Idriss Jamil

Idriss J Aberkane

At least almost all Collatz orbits attain bounded values.

Introduction

This communication introduces a simple yet powerful methodology to study the Collatz orbits, essentially consisting of analysing each natural number with respect to its position on the complete binary, ternary and quaternary trees over N, that is, trees defined respectively by the infinite iteration of all the possible compositions of the following operations on number 1:

• {•2; •2 + 1} binary tree • {•3; •3 + 1; •3 + 2} ternary tree 1 • {•4; •4 + 1; •4 + 2; •4 + 3} quaternary tree ...the binary tree thus generating an infinity of branches that has the cardinality of R (and the quaternary one, the cardinality of ℘ (R)) [START_REF] Aberkane | On the Syracuse conjecture over the binary tree[END_REF] . For any number, defining its neighbour- hood in terms of which branch (and of which length) it belongs to on each of these trees provides a framework to demonstrate very fruitful results that could actually be applied beyond the Syracuse problem, and more importantly, beyond discrete mathematics, for example in the study of the Julia sets of holomorphic functions.

Definitions

Note 2.1. For all intent and purpose we will define Syr(x) as "the next odd number in the forward Collatz orbit of x".

Whenever two numbers a and b have a common number in their orbit, we will also note a ≡ b, a relation that is self-evidently transitive: The set 2N + 1 is thus endowed with three unary operations without a general inverse and that are non commutative with G • S = V . Whenever we will mention the inverse of these operations, it will be assuming they exist on N.

Definition 2.4. Rank

The rank of an odd number is the number of consecutive end digits 1 in its base 2 representation, or equivalently, the number of times the action S has been applied to generate it (S is then defined on N), and any number o of rank 1 can be written S(e) where e is even.

Definition 2.5. Odd branch An odd branch is the infinite set of numbers {a; S(a); S 2 (a) . . .} where a is of rank 1.

Definition 2.6. Glacis

The glacis of bottom b is the infinite set of numbers {G(V (b)); G 2 (V (b)) . . .} where b is any odd number.

Definition 2.7. Root

The root r of any odd branch is its only number of rank 1.

Definition 2.8. Determinant

The determinant d of any odd branch of root r is G -1 (r). Any number of rank 1 admits a determinant.

Definition 2.9. Vertical, Verticality A number a of rank 1 always admits a V -1 (a) which is called its Vertical. To avoid any confusion, when ambiguous, we will call V (a) the V + of a. If the vertical of a is odd, we will call it Vertical odd, otherwise, it is Vertical even.

The Verticality of a number a of rank 1 is the vector

[ n b
] where b is either an even number or a number of rank 2 or more, and a = V n (b) We will say that a has a verticality of n and of bottom b.

Definition 2.10. Successal, Successality

A number a of rank 2 or more will be called Successal, and its successality is equal to its rank.

Definition 2.11. Glaciality

In a glacis of bottom b, the

glaciality of S(b) is set to -1, that of V(b) is set to 0 and that of G n (V (b)) is set to n.
To aggregate the information of the bottom b of its glacis to any glacial number (that is, a number that can be written as G(x) where x is odd), its glaciality will be the vector Note 3.4. The Syracuse action over an odd number is equivalent to adding 1 to it, then the half of the result, then -1. How many times one can add an half to the number +1 directly depends on the length of the immediate even branch of the binary tree that is to its right.

S(a) = 4d -1 3S(a)+1 2 = 12d-2 2 = 6d -1 = G(3 • d) Syr(S(a)) = G(3 • d)
Let us take Mersenne numbers for example, that are defined as 2 n -1. One can Syracuse them consecutively a number of time that is proportionate to their rank-1, indeed, 31, which is written 11111 is of rank 5, because 32 = 2 5 so if you repeat the action "add to the number the half of itself" which is equivalent to a multiplication by 3 2 this will yield an even result exactly four consecutive times.

Thus, any ascending orbit in Syracuse concerns only numbers a of rank 2 or more, and is defined by

(a + 1) • ( 3 2 ) rank-1 -1
because the rank is strictly equivalent to the length of the next even branch of the binary tree on the right of the number, defining how many consecutive times the action • 3 2 will yield an even number. Lemma 3.5. Let a be an odd number of rank 1 that is vertical even, then 3a is successal, and 9a is vertical even. Let a be an odd number of rank 1 that is vertical odd, then 3a is successal, and 9a is vertical odd.

Proof. If a is vertical even it can be written 8k + 1 ∀k 3a = 24k + 3 and this number admits an S -1 that is 12k + 1, which is an odd number, therefore 3a is Successal 9a = 72k + 9 and this number admits a V -1 that is 18k + 2, an even number indeed. Now if a is vertical odd, it can be written 8k + 5 ∀k 3a = 24k + 15 and 9a = 72k + 45, 3a admits an S -1 and 9a admits a V -1 , respectively 12k + 7 and 18k + 11 and they are both odd. Theorem 3.6. (regular quaternary equivalence) Let a be a number that is vertical even, then (a) ≡ S(a) and S k (a) ≡ S k+1 (a) for any even k. Let a be a number that is vertical odd, then S(a) ≡ S 2 (a) and S k (a) ≡ S k+1 (a) for any odd k.

We will call these relations merging alternate pairs of odd branches "regular quaternary equivalences" or qe.

