supported on the set {(a, b) ∈ X × X : φ(a) = φ(b)};

• P is supported on the diagonal of X × X if and only if the factor map φ is an isomorphism.

Let a = (ω, e, x) and b = (ω , e , x ) be two elements of X such that φ(a) = φ(b). We then have ω(ω, e) = ω(ω , e ) and x = x . In other words, a and b describe two trajectories in the towers, starting from the same real number x and with the same transformations ω, but possibly following different floors e and e . Proving that P is supported by the diagonal of X × X consists in proving that for P-almost every (a, b) ∈ X × X, the two sequences of floors e and e are actually equal.

The idea of the proof consists in taking two trajectories in X starting from the same real number x, with the same transformations, and proving that they end up in the same floor. Once these trajectories come at the same time in the same floor, they coincide in the future, and in the past by injectivity.

We firstly work on the extension X. Let k ∈ N and consider the set of positive measure

C k := (ω, e, x) ∈ X : x < 1 β k , ω 0 = ω 1 = • • • = ω k-1 = g .
The set C k describes the points of X whose real component is close to 0, and on which will be appplied k times in a row the greedy transformation. The label e does not take part in the definition of C k , which means that the event C k is measurable with respect to the factor X: given two trajectories with the same real component and the same sequence of transformations, the event C k happens almost surely (by ergodicity of K and because C k is of positive measure) and at the same time for both trajectories. Knowing that C k is realized actually gives some information about the label. We will prove that for large k and conditionally to C k , both trajectories are in a floor of the greedy tower whose interval is large, with high probability. Let A k be the union of the floors of G whose length is less than 1 β k . Let B k be the complementary set of A k in G. Then the set C k can be divided into three parts:

C k ∩ L, C k ∩ A k and C k ∩ B k . The set C k ∩ B k is
particularly interesting since a point in this set will end up in the greedy base E g in at most k iterations: such a point will remain in the left part of its floor under the action of K, and the length of the floor will eventually be larger than 1 β , allowing the point to go to E g in less than k iterations (see Figures 3 and4). In the following, we prove that for large k and knowing C k , the event C k ∩ B k happens with high probability.

We first introduce this technical lemma:

Lemma 22. The intervals of the lazy tower L never contain 0.

Proof. This property holds by construction of the extension K. Indeed, if an interval of a floor of L contained 0, it would have to be the image by T g of an interval whose lower bound is 1 β . However, if the lower bound of an interval of L equals 1 β , we are in the case where the part of the floor satisfying x ∈ [ 1 β , 2 β [ is sent to the greedy base when applying T g .

For any label e ∈ E, we set Y e := Ω × {e}, and we have E e = Y e × I e . We then set Y := e∈E Y e , which is represented vertically in the figure 2. We note ν the measure on Y such that, for any e ∈ E,

µ |Ee = ν |Ye ⊗ λ |Ie .
Finally, we note π X,Y the projection from X onto Y .

Let ε > 0 be small enough, so that 1 -1+2(β-1) (β-1)µ(G) ε > 0.5. Let N ∈ N be such that the measure µ of the floors beyond the level N in L is less than ε. By the previous lemma, there exists k 0 ∈ N such that no floor of L up to the level N has an interval whose lower bound is less than 1 β k . As previously said, we have:

µ(C k ) = µ(C k ∩ L) + µ(C k ∩ A k ) + µ(C k ∩ B k ). (12)
We want to estimate each term of the right hand side of equation ( 12). We will prove that for large k, C k ∩ L and C k ∩ A k have a small measure, and that C k ∩ B k represents more than half of the measure of C k .

Lemma 23. For k large enough, we have

µ(C k ∩ L) 2p k β -1 β k ε.
Proof. Let k k 0 . A floor of L can intersect C k only if the lower bound of its interval is less than 1 β k . The integer N is fixed such that every floor below the level N does not contain 1 β k . Therefore, only the floors beyond the level N can intersect C k . Additionally, for any e ∈ E, we have, by definition of the measure µ on each floor:

µ (ω, e, x) ∈ E e : ω 0 = ... = ω k-1 = g, x < 1 β k = p k µ E e ∩ x < 1 β k .
We then have:

µ(C k ∩ L) = p k e∈E,e(0)= ,|e| N +1 ν(Y e )λ I e ∩ 0; 1 β k .
For a floor E e of the lazy tower such that I e ∩ [0; 1 β k ] = ∅, we have

λ(I e ) > 1 β -1 - 1 β k .
For k large enough, we then have

λ(I e ) > 1 2(β -1)
.

It implies that

λ I e ∩ 0;

1 β k 1 β k 2 β -1 β k λ(I e ) hence µ(C k ∩ L) 2p k β -1 β k ε.
Lemma 24. For k large enough, we have

µ(C k ∩ A k ) p k β k ε. Proof. Let k ∈ N. µ(C k ∩ A k ) = p k e∈E,Ee⊂A k ν(Y e )λ I e ∩ 0; 1 β k .
If E e is a floor of A k , then λ(I e ∩ [0;

1 β k ]) 1 β k . Therefore, µ(C k ∩ A k ) p k β k ν(π X,Y (A k )).
We have

k 1 A k = ∅, hence π X,Y   k 1 A k   = ∅.
Since the intersection of the A k is decreasing, we have

k 1 π X,Y (A k ) = ∅. It implies that lim k→∞ ν(π X,Y (A k )) = 0.
We can choose k large enough, so that ν(π X,Y (A k )) < ε. Therefore,

µ(C k ∩ A k ) p k β k ε.
Proposition 25. For k large enough, we have

µ(C k ∩ B k ) µ(C k ) > 0.5.
Proof. Let k ∈ N large enough such that Lemmas 23 and 24 are both satisfied.