Proof. If a is vertical even then it can be written as G(d) where d is necessarily vertical (odd or even) so by lemma 3.5 we have that 3d is successal and by lemma 3.2 we have Syr(S(a)) = G(3d) so it is necessarily vertical odd (since 3d is successal) so Syr(a) = V -1 (Syr(S(a)) and therefore a ≡ S(a)

This behavior we can now generalize to the n, because if a is vertical even and of determinant d, then the lemmas we used also provide that Syr n (S n (a)) = G(3 n • d) and therefore Syr n (S n (a)) will be vertical even for any even n because 3 n • d will be vertical something (even or odd, depending on what the determinant was) for any even n. Now if a is vertical odd it can be written G(d) and d is necessarily successal because

G • S = V .
Thus 3d is vertical (even or odd) and therefore Syr(S(a)) = G(3d) is vertical even. This qe theorem is a more elaborate, and now very useful building block of our demonstration, because it allows to place a relation of equivalence between every other pair of any odd branch of the binary tree, up to infinity. It is also based on the characteristics of increasing phases of the Syracuse orbits: any number of rank n is finitely turned into a number that has a vertical, which is either odd or even.

We should now interest ourselves with the decreasing phases of the Syracuse orbits2 , and they only concern the glacis.

Theorem 3.7. (glacial decreasing)

Let a be a vertical even number with a glaciality of

[ n b ]
where n is even, then a ≡ 3

n 2 (b)
Let a be a vertical even number with a glaciality of

[ m b
] where m is odd, then a ≡

S(3 m+1 2 (b))
Proof. if a is of glaciality

[ n b ]
then by definition a = 2 n+2 b + 1.

Then 3 • a + 1 = 3(2 n+2 b + 1) + 1) = 2 n+2 • (3b) + 4.
As this expression can be divided by no more than 4, we have

Syr(a) = 2 n 3b + 1, therefore the glaciality of Syr(a) is [ n -2 3b ] Note that if n = 2 then V -(Syr(a)) = V -(2 2 • (3a) + 1) = 2 2 • 1 4 • (3b) = 3b
which is of course an odd number. Therefore Syr(a) is vertical odd and V -(Syr(a)) = 3b thus we have proven that a ≡ 3b.

If n = 1 then a = 2 3 • b + 1 so 3(a + 1) = 2 3 • 3b + 4 therefore Syr(a) = S(

3b) and thus a ≡ S(3b).

From this we can generalise the progression of glacis numbers. Let b be any odd number, thus defining a glacios bottom. All "Variety S" numbers of its glacis are written

V (b • 2 2k-1 ) or S(b • 2 2k ) = 2 2k+1 • b + 1 and all "Variety V" numbers of its glacis are written V (b • 4 k ) or equivalently S(b • 2 2k+1 ) = 4 k+1 • b + 1.
Any glacis number g of order 2k or 2k -1 may be thus reduced to a glaciality of 0 or -1 by the following transformation:

(g -1) • ( 3 4
) k + 1 therefore we do have indeed that,

• for Variety S numbers :

2 2k+1 • b • ( 3 4
)

k + 1 = 2b • 3 k + 1 = S(b • 3 k ) • for Variety V numbers : 4 • 4 k • b • ( 3 4
) k + 1 = 4b • 3 k + 1 = V (b • 3 k )
As the obtaining of these equalities will fit any odd number b, we have that any glacis number of glaciality 2k will be finitely mapped to 4b

• 3 k + 1 = V (b • 3 k
) and that any glacis number of order 2k -1 will be mapped to 2b

• 3 k + 1 = S(b • 3 k ).
. Any glacis number merges either directly with a power of three of its bottom or with the successor of it.

General description of the Syracuse dynamic

The qe theorem allows to place an infinite amount of equivalences along the binary tree that are the result of quaternary properties. warped the binary tree so that V(a) is indeed Vertical to a. We have also not connected the rank 1 numbers to the even ones of which they are the successors so as to make glacis more easily visible. The bold black lines indicate qe equivalences and vertical equivalences, and so whenever numbers are joined by a connective series of those lines, their Collatz orbits merge. Connecting all of those equivalences together completely solves the Syracuse problem, and as we will see in the next section, this endeavour requires the introduction of a third dimension: that of the ternary representation of any number.

Ascending phases

The orbit of an odd number can only increase if it is of rank 2 or more. Odd branch numbers of rank n > 2 ascend with the progression +1 • 3 2 n-1 -1 and this allows to compress their orbit to the next glacis.

More particularly though, if a pair is not connected by the qe equivalence, it is because the rank 1 reduction of its smallest number by Syracuse is vertical odd. When we have a pair {a; S(a)} where a is vertical odd, what happens is that a is mapped to Syr(V -1 (a)) and S(a) is mapped to the glacis of bottom Syr(V -1 (a)) and this phenomenon, by which a pair of numbers that were related on the binary tree become separated we call an Avalanche. Avalanches account for absolutely all of the chaoticity of the Syracuse orbits, and whoever can perfectly predict their occurence and consequences has cracked all of the Syracuse orbits. Needless to say, this communication doesn't have this ambition.

An example of Avalanche can be understood on figure 2 by observing the pair {3; 7} of which the first Syracuse image is {5; 11} where 5 indeed is vertical odd. The avalanche is that 5 is mapped to the image of 1, which is 1, and 11 is mapped to 17, which is in the glacis of bottom 1. This happens to all such pairs, which we have called "buds" in a previous work.

The alacrious reader will not fail to notice that 17 is precisely a glacis number of Variety V , and this is not happening by chance: if a vertical odd number a is the finite vertical of an even number, then S(a) will be mapped as a variety V in the next glacis, and proportionally as high as a was vertical, and if the bottom of its verticality is odd, then S(a) will be mapped as a variety S. The easy reason for which it is so will be left for the reader to grasp, as we do not use it in the other demonstrations of this piece.

We have also noted that any power of 9 of a vertical number is either of the series "vertical odd" or the series "vertical even" and these two are parallel: one cannot obtain a vertical odd number by applying any power of nine to a vertical even one. Thus, the destiny of branches the determinant of which falls within the "vertical odd" or "vertical even" series is quite different, and this level of precision can also help understand the Syracuse dynamic better. Still, we shan't use it further in this communication, even though it is more significant than a simple curiosity.