We have:

µ(C k ∩ B k ) µ(C k ) = 1 - µ(C k ∩ A k ) + µ(C k ∩ L) µ(C k ) .
From Lemmas 23 and 24, we then have

µ(C k ∩ B k ) µ(C k ) 1 - p k β k ε + 2p k β-1 β k ε µ(C k ) 1 - p k β k ε 2p k β-1 β k ε µ(C k ∩ G)
.

On the other hand, For any floor E e of G, we have the minoration λ(I e ∩ [0, 1

β k ]) λ(I e ) β-1 β k hence µ(C k ∩ B k ) µ(C k ) 1 - p k β k ε + 2p k β-1 β k ε p k β-1 β k µ(G) = 1 - 1 + 2(β -1) (β -1)µ(G) ε.
The real ε is such that 1 -1+2(β-1) (β-1)µ(G) ε > 0.5, which gives the result.

All these computations are also valid on the extension X. Indeed, π -1 X,X (C k ) ∈ π -1 X,X (B), which implies μ(π -1 X,X (C k )) = µ(C k ). In the following we will still note C k instead of π -1 X,X (C k ). Since K is ergodic and µ(C k ) > 0, almost every trajectory in X encounters C k with frequency µ(C k ). Therefore, for P-almost every couple of trajectories in X, both trajectories simultanously encounter C k (since C k is measurable with respect to the factor X) with frequency µ(C k ) (since P is a self-joining of µ). In other words, we have, for P-almost every (a, b) ∈ X × X, for any n ∈ N ∪ {0},

• Kn (a) ∈ C k if and only if Kn (b) ∈ C k , • 1 N N i=1 1 C k ( Ki (a)) = 1 N N i=1 1 C k ( Ki (b)) → µ(C k ).
The ergodicity of K and Proposition 25 also imply that among the occurrences of C k , the event C k ∩ B k occurs with a frequency greater that 0.5 for the orbits of P-almost every (a, b) ∈ X × X under K. Therefore, for P-almost every (a, b) ∈ X × X, there exists n 0 ∈ N ∪ {0} such that Kn0 (a) ∈ C k ∩ B k and Kn0 (b) ∈ C k ∩ B k . Once the two trajectories are in C k ∩ B k simultaneously, they will end up in the greedy base in at most k iterations (note that even if one of the two points goes in the base before the other, it will "wait" for the second one to join it. Since its real component is small enough, the point is in the atom P 0 ). In conclusion, for P-almost every (a, b) ∈ X × X, if we note Open questions 1. Can we generalize the extensions of this paper to any β > 2? We must take into account the differents branches of the greedy and lazy transformations in the construction of the natural extension, especially the branches that induce a return to a base.

2. It seems natural to apply this kind of construction to other dynamical systems, either deterministic or random ones. It seems that it can be generalized to random systems with piecewise linear maps, even when the branches have different slopes. Indeed, with the construction of the towers, we can keep track of the exact branch that is being applied at each step.

However, systems based on continued fraction expansions should pose way more difficulties. In [START_REF] Janvresse | Dynamics of λcontinued fractions and β-shifts[END_REF], the authors highlight a bijection between the β parameter of the non-integer base expansions and the λ parameter of λcontinued fractions. The first question would be to identify the analog of the greedy and lazy transformations in this case, and then to closely study this bijection.

3. The description of the extension (X, µ, K) provides a better understanding of the random β-transformation. The extension shows off some nice renewal times at each passage to the greedy base, which provided a proof of the Bernoullicity of the natural extension. We can hope to get limit theorems as well thanks to this property. Indeed, in [START_REF] Young | Recurrence times and rates of mixing[END_REF], Young describes a certain class of dynamical systems with renewal properties on reference sets. Young shows that the tail distribution of the return time to this reference set is directly linked to convergence to equilibrium and limit theorems. In our setup, an estimation of the length of paths of C should help in this direction. Could we then obtain a central limit theorem for the digit sequences in base β following a fixed sequence ω? Or on average on ω?

4. Our construction heavily relies on the independent choice of the transformations at each step. Could we adapt some of these constructions to more general stationary measures on Ω? For example Markov measures?

Figure 1 :

 1 Figure 1: Extensions involved in this paper

Figure 2 :Figure 3 : ω 0 = g and t < 1 βFigure 4 : ω 0 = g and t 1 βFigure 5 :

 231415 Figure 2: Greedy and lazy towers

Figure 6 :

 6 Figure 6: ω 0 = and t 1 β(β-1)

Figure 7 :Figure 8 :

 78 Figure 7: Overview of the dynamics on a floor E e of the greedy tower, depending on the length of the floor

Figure 9 :

 9 Figure 9: Function from the class F n

Lemma 6 .

 6 The set P is a countable partition of E g , up to a set of zero measure.Proof. If we consider two distinct paths C, C ∈ C , then it is clear that E C and E C are disjoint. And since µ g -almost every a in E g has a finite return time, there exists C ∈ C such that a ∈ E C .

Figure 11 :

 11 Figure 11: Graph of the function f ε

  µ(C k ∩ G) = p k e∈E,e(0)=g ν(Y e )λ I e ∩ 0; 1 β k .

Acknowledgements: Many thanks to Karma Dajani for her inspiring suggestions and the interest she showed to my work. Thanks to Thierry de la Rue and Jean-Baptiste Bardet as well for the numerous discussions we had during my PhD thesis, and for their continuous support.

This number t is such that λ(A N (ω, x) ∩ [0, t]) = 1 β m λ(A N (ω, x)). Let F be the factor σ-algebra F := +∞ j=0 K -j g P. The partition P provides a Bernoulli process, we want to show that this process generates the induced system by proving the following proposition: Proposition 10. The σ-algebra F is (up to zero measure sets) the Borel σalgebra on E g . In other words, the partition P is a generator of the induced system, which is isomorphic to a one-sided Bernoulli shift.

Lemma 11. F contains the σ-algebra generated by the ω-component.

Proof. Let N ∈ N. As said earlier, every atom of N j=0 K -j g P is of the form [ω 0 , . . . , ω r-1 ] r-1 0 × J C , with r N . Therefore, for any N ∈ N, (ω 0 , . . . , ω N -1 ) is F-measurable.

To prove Proposition 10, it suffices to show that for any continuous function ϕ :

The existence of the regular conditional probability implies that for almost every (ω, x) ∈ E g , there exists a measure µ (ω,x) on E g such that for any positive continuous function ϕ : We finally note X := G L. We set:

the projection from X onto X.

Let a = (ω, e, x) ∈ X. Then π X,X (a) = (ω, e 0 , x). We have K(ω, e 0 , x) = (σ(ω), e , T ω0 (x)), where the label e depends on e 0 , ω 0 and x, according to the previous construction. By construction of K, we obviously have e 0 → e in the graph G.

Applying the new dynamics K on a consists in concatenating the new label e to the sequence e (we note e • e this new sequence), in addition to shifting the sequence ω and applying T ω0 to x: we retain every past labels in memory.

In other words, the dynamics K is defined on X by

,

where e is the label associated with the point K(ω, e 0 , x) ∈ X.

The projection π X,X is measurable and we have by construction

Note B the Borel σ-algebra on X and B the Borel σ-algebra on X. Let us prove that

The inclusion n 0 Kn (π -1 X,X (B)) ⊂ B is clear. Let k and n be two non-negative integers. We consider a set A of B of the

We then have:

, and proves [START_REF] Kempton | On the invariant density of the random β-transformation[END_REF].

We can now define the measure μ on X.

a = (ω, e, x) and b = (ω, e , x), there exists an integer M = M (ω, x, e, e ) such that e M = e M = (g).

Let δ > 0. There exists an integer M δ such that

Therefore, we have P(e M δ = e M δ ) 1-δ. Since P is K × K-invariant, we deduce that for any integer k ∈ Z and any δ > 0,

We proved that P is supported by the diagonal of X × X, which implies that φ is isomorphism. Therefore, ( X, μ, K) is a natural extension of the initial system, which concludes the proof of Theorem 21.

Bernoullicity

We know that the natural extension of the initial random system is ergodic. In fact, we have the following stronger result:

Theorem 26. The natural extension of the random β-transformation is isomorphic to a Bernoulli shift.

To prove this theorem, we introduce a generating partition of the two towers. We then prove that this partition is weak-Bernoulli, by an argument on countable state space Markov chains of Ito, Totoki and Murata [START_REF] Ito | Remarks on the isomorphism theorems for weak Bernoulli transformations in general case[END_REF], which implies the theorem. In the following, we note μg := μ(•| Ẽg ), and Kg the induced transformation of K on the base Ẽg .

Lemma 27. Let (X, µ, T ) be an ergodic dynamical system, and A a measurable set of X with positive measure. Note ( X, μ, T ) a natural extension of the system, and π a factor map from X onto X. Then the system induced by T on π -1 (A)is a natural extension of the system induced by T on A.

Proof. We start by introducing the necessary notations. The natural extension of the system (X, µ, T ) can be described by a two-sided shift T on the set

with the measure μ defined by

Note à := π -1 (A) = {(x n ) ∈ X, x 0 ∈ A} and ( Ã, μ Ã, T Ã) the system induced by T on Ã. The transformation T à is defined by T Ã(x) = T r A (x0) (x) for every x ∈ Ã, where r A (x 0 ) is the return time of x 0 in A, which is the same as the return time of x in Ã.

We denote by (A, µ(•|A), T A ) the system induced by T on A. The natural extension of this induced system can also be described by a two-sided shift T on the set A = {x = (x n ) n∈Z ∈ A Z , ∀n ∈ Z, T A (x n ) = x n+1 }, with the measure µ defined in the same way as μ.

We want to prove that the two systems ( Ã, μ Ã, T Ã) and (A, µ, T ) are isomorphic. We define the map ϕ as:

The map ϕ consists in keeping only the terms of the sequence x that belong to A.

We clearly have that ϕ • T Ã = T • ϕ and that µ is the pushforward measure μ Ã • ϕ -1 : the map ϕ is a factor map from ( Ã, μ Ã, T Ã) to (A, µ, T ).

We now prove that ϕ is an isomorphism. Let y = (y n ) n∈Z ∈ A. Let n ∈ Z. From y n , we can construct the finite sequence of the iterates of y n s(y n ) := (y n , T (y n ), . . . , T r A (yn)-1 (y n )), stopping juste before y n+1 = T r A (yn) (y n ). Then the sequence x obtained by concatenating the sequences s(y n ) for all n ∈ Z is by construction a sequence of à such that ϕ(x) = y.

Let us prove that ϕ is one-to-one. Let w, x ∈ Ã such that ϕ(w) = ϕ(x). Then w 0 = x 0 , which implies that for any n ∈ N, T n (w 0 ) = T n (x 0 ), i.e. x n = w n . Let j 0 := max{j < 0, T j (w) ∈ Ã}, and j 1 := max{j < 0, T j (x) ∈ Ã}. The existence of the integers j 0 and j 1 is assured by Poincaré's recurrence theorem applied to T -1 . Note that T j0 (w) = T -1 Ã (w), and T j1 (x) = T -1 Ã (x). Since ϕ(w) = ϕ(x), we get that T j0 (w) = T j1 (x), i.e. w j0 = x j1 . The integer -j 0 is the return time of w j0 in A, which implies that j 0 = j 1 , and w n = x n for n j 0 . By induction, we prove that w = x, and ϕ is an isomorphism, which concludes the proof.

Corollary 28. The system ( Ẽg , μg , Kg ) is a natural extension of the system (E g , µ g , K g ).