Descending phases

Any odd number can only decrease in Syr if it is of rank 1, and then it intersects either the Successor or the Vertical of a power of 3 of the bottom of its glacis. Note that in so doing, it always encounters the consequences of another qe on the branch it meets: either the power of three of the bottom of the glacis can be proven to merge with its successor, either it is vertical odd or it is its predecessor that is merging with it.

Here too, powers of 3 of the bottom of the glacis, just as being the case with odd branch determinants had useful implications to elaborate more advanced theorems, can either be of the "vertical odd" or "vertical even" sequences.

Numbers of even glaciality n decrease following the dynamic -1 • ( 3 4 )

n 2 + 1 and those of odd glaciality m, -1 • ( 3 4 ) m+1 2 + 1.
Thus, whenever one can prove that an odd number merges with its triple, one proves that it merges with its first number of even glaciality Also, whenever one can prove that an odd number merges with the successor of its triple, one proves that it merges with its first number of odd glaciality.

We cannot stress enough the importance of this pair of results.

5 Using the ternary tree to connect the quaternary equivalences

Definitions

In the previous sections we have mostly identified numbers by their position in the complete binary and quaternary trees over N. Using statements of the kind "a is vertical even" is a typical example of crossing binary and quaternary properties to identify specific characteristics of a number. We will now expand this methodology by adding the ternary tree, which elements we will identify with the following definitions:

Definition 5.1. Ternary, Ternarity A number b is ternary or of type B, if it can be divided by 3. Its ternarity is the total number of times it can be divided by 3, to which we will add the information of the non-ternary number resulting from this finite operation, thus the full ternarity of any ternary number that can be written 3 n • x where x is non ternary is

[ n x ]
For all intent and purpose, when we will refer to just the "ternarity" of a B type number (as opposed to "full ternarity") we will just be meaning n alone.

Definition 5.2. 1-ternary, 1-ternarity A number c is 1-ternary or of type C, if its base 3 representation ends with digit 1.

The number of times one can remove a consecutive end digit 1 (we call this operation C -1 ) is its 1-ternarity, to which we add the information of the number resulting from it. Thus, a number c that can be written x 1 . . . Numbers of a ternarity of 2 or more (numbers that can be divided by 9, that is) are also colored in gold. To explain again the previous definitions: 27 is a Bup and 63 is a Bdown for example. Any glacis number of glaciality above 0 is "up": 17 is an Aup for example, and 19 a Cdown. Though exotic, these names are absolutely essential to the results we obtain, and their very use is a pure result of our methodology: the "up" and "down" properties come from the study of the intersections of the binary and quaternary trees, and the A,B,C ones, from the ternary.

1

Golden Automaton

We shall now define an algorithm that can mend together infinitely many quaternary equivalences, and actually at least almost all of them. This algorithm we call the Golden Automaton or GA.

1. Start from the equivalence 1 ≡ 3 ≡ 5

2. Whenever a new type B number is found in the network, divide it by 3, and always prioritize the Bups for the first round, the only B is 3, so the GA notes that 1 ≡ 3

3. project the equivalences in the corresponding glacis at round 1, there is only one equivalence: having that 1 merges with its triple means it also merges with its first glacis number of the V Variety, thus giving 17 ≡ 1 4. whenever a new type A glacis number is found in the network, apply A -1 up to a non A type 3 , which is added to the network. If this generates an equivalence of the type B ≡ S(B), prioritize it.

at round 1, this gives 17 ≡ 7. Also by the qe 7 ≡ S(15) and 15 is of type B, but not a Bup. so it will not be prioritized by the GA. Still, we now have 3 ≡ S( 3), which will be put to use in the glacis of bottom 1 5. project the equivalence in the corresponding glacis 3 ≡ S(3) → 1 ≡ 9. 9 is ternary, and being a glacis number, it is a Bup, so it will now prove two glacis equivalences together, namely those of 49 and 35 (each of them bringing two Bdowns) and equivalently, 65 and 33.

6. After checking the Aups and prioritizing those of the highest 2-ternarity (they are easy to spot, as they can be written G(B) where B is of high ternarity) check all the Bdowns added to the network, including the ones that are the verticals of an A type. Always prioritize the lowest numbers in any case.

7. repeat. Proof. The moment a Bup is brought to its growing equivalence network, the GA can finitely prove that a number of either type A or C merges with two consecutive glacis numbers. Also, if the Bup is of a ternarity above 1, it will additionally prove that two glacial type C merge with a type B bottom, for example 1 ≡ 9 means 65 ≡ 33 ≡ 1 but also 49 ≡ 25 ≡ 3 because 9 is of ternarity 2. Exploiting the first Bup, there will always be an Aup and Bup that will be freshly grafted to the equivalence network, either in the descending glacis order (B; A) if the bottom is of type C, and

(A; B) if it is of type A.
The more ternary the Bup, the higher-reaching the equivalence. That a Bup be brought in the network guarantees a full new equivalence of the type x ≡ 3 k x ≡ S(3 k ) which will always reconnect a new Aup and a new Bup. Thus, the grafting of a new Bup to the network always implies the grafting of another one. So it cannot stop.

Can it loop now? Any Bup the GA gets, it does by either operations

G • G • V or G • V ,
and then it will also get an Aup in the process. The equivalence it obtains is then that Bup

3 merges both with G • G • V ( Bup 3 ) and G • V ( Bup 3 )
this progression is strictly increasing, precisely because glacis numbers have a strictly decreasing progression in Syracuse.

Theorem 5.2. The Golden Automaton connects at least almost all orbits.

Proof. So we have an algorithm that never stops, cannot loop and that recoups an exponentially growing diversity of numbers within the ternary tree over N. Also, all the ascending Syracuse orbits lead to numbers that can be written G(B), and all the descending ones, to numbers that are either B or S(B). Remember too, that, regarding the title of this communication, whenever an orbit is grafted by the GA, it is never almost bounded, but bounded, period. Besides, we have not at all fully exploited all the Bdown the GA grafts to its equivalence network, and which new equivalences they also prove. Even though each of them is initially less powerful than the grafting a Bup (ie. in its first round of exploitation, a freshly grafted Bdown either offers the grafting of a new Aup or that of a new Bup) each round of running the GA brings proportionally much more of them, and this for simple reasons kind

• whenever the GA grafts a number of glaciality above 2, it also grafts its next glacial image, which is necessarily of type C. For example, as we have already seen, grafting 65 and 33 implies the grafting of the type C 49 and 25 (their next glacial images) and therefore that of the Bdowns 99 and 51 which are their successors, which which they merge by virtue of the qe.

• whenever the GA grafts a type A number, each application of A -1 produces either another A type which vertical is then a Bdown, or if it is of a 2-ternarity of 1, directly either a Cup or a Bup. As any glacial type A can be written G(B), the ternarity of its determinant gives the number of times A -1 can be applied (and so how many new Bdowns it gives, minus one). For example: 17 being the glacial image of 7, and G(9), A -1 can be applied only twice to it, so only one type A number is also grafted (11 that is) thus grafting V (11) which is 45.

Thus, Bdown that is grafted to the ternary tree (along with its third) always initiates the adding at least of an endless other series of Bdowns and of Bups too, this also, for elementary algebraic reasons:

• Any freshly grafted Bdown allows the grafting of either a glacis A or a glacis B depending on whether this Bdown can be written 3 n • c (then it grafts an Aup) or 3 n • a (then it grafts a Bup).

• (in which case it finitely maps to a Cup). Thus the play again through the Bdown series is guaranteed for the Golden Automaton.

• if it's a Bup, it now grafts both a glacis A and a glacis B, the play again is also guaranteed.

Thus the GA does not stop, both through the grafting series that started with the first Bup (namely 9, then 33) but also to the one that started through the many Bdowns after it (15, 21, 45...). The existence of the qe also proves that the only problems that need to be solved to fully demonstrate that all Collatz orbits are bounded are the A-A avalanches, that is, the demonstration that all glacis numbers of type A merge with their glacis Bottom, which would allow to prove that all the numbers of any odd branch merge. The Golden Automaton not only guarantees that an infinite amount of those A-A avalanches are solved but also that, for the progression of it to 3 n , when n goes to infinity, at least almost all are solved.

(

(

( 9 ) (6 5) ( 1 7)(7 ) ( 9 ) ( 9 ) ( 9 )
(

(103

) ( 10 3)
(137)

(

( 1 6 1 )

(1 61 )

(16 1) (127)
(

( The most problematic numbers at any stage 2 n remain the Mersenne numbers and of course, the ones leading to them, yet the Golden Automaton grafts them very easily. All the problematic Mersennes are Cups like 31, 127 and the like. Since the determinant of the Mersenne branch is 1, at any level of the binary tree, their first glacial image will always be written G(3 n ). For example, the first forward glacial image of 31 is G(81) = 161. This is the reason 31, for example, cannot cycle when reaching its first glacis: being of high rank, it has to be written as the glacial of a proportionately high power of 3 of its determinant (1), which just can never be a vertical of 1 (no power of 3 of 1 can be written V n (1)). Also, since multiplying a number of a high verticality by 3 always gives a number of proportionate successality, it means that the precursors of the Mersenne Cups will always be written G(V n (1)), for example the precursor of 7 is 9 and that of 31 is G(21) = 41. Logically, the precursor of 127 is G(85) = 169, that of 511 is G(341) = 611.

Note that it is equally easy to demonstrate that the precursors of the "Anti-Mersenne" of type A, that is, the type A numbers that can be written 2 n + 1 or equivalently the type A numbers of the glacis of bottom 1, can always be written S(V n (1)): 11 leads to 17, 43 leads to 65, 171 leads to 257 etc.

As 729 = 3 6 is of Variety S, G(729) ≡ 127 is of Variety V, which makes it much easier to solve as it brings its solving to the problem of grafting 91 • 3 = 273, which is itself of Variety V. This is why 127 is proven to merge with 15 before we even prove that 31 converges as well.

31 indeed is a much harder number to graft abstractly because its first glacial image is of Variety S, meaning it requires the grafting of many more ternary numbers and their successors to achieve its grafting, but it works nevertheless. It is easy to prove that if the first glacial image of 31 is of Variety S, then that of 127 is of Variety V, but that of 511 will be of variety S too. Hence: that 31 is "long" to solve means 127 is "short", and thus 511 is "long" but 2047 is "short". 31 was "long" also because 7 was "short": 9 is of variety S, hence the first glacial image of 7 is of variety V, and easy to graft back to 1.

One can very easily infer from Figure 5 that with the infinity of avalanches the Golden Automaton solves, including those of the dreaded Mersenne numbers, at least almost all orbits attain, not almost bounded but bounded values, period. The emphasis on at least means it could be proven, through more abstract methods, that no Collatz Orbit can escape a finite development of the Golden Automaton, a statement this article will leave as a conjecture for now.

Conjecture 5.3. For any Collatz orbit there is an arbitrarily long, but finite, development of the Golden Automaton that grafts it back to 1.

Conclusion

Studying the intersections of the complete binary, quaternary and ternary trees over N yields simple yet very powerful results regarding the Syracuse dynamic. As we have already pointed out in Aberkane (2020) this methodology could be extended to other discrete dynamic systems, diophantine problems and even to problems involving nondiscrete environments.