Proposition 29. The partition P := π -1 (P) is a generator of the system ( Ẽg , μg , Kg ).

) n 0 can be viewed as the P-name of the point a. Note φ P the application defined almost everywhere on E g by φ P (a) = (C a n ) n 0 . Note ν the image measure of µ g by φ P and σ the left shift on C N∪{0} . The application φ P is a factor map from the system (E g , µ g , K g ) to the one-sided Bernoulli shift (C N∪{0} , ν, σ). From Proposition 10, the partition P is a generator of the system (E g , µ g , K g ). Therefore, the application φ P is an isomorphism.

We can construct a natural extension of the system (C N∪{0} , ν, σ) the same way we did for K. Doing so, we construct the system (C Z , ν, σ), which is a two-sided Bernoulli shift on C Z . Since the natural extensions of a system are isomorphic to one another, we deduce from the previous lemma that the systems (C Z , ν, σ) and ( Ẽg , μG , Kg ) are isomorphic, thus concluding the proof.

From this partition P, we construct the set

This family is obtained by "unfolding" the partition P on the two towers: each path C of C is not only described by the first atom ẼC , but by the sequence of successive images of this atom by K. Lemma 30. The set P X is a partition of X.

Proof. Two distincts sets of P X are clearly disjoint.

On the other hand, let (ω, e, x) ∈ X. Almost surely, There exists k ∈ N∪{0} such that e -k = (g) (applying Poincaré recurrence theorem to K-1 ). Let k 0 := min{k ∈ N ∪ {0}, e -k = (g)}. The integer -k 0 corresponds to the last visit of the greedy base in the past orbit of (ω, e, x). Almost surely, there exists a unique C ∈ C such that K-k0 (ω, e, x) ∈ ẼC . Since k 0 is minimal, we have 0 k 0 |C| -1, and (ω, e, x) ∈ Kk0 ( ẼC ).

Proposition 31. The partition P X is a generator of the system ( X, μ, K).

Proof. The goal is to prove that almost every point in X is uniquely determined by its P X -name.

Let (ω, e, x) ∈ X. Almost surely, we can define the integer k 0 as in the previous proof:

We note a := K-k0 (e, x) ∈ Ẽg . Almost surely, a is characterized by its Pname (C n ) n∈Z . To each path C n corresponds a unique sequence of length |C n | of atoms of P X : the sequence ( Kk ( ẼCn ), 0 k |C n | -1). Therefore, a is characterized by the unique sequence (P n ) n∈Z of elements of P X such that for any n ∈ Z, Kn (a) ∈ P n .

We deduce that the point (ω, e, x) is uniquely determined by the sequence (P n+k ) n∈Z .

We then define the isomorphism ϕ, which associates to almost every point in X its P X -name. We note η the image measure of μ by ϕ and σ the left shift on P Z X . Then ϕ is a factor map from ( X, μ, K) to (P Z X , η, σ). The measure η is the Markov measure generated by the invariant distribution (μ(P )) P ∈P X and the following transition probabilities:

• From an atom Kk ( ẼC ) with C ∈ C and 0 k |C| -2, the process goes to the atom Kk+1 ( ẼC ) with probability 1.

• From an atom K|C|-1 ( ẼC ) with C ∈ C , the process can go to ẼC for any C ∈ C with probability µ g (E C ).

Therefore, we can identify the natural extension ( X, μ, K) with a Markov shift on P Z X . This Markov shift is irreducible and recurrent, since K is ergodic. Moreover, it is aperiodic (the atom [g] 0 × {(g)} × [0, 1 β [ can appear twice in a row for example). Proposition 2 p. 579 of [START_REF] Ito | Remarks on the isomorphism theorems for weak Bernoulli transformations in general case[END_REF] implies that the partition P X is weak-Bernoulli, and the system ( X, μ, K) is isomorphic to a Bernoulli shift. This concludes the proof of Theorem 26.


Francisco Fabra

Anna Maria Vegni
email: annamaria.vegni@uniroma3.it

Valeria Loscrí
email: valeria.loscri@inria.fr

Carlos T Calafate
email: calafate@disca.upv.es

Juan Carlos Cano
email: jucano@disca.upv.es

Pietro Manzoni
email: pmanzoni@disca.upv.es

Collision-free Cooperative UAV Protocols for Sustainable Aerial Services

UAVs are offering many global industry sectors the opportunity to adopt more sustainable business models. By offering innovative ways of managing resources, and newer opportunities to address key challenges in many areas, UAVs are expected to play a relevant role in the upcoming 6G networks scenario. As an example of their deployment, in smartagriculture, UAVs are used to collect data such as weather and soil moisture, enabling more effective land management. They can also be applied to the inspection of infrastructure projects or equipment such as onshore and offshore wind turbines, reducing the need for potentially dangerous human inspection, or, finally, for the distribution businesses and services.

All these new applications tend to require the cooperation of groups, namely "swarms" of UAVs, in order to provide collaborative sensing and processing solutions. New requirements are then imposed in terms of safety, coordination, and operation management. In this paper, we provide an overview of some of the technical challenges that multirotor UAVs are still facing in terms of aerial coordination and interaction. In this regard, we focus on recent developments available in literature and present some contributions we realized during the past few years that address UAV interaction to achieve collision-free flights and swarm-based missions. Based on the analysis provided in this work, we eventually provide an insight on the challenges still open that need to be solved in order to enable effective UAVbased solutions to support sustainable aerial services.

I. INTRODUCTION

Recently, the concept of Internet of Things (IoT) is evolving to Internet of Everything, assuming a continuous increase of heterogeneous devices, from sensors for smart-agriculture and smart-farming to autonomous and unmanned ground vehicles, devices for smart home and smart cities, and also devices for health and wellness applications, including nanorobots for nanomedicine [START_REF] Miraz | A review on Internet of Things (IoT), Internet of Everything (IoE) and Internet of Nano Things (IoNT)[END_REF].

As depicted in Fig. 1, a general overview of IoT networks environment considers the coexistence, and also the coordination of a plethora of devices communicating with each other. In such a scenario, aerial IoT [START_REF] Lagkas | UAV IoT Framework Views and Challenges: Towards Protecting Drones as "Things[END_REF] is an emerging research area where the different benefits of Unmanned Aerial Vehicles (UAVs) are leveraged to assist in the creation of a richer heterogeneous IoT ecosystem, comprised of not only ground networks, but also ocean, aerial and space networks.

UAVs, popularly known as drones, are semi-autonomous or fully autonomous unmanned aircrafts that have embedded sensors, cameras, and communication equipment. UAVs have brought several benefits in the field of sustainable development. First of all, UAVs with cameras are a very useful tool when flying over large areas of land. They can quickly acquire images of cultivated areas, forest zones, areas with fire hazards, etc. with a minimal emission of pollutants and being able to reach the point of interest faster before the event becomes too serious. Also, UAVs have been largely deployed in solar power plants and wind farm applications. With the support of UAVs, technicians can fly over solar power plants and wind turbines to check for technical failures, material leaks or malfunctions without having to relocate personnel to each element of the installation. In this way, it can be avoided to put in danger people life, while saving fuel and time. Finally, with the spread of the COVID-19 pandemic, UAVs have also been validated as a solution to counter this epidemic with specific tasks such as disinfection of large areas, thermal image collection, quarantine patient identification and monitoring, and contact detection [START_REF] Chamola | A comprehensive review of the covid-19 pandemic and the role of iot, drones, ai, blockchain, and 5g in managing its impact[END_REF].

Acting as supporting nodes for communications, UAVs can be deployed on demand, and can benefit from a wider communications range and better line-of-sight (LOS) features than ground network infrastructures. Communications among such heterogeneous devices should be guaranteed in order to enhance network performance and reduce connectivity issues, especially in harsh environments. A simple example relays on ground-to-aerial (G2A) wireless links for a rural IoT setting, where coverage issues require a UAV to be deployed in order to act as a mobile gateway for the different ground sensing devices. More complex deployments include those situations where UAVs are also acting as mobile sensors, gathering data and possibly processing and transmitting it in real time via aerial-to-ground (A2G) and aerial-to-space (A2S) links.

A detailed analysis on the impact of UAVs on CO2 emissions and cost has been provided in [START_REF] Chiang | Impact of drone delivery on sustainability and cost: Realizing the uav potential through vehicle routing optimization[END_REF], where the authors demonstrate that using UAVs for last-mile logistics is not only cost effective, but also environmentally friendly.

Finally, through swarm features, UAVs show a collaborative behavior by leveraging on communication facilities, with the aim of achieving a common goal [START_REF] Tahir | Swarms of unmanned aerial vehicles: A survey[END_REF]. When deployed as a swarm, UAVs can deliver multiple virtual/augmented reality immersive communication sessions to remote users, in order to assist more efficiently in these types of scenarios [START_REF] Lan | Development of uav based virtual reality systems[END_REF].

Despite the many advantages, the use of UAVs involves a number of critical issues to be considered, such as privacy, security, and flight safety [START_REF] Mohammed | UAVs for smart cities: Opportunities and challenges[END_REF], especially in urban environments where the consequences of any flight disruption are typically much more severe due to the risks of injuries for citizens. To address such issues, several efforts are taking place worldwide to make UAV flights safer. For instance, U-space is an European initiative that aims at making UAV traffic management safer and more secure. It attempts to provide an appropriate interface with manned aviation and air traffic control, to facilitate any kind of routine mission in all classes of airspace, thus achieving the ambitious Single European Sky (SES) goal. The SESAR Joint Undertaking [8] partnership was set up in order to manage this large scale effort, coordinating and concentrating all European research and development activities focused on air traffic management. This way, a wide range of UAV missions that are currently restricted will be possible in a near future thanks to a sustainable and robust European ecosystem that is globally interoperable.

In this paper, we will present an overview of main open challenges of UAVs, related to both their aerial coordination and interaction, as well as their communication capability, in the context of sustainable aerial applications.

This paper is organized as follows. Section II presents an overview of the main challenges UAVs face when deployed for sustainability services. In Section III, we describe some collaborative UAV solutions with particular emphasis to applications where UAV swarms are mainly applied, and the respective solutions, along with UAV crash avoidance strategies to provide collision-free flights. Section IV follows, dealing with some insights on the main gaps still existing with the current solutions, and we sketch the main future research directions. Finally, conclusions are drawn at the end of this paper.

II. UAV CHALLENGES FOR SUSTAINABLE AERIAL OPERATIONS

In the coming years, UAVs are expected to become ubiquitous, flying around in both indoor and outdoor environments. Such a massive number of UAVs raises new challenges that deserve scrutiny, like dynamic and flexible geofencing to achieve flexible aerial operations, energy efficiency and resource management, as well as security issues. In the following, a brief overview of the main UAV challenges is presented.

Dynamic geofencing for flexible aerial operations. Geofencing is essential to ensure that UAVs comply with airspace restrictions in order to efficiently use the low-altitude airspace, while keeping it safe for all. In particular, standard geofences are those that prevent UAVs from entering no-fly zones, G2A/A2G (ground-to-aerial/aerial-to-ground), G2S/S2G (ground-to-space/space-to-ground), A2S/S2A (aerial-to-space/space-to-aerial), O2G/G2O (ocean-toground/ground-to-ocean), O2A/A2O (ocean-to-aerial/aerialto-ocean), G2O/O2G (ground-to-ocean/ocean-to-ground), and O2S/S2O (ocean-to-space/space-to-ocean). and to keep them away from protected areas and critical infrastructures. Dynamic geofences can also be defined by a UAV operator to create a temporary restricted airspace that avoids other UAVs to enter that space during a mission. While the former type of fencing has little flexibility, it has the advantages that their number is reduced, and the locations of such fences are well known. On the contrary, with the proliferation of UAVs, it becomes possible that many UAV operations take place in a same area (e.g., urban scenarios), and so a strict geofence per UAV operation may become too restrictive, or severely affect the sustainability of operations.