Does the Golden Automaton's exponential grafting of new equivalences allow to demonstrate that actually all the Collatz orbits are finitely grafted to it? This will be left to another document although the curious reader may very well establish their own rigorous demonstration in that sense already. [START_REF] Bowling | Heads-up limit hold'em poker is solved[END_REF]

1 Definition 2 . 2 . 1 Definition 2 . 3 .Figure 1 .

 1221231 Figure1. Representation of operations V, G and S on all odd numbers from 1 to 31. The set 2N + 1 is thus endowed with three unary operations without a general inverse and that are non commutative with G • S = V . Whenever we will mention the inverse of these operations, it will be assuming they exist on N.

Lemma 3 . 1 .Lemma 3 . 2 .

 3132 If a = V (b) and b is odd, then Syr(a) = (Syr(b)) and we will note a ≡ b Proof. If a is written 4b + 1 then 3a + 1 = 12b + 4 = 4(3b + 1) therefore a ≡ b This lemma is quite trivial and therefore in no way original, but it is an essential building block nonetheless. Let a be a number of rank 1 and of determinant d, then Syr(S(a)) = G(3•d) Let a be a number of rank n in an odd branch of determinant d, then Syr n-1 (a) = G(3 n-1 • d) Proof. If a is of determinant d then a = 2d -1, and of course d is odd.

2 = 6a + 5 SCorollary 3 . 3 .

 2533 Now let's generalize to the n Note that if Syr(S(a)) can be written G(3 • d) it is also of rank 1, whereas S(a) was of rank 2, therefore, the action of Syracuse has made it lose one rank. All we thus have to prove now is that Syr(S 2 (a)) = S(Syr(S(a))) under those conditions 3•(S 2 (a))+1 (Syr(S(a))) = S(3a + 2) = 6a + 5 = Syr(S 2 (a)) If a is of rank n > 1, Syr(a) is of rank n-1, and Syr(S(a)) = S(Syr(a))

Figure 2 .

 2 Figure 2. A representation of the intersection of the binary and the quaternary trees: as only the quaternary operation •4 + 1 really matters to the representation, we have just

Figure 3 .

 3 Figure3. All odd numbers from 2 0 to 2[START_REF] Heule | Solving and verifying the boolean pythagorean triples problem via cube-and-conquer[END_REF] . Type A are circled in teal, B in gold and C in purple. Numbers of a ternarity of 2 or more (numbers that can be divided by 9, that is) are also colored in gold. To explain again the previous definitions: 27 is a Bup and 63 is a Bdown for example. Any glacis number of glaciality above 0 is "up": 17 is an Aup for example, and 19 a Cdown. Though exotic, these names are absolutely essential to the results we obtain, and their very use is a pure result of our methodology: the "up" and "down" properties come from the study of the intersections of the binary and quaternary trees, and the A,B,C ones, from the ternary.

Figure 4 .Theorem 5 . 1 .

 451 Figure 4. The golden branches on this figure are only a subset (and precisely: the Bup series after exploiting 3) of the equivalences the GA proves to reach them: assuming number 3 already grafted to the network (since we start from 1 ≡ 3 ≡ 5), they are the ones generated by only considering the Bups, and not the (many) more opportunities offered by connecting Bdowns, which are of lesser immediate interest but of much higher frequency and actually always yield either a new Aup or a new Bup to graft as well.

Figure 5 .

 5 Figure5. Note that the development of the Golden Automaton mimics the one of the famous "Collatz Seaweed", which is of course no accident. This more complete description of the Automaton, grafting the qe together as it progresses from 1 ≡ 3 ≡ 5, goes all the way to grafting 127 and 31. Each branch indicates which number generates it, for example 9 is grafted to 5 because 7 is grafted to 3, and 9 being a Bup it grafts 33 and 65. Never before has such a large proportion of any odd numbers 4 up to to 2 n + 5 been demonstrated to finitely merge with 1, and with an algorithm that has of course nothing to do with the brute force computation of each orbits, in that not only are each of its steps finite and predictable, but the demonstrating that all numbers from 2 0 to 2 n + 5 converge has not the algorithmic complexity of calculating each single orbits. In this figure are shown the grafting of 31 and 127, thus, even though all the consequences of this grafting are not represented, proving the merging of all numbers up to 261.

  if an Aup is thus grafted, it finitely connects either a Bup or a Cup and therefore a Bdown (as in what happens when connecting the pair 7 ≡ 15), because this Aup, in base 3, can only be written either b 2 . . . 2

n (in which case it finitely maps to a Bup) or c 2 . . . 2 n

Interestingly enough, it is because of considerations regarding the cardinality of the set of all branches of the complete ternary tree over N that the founding observations leading to this communication were made, initially over considerations regarding[START_REF] Feferman | Is the continuum hypothesis a definite mathematical problem[END_REF]

remember that we are still only considering odd numbers when we write "decreasing phases", and still defining Syr(a) as "the next odd in the orbit of a"

because it is easy to prove that if all numbers up to 2 n -1 converge then so do all odds until at least 2 n +

Indeed, 2 n + 1 being glacial, always have a smaller number than 2 n -1 in its orbit, from n=4 onward, 2 n + 3 is always the successal of a glacial and 2 n + 5 is always a vertical odd number

The author encourages the ever alacrious reader to verify it by a method similar to Heule et al. (2016)'s "cube and conquer".


Isabelle Nicolaï
email: isabelle.nicolai@centralesupelec.fr

Rémy Le Boennec
email: remy.leboennec@vedecom.fr

Chapter 7 Smart mobility providing smart cities Authors

Keywords: smart mobility, smart city, disruptive innovation, eco-innovation, territory, public policy, governance

By 2050, 70% of the world's population will live in or around a city. Cities already generate 70% of energy-related greenhouse gas emissions. The future of urbanisation will be smart, in which land use is optimised and the transport system is more efficient and environmentally friendly, providing affordable mobility services to ensure the well-being of people in the city.

In a smart city, urban and transport planning should be co-conducted harmoniously in order to create a new transit-supportive city. After defining our vision of smart mobility, we will present and analyse the links between the transport system, disruptive innovation, and the role of public policies in change management. In this chapter 6, we focus on the way to organise the co-conception of smart mobility, defined as a disruptive eco-innovation, in a local territory. The development and diffusion of innovations within the mobility ecosystem significantly disrupt usages and modify market boundaries. Implementation conditions to achieve a widespread adoption of smart mobility are discussed and the role and decisionmaking methods of territorial actors are considered.

Introduction: The lack of transport systems in the least dense territories

In order to work or study, to consume and to build relationships, people have a fundamental need to travel. The mobility of people and goods strengthens exchanges and enables access to much larger territories [START_REF] Didier | Infrastructures de transport, mobilité et croissance[END_REF]Prud'homme 2007, Le Boennec 2013). People move in space: within cities, from the suburbs to the city, to the country, between two cities, two regions or two countries. What mobility has in common in these heterogeneous spaces is that it provides people with better accessibility to jobs, goods and services. Accessibility is thus defined as the number of travel opportunities for a given distance [START_REF] Pouyanne | L'interaction entre usage du sol et comportements de mobilité. Méthodologie et application a l'aire urbaine de Bordeaux[END_REF] or for a given travel time [START_REF] Deymier | Analyse spatio-temporelle de la capitalisation immobilière des gains d'accessibilité: l'exemple du périphérique Nord de Lyon[END_REF]. Accessibility gains occur when a new transport infrastructure enables the individual to reach the same destinations in a shorter time, or destinations further away with no increase in time, thus offering new travel opportunities.

Accessibility has extremely different characteristics depending on the territory considered, influencing the modal decisions of households. In 2013 in Paris, France, 100% of inhabitants were served by a structuring transport accessible at less than one kilometre from their home [START_REF] Marks | People Near Transit : Improving Accessibility and Rapid Transit Coverage in Large Cities[END_REF]. This privileged situation is correlated with a high population density coupled with a public transport network that is historically well connected (particularly the metropolitan network). If one considers the inner circle of Paris (the three departments around the city centre), then the outer Parisian circle (composed of the remaining territories of the Ilede-France Region), this level of cover of inhabitants by structuring transport falls dramatically to 66%, then only 13%, respectively. This heterogeneous accessibility to a public transport network within the same urban area1 constitutes a direct manifestation of the rapidly decreasing density gradients from the city centre. It is much easier to connect a territory effectively where the housing density, and thus the population, is high (generally, the central municipality of an urban unit). Conversely, in less dense zones, a transport network, especially structuring (underground, tram, train or bus rapid transit), necessarily favours certain zones, with some irreversibility due to the size of the infrastructure deployed. This less favourable situation usually corresponds to municipalities that are peripheral to the urban unit, or to municipalities of the urban area around the urban unit (commonly called the peri-urban space). Regarding rural municipalities, they are rarely served other than by a bus network of varying size and frequency. Such a heterogeneity in public transport provision can be seen in most European conurbations, where the collective housing is largely concentrated in the urban unit; individual housing is more often located in the peripheral municipalities, on building plots whose size is positively correlated with their distance from the centre, consistent with the urban economics theory [START_REF] Alonso | Location and land use. Toward a general theory of land rent[END_REF][START_REF] Fujita | Urban economic theory: land use and city size[END_REF][START_REF] Boennec | Externalité de pollution versus économies d'agglomération: le péage urbain, un instrument environnemental adapté?[END_REF]. In emerging countries, very rapid urbanisation, often poorly controlled, does not always allow a sufficiently thought out coordination between urban planning and transport systems [START_REF] Marks | People Near Transit : Improving Accessibility and Rapid Transit Coverage in Large Cities[END_REF].

It is thus the urban sprawl that is at the origin of the contrasting accessibility to the public transport network. This sprawl has only one consequence: in the less dense housing areas, peripheral municipalities of urban areas and rural municipalities, the private car is often the only solution for daily journeys [START_REF] Brownstone | The impact of residential density on vehicle usage and energy consumption[END_REF] 2 . Thus, the car is used daily by 10% of Paris inhabitants, 26% of those in the inner circle, but 56% of the outer circle inhabitants (source: Chronos/L'ObSoCo study, "L'Observatoire des mobilités émergentes", The Observatory of Emergent Mobilities, September 2016). Similar travel patterns are found in the rest of France; while 54% of city centre inhabitants think that they can do without their (owned) car completely, this can be envisaged by only 27% of people in conurbations of 2 to 20,000 inhabitants. As a result, in all territories taken together, 85% of journeys in 2016 were still undertaken in private cars. This contrasting accessibility, and the heterogeneity of opportunities that it offers, is often, in an urban environment, the legacy of functionalist urban planning of the 1950s to 1970s, at least in France. Ways of living, working and consuming often concern separate spaces, which may be far from each other even within the same urban unit. Moreover, the aspiration to own an individual house for a large number of households, coupled with easier home ownership, has increased land use and urban sprawl beyond the urban unit. In these peri-urban territories, accessibility is limited, including by private car in rural municipalities, with access times to major services (leisure, studies, health) systematically longer than the average (source: Insee Première n° 1579, January 2016).

In these patterns of development in which the private car dominates, the negative externalities associated with an exclusive use of the automobile are reinforced. These external effects are known: traffic noise, greenhouse gas (GHG) emissions, and air pollution [START_REF] Verhoef | The economics of regulating road transport[END_REF]. Although most are suffered relatively little by the inhabitants of less dense zones, GHG emissions at least, due to their worldwide dimension, constitute a universal problem that public policies must take into account in their mobility and urban development plans.