An example is provided in Fig. 2 (a) where UAV #2 has to take a long detour to avoid entering the geofenced area defined by UAV #1, which results in additional time and energy consumption. The problem is even more severe in Fig. 2 (b), where the target for UAV #2 is inside the area temporally reserved by UAV #1, meaning that UAV #2 cannot complete its mission until the reservation comes to an end. Yet, notice how, in both cases, UAV #1 is in fact operation away from the direct path followed by UAV #2 at the time it is approaching, meaning that the crossing of the geofenced area would not represent a problem for none of them.

To make such optimization possible, an automatic detection and avoidance of potential collisions between UAVs becomes quite relevant. Hence, the crash avoidance problem, and the computation of optimal paths, have to be jointly considered with the requirement of energy consumption minimization.

The challenge described above can also be extended to the swarm case. For instance, if multiple UAVs from a same operator have to do the same mission, it becomes much faster to make them fly together as a swarm, thereby being able to release the aerial space much sooner than having the UAVs fly sequentially in independent missions. Furthermore, in those cases, avoiding collisions with a swarm of UAVs instead of a single UAV becomes much more complex. Inside the swarm itself, UAVs should also avoid collisions with their neighbors. In this regard, planning optimal trajectories by avoiding collisions with other UAVs and obstacles remains a critical and open issue [START_REF] Mu | Formation control and obstacle/collision avoidance with dynamic constraints[END_REF].

Aerial sensing and data relaying. The capabilities of UAVs enable the adoption of novel IoT applications. From the sensing perspective, they can lift sensors of low/moderate weight, and provide high-altitude measurements that prove to be priceless in many contexts, including fire detection, precision agriculture, wildlife monitoring, and every application benefiting from aerial cameras. Nowadays, these devices can also include embedded units with edge computing capabilities, allowing data to benefit from processing at the edge, on the UAV itself, effectively reducing communication requirements, and even enabling a faster response time in some contexts.

From the data relaying perspective, UAVs are shown to be highly beneficial in many aerial IoT contexts by acting as mobile sinks for sensors located in place with littleor also, no-wireless coverage, or merely as message relays towards the wireless infrastructure via multi-hopping, or by acting as data mules from a Delay Tolerant Networking (DTN) perspective.

Identification of Key Parameters and Performance Analysis of UAVs. In a complex IoT ecosystem with collaborative UAVs exchanging a high number of messages, including both data and coordination messages, the correct identification of the key parameters, and their inclusion the system design, plays a fundamental role. In order to correctly evaluate the performance of a UAV-based aerial IoT system, Quality of Service (QoS) metrics should be considered, such as coverage, reliability, connectivity, etc. In most cases, trade-offs need to be considered related to the specific IoT scenarios/applications, by developing event-driven/mission-driven solutions [START_REF] Nawaz | UAV Communication Networks Issues: A Review[END_REF]. Of course, network performance depend on the particular communication technology adopted. As an instance, in case of Free Space Optics (FSO) data links, weather conditions and the atmospheric environment can cause fluctuations in amplitude and attenuation of the optical signal.

Energy Efficiency and Resource Management. Beside evaluating the impact of the main parameters on the global system, important constraints such as the limited available energy should be considered for the implementation of effective IoT solutions. In particular, resource management for UAVs is characterized with specific challenges due to different factors such as (i) unique mobility characteristics, and (ii) stringent energy and flight limitations. An effective path plan has to consider all these features. However, such factors increase the complexity of these systems, requiring the resource allocation process to be optimized [START_REF] Yang | Energy Efficient Resource Allocation in UAV-Enabled Mobile Edge Computing Networks[END_REF].

III. COOPERATIVE SOLUTIONS FOR COLLISION-FREE FLIGHTS

The previous section highlighted some challenges in different areas related to sustainable aerial flights. This section provides an overview of some contributions addressing the first challenge i.e., dynamic geofencing for flexible aerial operations. We put evidence on how UAV coordination mechanisms enabled by wireless communications can effectively help at having flexible and dynamic geofencing that can avoid strict reservations of air space by detecting and avoiding possible crashes when UAVs cross each others paths. In addition, we also show how to create and maintain UAV swarms undertaking both autonomous missions and manually guided swarm missions in order to minimize the UAV flight times, thereby reducing the impact of dynamic geofencing by releasing the airspace sooner.

In the following, we start by addressing the problem of dynamic geofencing based on "sense and avoid" techniques.

A. Dynamic geofencing through sense and avoid techniques

Among the different areas where UAV flight safety is being considered, the development of sense and crash-avoidance mechanisms to enable dynamic and flexible geofencing has not yet been fully addressed. This problem is applicable to all types of contexts, and applies to both single UAVs performing a mission, or a group of UAVs acting as a single swarm, and often requires taking evasive actions to avoid crashes [START_REF] Mohammed | UAVs for smart cities: Opportunities and challenges[END_REF]. To address this challenge, we proposed the Mission Based Collision Avoidance Protocol (MBCAP) [START_REF] Fabra | A Distributed Approach for Collision Avoidance between Multirotor UAVs Following Planned Missions[END_REF], i.e., a collision avoidance solution that relies on wireless communications between nearby UAVs performing planned missions. In particular, MBCAP-enabled UAVs will constantly broadcast their future positions and, whenever two UAVs determine that their flight trajectories overlap in time and space, they will stop to quickly negotiate, and execute the process to safely go through the critical area, while giving higher priority to one of the UAVs according to any specified criteria.

MBCAP has been implemented in real UAVs. Experimental results, along with large-scale simulation experiments, have validated the effectiveness of our proposed protocol, and evidenced the low overhead introduced both in terms of channel occupation and mission delays. Through practical validation, we have observed that MBCAP is able to avoid UAV crashes, defined as having an inter-UAV distance lower than 4 m, when varying the total number of UAVs in an area sized 5 × 5 km 2 . Table I collects the performance of MBCAP, which shows an effective behavior even when the number of UAVs reaches high values. In particular, we find that the chances of collision remain in general very low, only surpassing the 1 crash per hour threshold for 100 UAVs. A video summarizing our experiments is available online at: https://youtu.be/bEdcsPX1hXY.

Further results of the MBCAP technique are reported in Fig. 3, which depicts a Google Earth 3D view that shows (i) the path followed by the real multicopters with a red line, (ii) the path of the virtual high-priority UAV with a blue line, and (iii) the route of the virtual low-priority UAV with a black line. The green arrows indicate the direction the UAVs are moving towards before detecting the collision risk, also marked with a green circle. We can observe that the paths followed in simulation and in real experiments are quite similar and mostly overlap. A video showing our real experiments is also available online at: https://youtu.be/xHnMuMOd9C0.

Overall, we find that MBCAP technique can effectively allow multiple UAVs to operate without air space segregation, paving the way for more flexible and sustainable aerial operations, especially in environments characterized by a high volume of UAV activities.

B. UAV Swarm Solutions

Although there are already some solutions for the automation of UAV swarm flights, in certain situations the support for coordinated missions is required. Examples of such situations may include rescue operations and environmental monitoring, as well as applications for large-scale agriculture Fig. 3: MBCAP simulation vs. real experiment in a perpendicular crossing. in search of pests or weeds, wild life recordings, or border surveillance, among others. In these cases, the coordination of all UAVs conforming the swarm is critical when carrying out the mission to avoid collisions and promote network stability. Such process typically relies on pre-planned missions where communications among UAVs are critical to enable near-realtime responsiveness so as to maintain the consistency of the swarm.

Mission planning for collaborative UAVs is a complex aspect that involves a compromise between (i) the necessary level of centralization or decentralization, (ii) the level of abstraction at which plans are generated, and (iii) the level to which such plans are distributed among participating units.

Regarding applications where UAV guidance must be manual and not following a pre-planned mission, UAVs that make up the swarm have to dynamically adjust their routes in order to follow the master UAV acting as the leader of the swarm. Such a solution may be required in scenarios such as search & rescue [START_REF] Aljehani | Multi-UAV tracking and scanning systems in M2M communication for disaster response[END_REF], fire tracking, or the monitoring of disaster areas. In all these cases, the pilot must respond to visual stimuli in real-time, and adapt the UAV course accordingly. Furthermore, there are situations where, in addition to manual guidance, there is the need to carry multiple items or sensors that go beyond the lifting capacity of a single UAV.

In terms of sustainability, having UAVs flying as a flock enables faster operations involving multiples UAVs, which has the potential of minimizing the usage of aerial space in order to release it as soon as possible for other operations in the same area. We now proceed to address the swarm communication problem, and then two proposals to create swarms efficiently.

1) Swarm communications: The reliability of wireless communications is one of the main problems in the creation and maintenance of swarms. Through direct wireless links, collaborative UAVs can build an infrastructure-less wireless network. However, collaborative UAVs solutions present some limitations, mainly depending on the heterogeneous nature of devices. Developing applications for UAVs with heterogeneous devices, showing different energy levels, storage features, communication, sensing and processing capabilities, represents a complex task.

The distance separating neighboring UAVs must also maintain consistency to avoid both collisions and interruption of communications, which hinders synchronization, causing delays to the entire process, or even a reduction of the number of UAVs in the swarm.

Other of the key challenges in UAV-based communications is the backhaul throughput. Hanna et al. [START_REF] Hanna | Distributed UAV Placement Optimization for Cooperative Line-of-sight MIMO Communications[END_REF] proposed an approach for the optimization of the UAV swarm positions, in order to achieve a high multiplexing gain in MIMO backhaul with LOS. They developed two distributed algorithms to estimate UAV positions such that each UAV moves for a minimal distance to achieve the highest capacity in a LOS MIMO channel. Another work [START_REF] Hanna | UAV Swarms as Amplify-and-Forward MIMO Relays[END_REF] considered the use of a UAV swarm as an amplify-and-forward MIMO relay to provide connectivity between an obstructed multi-antenna equipped source and destination. Authors also proposed a simple near-optimal approach to find the positions that optimize the capacity for the end-to-end link given that the source and the destination have uniform rectangular arrays.

2) The MUSCOP protocol [START_REF] Fabra | An UAV Swarm Coordination Protocol Supporting Planned Missions[END_REF]: The first approach is a solution able to provide UAV coordination to maintain the desired flight formation when carrying out planned missions. MUSCOP uses a centralized approach where the master UAV synchronizes all UAV slaves each time they reach an intermediate point in the mission. Our protocol allows different formations to be created around the leader (e.g., matrix, linear and circular). In addition, the MUSCOP protocol has been validated using our ArduSim simulation framework, which allows to perform realistic experiments in two types of environments, i.e., ideal and lossy wireless channel, in order to validate the formations with different numbers of UAVs. In an illustrative video (https://youtu.be/VLMsbL5B6tA) we run three experiments with different flight formations on the ArduSim simulator.

Fig. 4 shows the MUSCOP time offset when varying the distance among UAVs, which represents the relative delay swarm members have with regard to the reference, in this case provided by the master UAV. In general, it is observed that the mean time offset increases as the separation distance increases. This is expected since, in a lossy channel, large distances will impair or harm the synchronization among UAVs, thus becoming a critical problem. It is also worth mentioning that, for up to 300 m between contiguous UAVs, the time offset does never exceed 1 s threshold, which is a significant achievement.