We will begin by considering the difficulty of changing local planning policies for a better coordination with local mobility policies (Section 2). We will then see how, in a smart city, innovation enables such limitations to be, at least partly, overcome (Section 3). Nevertheless, deploying a disruptive innovation in a territory demands the concordance of a number of conditions, the nature of which we shall consider in Section 4. In the context of emerging offers and more difficult decisions for public actors, we will discuss in Section 5 the different techniques for assessing mobility policies before concluding. Section 6 concludes the sixth chapter of this book.

Reducing inappropriate urban forms: expensive and long-term public policies

There are many territorial scales required to correct the negative externalities of road transport; in fact, they are complementary. In its White Paper on Transport (2011), the European Commission drew up the "roadmap for a single European transport area -Towards a competitive and resource-efficient transport system". The states developed the European objectives into national policies (bonus-penalty systems for the purchase of vehicles, regulatory framework for trying out restricted traffic zones). On a local scale, the municipality authorities often seem relevant for the development, implementation and evaluation of local policies, whose objectives are often quite similar from one document to another3 , but whose range of operational measures is supposed to take into account the special features of territories in connection with other planning documents concerning transportation and landuse.

In France, the constitution of metropoles on 1 January 2016 aimed to resolve some governance problems that had been observed. The non-simultaneous implementation of multiple planning documents may have resulted in an insufficiently consistent vision of the development of a territory to the relevant boundaries (the city centre and its suburbs constituting the urban unit). The legislators have sought to change these historic practices by recently encouraging the adoption of urban plans at this scale. In 2016, the majority of the exurban communities that had become metropoles intended to adopt such a document. Urban planning at this scale should ensure a better visibility in favour of a concerted development of housing, employment and service areas. In parallel, the mobility policies were themselves strengthened at the municipality level by the NOTRe law of August 2015. For example, since 1 January 2016, the new conurbations reaching the threshold of 50,000 inhabitants have jurisdiction over transport in their territory by becoming Autorités Organisatrices de Mobilité (AOM) (Mobility Organising Authorities). This scheme is further facilitated by the legislators who encourage the existing municipalities to come together in order to reach higher population thresholds. Thus, with some exceptions, on 1 January 2017 those new (inter-)municipalities should cover at least 15,000 inhabitants. This rationalisation, of an unprecedented size and speed in France4, originating from greatly constrained budgets since 2009, has led to a reinforced integration of the governance of development and mobility policies at the local level. Moreover, the territorial projects, often pre-existing and contractualised with the higher authorities, have increased the financial means as well as the expertise available to the conurbations on the question of mobility. Thus, Transit Oriented Development contracts (TOD) have been decided jointly to serve territories of average density. The fundamental principle of such a contractualisation envisages specific subsidies from the higher authorities5 for the deployment of a public transport network with a high level of service, through a commitment from the local authorities to increase the housing density in the town centres served. In addition, multimodal exchanges, park-and-ride car parks, bike lanes and footpaths conceived jointly can also be supported. As well as financial aspects, the role of the higher authorities may include sharing and disseminating experiences of the same type, which have already been implemented in its territory or elsewhere.

The traditional cooperative modes mostly concern public transport and intermodality. However, while the developments for active modes are fairly expensive for local authorities, which may contractually agree to share the financing, the deployment of a public transport network with a high level of service in territories less dense than a metropole raises the question of available funding sources, especially as the share likely to be covered by the user is substantially less than it could be in a densely urbanised environment.6 Moreover, the timescales needed for such projects are long: while a TOD project might be on a 5-year scale, the deployment of an urban planning zone on the future site often takes from 15 to 20 years. As a result, the very gradual arrival of housing, employment and services in these zones compromises even more the profitability of the transport provided during its first years of operation.

For both these reasons (financing and long timescales), many local authorities now prefer to consider the alternative, or complementary, opportunity of introducing light mobility offers. Moreover, the idea of a Mobility Organising Authority, which has replaced that of the Transport Organising Authority in France, aims to take into account these sometimes rapid changes. The advent of Mobility Organising Authorities thus envisages "extended authority in the fields of the shared use of the automobile (car-sharing, car-pooling), active modes and urban logistics". This promotion of the shared use of the car and multimodality offers a renewed regulatory framework for multimodal information platforms, which are part of the smart city, one of whose goals is to encourage these new mobility practices.

Innovation as a support of the ecomobility market within the smart city

From an innovation perspective, the transport sector, based on a combination of new energy systems and the spread of internet (typical of the third industrial revolution of Rifkin ( 2011)) turns today toward clean vehicles and connected mobility solutions. We thus refer to sustainable mobility for all the available offers -individual or collective, public or privatethat contribute to responding positively to economic and ecological issues in terms of reducing GHG emissions [START_REF] Pillot | Vers une mobilité décarbonée: quels écosystèmes d'affaire, quels positionnements stratégiques? Rapport Institut Transition Energétique VEDECOM[END_REF]. This trend toward a new technological paradigm applied to local territorial development is expressed in the idea of "smart mobility", as a component of a "smart city".

Since the 1990s, the smart city concept has become increasingly popular in international scientific studies and national public policies have adopted this type of territorial development. There are many definitions of the smart city and we propose the following typology of its analytical models (for more details about the many evolutions of the concept, see the works of [START_REF] Caragliu | Smart cities in an Open World[END_REF]).

The first approach defines the smart city as a model based on the data needed to manage and plan the city: "Smart Cities initiatives try to improve urban performance by using data, information and information technologies (IT) to provide more efficient services to citizens, to monitor and optimise existing infrastructure, to increase collaboration among different economic actors, and to encourage innovative business models in both the private and public sectors." [START_REF] Marsal-Llacuna | Lessons in urban monitoring taken from sustainable and livable cities to better address the Smart Cities initiative[END_REF].

Consequently, the objective of a smart city is to organise its activities in order to implement and interconnect technologies, devices and services as efficiently as possible using information technologies [START_REF] Hancke | The Role of Advanced Sensing in Smart Cities[END_REF].