3) The FollowMe protocol [START_REF] Fabra | Automatic system supporting multicopter swarms with manual guidance[END_REF]: The other solution is the FollowMe protocol [START_REF] Fabra | Automatic system supporting multicopter swarms with manual guidance[END_REF], able to define and maintain the formation of UAVs in a swarm. In particular, it is applicable to the specific case where a real pilot controls the swarm leader, and the other UAVs (i.e., slaves) automatically follow it in real time. When all the slaves are detected by the master UAV, and the user starts the setup step of the simulation, a message is issued by the master UAV, including its coordinates and their theoretical position in the flight formation. Once the master UAV reaches the same altitude as the rest of the UAVs in the swarm, it periodically broadcasts another message during its flight, which includes the current 3D location and heading of the master. Each time a slave receives this message, it calculates a new target location, and issues a command to the flight controller to move to the designated location in the formation. Finally, experimental results under different conditions show that the FollowMe protocol is able to adequately maintain the swarm formation and dynamically respond to the pilot's commands, with a lag of only a few seconds in the worst case, avoiding collision among UAVs as long as some minimal distances towards neighboring UAVs are defined.

IV. LESSONS LEARNT AND FUTURE RESEARCH

DIRECTIONS

Bridging the research in the UAV field and the related IoT areas remains a challenge to be tackled due to the many issues that arise. From our experience, real UAV flights introduce several problems, including (i) deployment issues, (ii) flight restrictions, as well as (iii) development and communication issues.

In terms of deployment, carrying, assembling and preparing a high number of UAVs for a flight is quite time consuming, and can derive in crashes when attempting to take off a large number of UAVs. It follows that having more compact UAVs with minimal assembly time is important, as well as having self-arranging algorithms to automatically organize takeoff procedures. In fact, the latter is a complex problem when scaling up to hundreds of UAVs, and remains an open challenge. Similar considerations apply for the UAV development, as commercial UAVs are characterized in their majority by being closed platforms that are not easily amenable to enhancements, hindering any development efforts in most cases, or merely offering a limited API. It becomes critical to promote open platforms that make UAV development possible and straightforward.

In terms of flight restrictions, many researchers disregard the current regulations regarding flight altitude (maximum of 400 feet in most countries), as well as the presence of physical obstacles, along with battery limitations, which limit flight times to a maximum of 30 minutes using current technology. Hence, in the context of aerial IoT and IoT-supporting UAVs, efficient path planning emerges as a critical issue.

Finally, regarding communication issues, it is worth pointing out that current commercial UAVs lack a technology that enables these UAVs to communicate with each other, a requirement that is critical to support any collaborative approaches such as the ones described earlier. In fact, there is currently no standard for UAV-to-UAV communications, meaning that experiments such as the ones performed are only made possible by relying on ad hoc wireless settings that are not generic enough to enable a seamless adoption. Additionally, commercial UAVs also lack the integration of wireless technologies that are more typical of IoT environments as only WiFi interfaces are usually available, and so additional interfaces must be equipped, a requirement that often can only be addressed via building custom UAVs.

V. CONCLUSIONS

UAVs and their associated applications represent nowadays some of the most exciting and promising research fields. In the upcoming years, with the advent of 6G networks, they are expected to provide several social benefits with a plethora of solutions for large-scale agriculture, environmental monitoring, or even contact monitoring and tracing in case of pandemics, such as COVID-19.

This paper presented the main challenges for sustainable and collision-free operations of UAVs, with an emphasis on those solutions where wireless communications are able to leverage collaborative UAV applications to improve the performance of different aerial scenarios. Specifically, we focused on a sense and crash-avoidance solution with general applicability, including experimental results carried out directly on field, as well as swarm management protocols, including some solutions that have been developed in our research group. Experiments have shown it is feasible to achieve collision-free flights with a low communications overhead, and introducing minimal delay to the planned missions. Overall, we believe such contributions will favor the deployment of autonomous UAV flights, which is a basic requirement for performing all sorts of tasks, and to promote a sustainable UAV ecosystem.

Fig. 1 :

 1 Fig. 1: Schematic of ubiquitous IoT with integrated and overlapping ground, ocean, aerial and space networks. Legend:G2A/A2G (ground-to-aerial/aerial-to-ground), G2S/S2G (ground-to-space/space-to-ground), A2S/S2A (aerial-to-space/space-to-aerial), O2G/G2O (ocean-toground/ground-to-ocean), O2A/A2O (ocean-to-aerial/aerialto-ocean), G2O/O2G (ground-to-ocean/ocean-to-ground), and O2S/S2O (ocean-to-space/space-to-ocean).

Fig. 2 :

 2 Fig. 2: Examples of geofencing issues.

Fig. 4 :

 4 Fig.4: MUSCOP time offset when varying the inter-UAV distance,[START_REF] Fabra | An UAV Swarm Coordination Protocol Supporting Planned Missions[END_REF].

TABLE I :

 I Number of UAV crashes and deadlocks avoided through MBCAP (mean value per hour of simulation).

	Number of UAVs	25	50	75	100
	Collisions expected	6.5	16.5	45.5	84.25
	Risks detected	23.08	105.08	249.08	438
	Crashes (distance < 4 m)	0.08	0.08	0.58	1.08
	Deadlocks avoided	0	0.33	0.25	0.58

ACKNOWLEDGMENTS

This work was partially supported by the "Ministerio de Ciencia, Innovación y Universidades, Programa Estatal de Investigación, Desarrollo e Innovación Orientada a los Retos de la Sociedad, Proyectos I+D+I 2018", Spain, under Grant RTI2018-096384-B-I00.