In this model, the smart city is conceived as an information system, which instrumentalises and interconnects its assets such as buildings, the energy or water network, and transport. According to [START_REF] Harrison | Foundations for Smarter Cities[END_REF], "instrumented" refers to the capability of capturing and integrating live real-world data through the use of sensors, meters, appliances, personal devices, and other similar sensors. "Interconnected" means the integration of these data into a computing platform, which enables the communication of such information between the various city services.

Gradually, this very technological approach has been replaced by more open definitions of the smart city that take into account the social capital dimension and its relationships with urban development. Thus, in the second model, we have a definition of the smart city through its governance and its ability to be resilient. The idea of the model is to move from a "connected" city to a "smart" city. "Smart" refers to the inclusion of complex analytical, modelling, optimisation, and visualisation services in order to make better operational decisions. The model emphasises the role of human capital/education, social and relational capital, and environmental interests as important drivers of urban growth (Leydesdorff et al. 2010[START_REF] Komninos | Smart Cities and the Future Internet in Europe[END_REF].

A smart city in this perspective is one that organises the conditions for the commitment of all the stakeholders of its ecosystem in the decision process. Resilience is thus reflected by the quality of people and communities, to be connected, to manage, and to be informed [START_REF] Albino | Smart Cities: Definitions, Dimensions, Performances and Initiatives[END_REF][START_REF] Repko | Smart Cities Literature Review and Analysis Smart Cities[END_REF]. Resilience is a factor of governance, risk assessment, knowledge and education, risk management, vulnerability reduction, and disaster preparedness and response [START_REF] Baron | Do we need Smart Cities for Resilience[END_REF][START_REF] Twigg | Characteristics of a Disaster Resilient Community[END_REF].

However, in a more operational logic with a territorial application of the smart city, it seemed necessary to propose a third model, which defines how to produce smart city strategies (Ben Leitafa 2015). To meet these objectives, we propose, in line with the studies of R. Giffinger, to consider the following levers to make cities smart:

develop new efficient services in the transport-mobility sectors; responsible housing and urbanisation; smart materials and energy networks, manage information systems in real and multiple time to help in decision-making (citizens, administrations, organisations), promote renewed governance and the financing of new services.

Based on this operational logic, we can propose a third so-called architectural approach, which breaks down the systems and dimensions of the smart city in an organic way. The literature, particularly the studies of [START_REF] Giffinger | Smart cities Ranking of European Medium-sized Cities[END_REF] and Dirks et al. (2009), proposes six components of the smart city that correspond to six dimensions of urban life that must be made efficient: industry, education, democracy/governance, logistics and infrastructures, efficiency and sustainability, safety and quality. These six components can be represented as follows: In this context, we use the definition proposed by [START_REF] Caragliu | Smart Cities in Europe[END_REF]: "We believe a city to be smart when investments in human and social capital and traditional (transport) and modern (ICT) communication infrastructure fuel sustainable economic growth and a high quality of life, with a wise management of natural resources, through a participated governance".

In each sector, there is a supply of and a demand for connected and smart systems with specific characteristics to be analysed in various approaches related to eco-innovation 7 (Rennings 2000; Cecere and alii 2014).

In the eco-innovative mobility system, there are many evolutions that impact the most typical drivers of eco-innovation classified as "market pull", "technology push" and "institutional factors and policy measures" (Horbach and alii, 2012;[START_REF] Nemet | Demand-pull, technology-push, and government-led incentives for nonincremental technical change[END_REF]:

-Changing travel habits and the demand for services to increase convenience, multimodality and predictability will require mobility solutions as well as business model transformation. -Concerning the supply, one observes, in an expanded automobile ecosystem, the integration of companies specialising in other sectors such as telecoms [START_REF] Faucheux | Business models and the diffusion of eco-innovations in the eco-mobility sector[END_REF]. Similarly, the vehicle has an evolving role beyond that of a means of 7 We adopt the consensual definition of eco-innovation proposed initially by [START_REF] Kemp | Final report MEI project about measuring eco-innovation[END_REF] : "Eco-innovation is the production, application or exploitation of a good, service, production process, organizational structure,or management or business method that is novel to the firm or user and which results, throughout its life cycle, in a reduction ofenvironmental risk, pollution and the negative impacts of resource use (including energy use) compared to relevant alternatives". See [START_REF] Horbach | Empirical determinants of eco-innovation in European countries using the community innovation survey[END_REF] for an overview of the eco-innovation literature.

travel toward a means of energy storage and production. Thus, a transport fleet of rechargeable electric vehicles enables electricity to be bought or sold on a smart grid.

-A system-level approach is critical: sustainable improvements in a city's mobility performance require simultaneous improvements in urban sustainable systems, smart infrastructures, new urban services and associated business models. In the management of a "smart mobility" system, there are additional constraints to take into account. When analysing the demand for "smart mobility", we can identify the following specific features of the services proposed:

i. user-focused service with an analysis of user preferences depending on the specific temporal and geographic contexts ii.

a service available at every point in the territory and from all the facilities iii.

an integrated service with data coming from and supplying different applications according to the principle of collaborative participation of the actors. From the point of view of the supply of "smart mobility", the actors, such as local authorities and companies that have to be taken into account in the eco-innovation eco-system (Ghisetti and alii 2015), must offer services:

i. inter-connected within a platform that is open to heterogeneous facilities, ensuring connectivity at every point in the territory ii.

with great efficiency: the organisation of such a network must be as cost-efficient as possible iii.

with satisfactory energy efficiency to valorise sustainable applications iv.

reliable in terms of a guaranteed connectivity in all situations, even exceptional ones.

In order to examine how the literature can transcribe the special features of a "smart mobility" system, we will deal with the quality of its eco-innovation to present how the new mobility solutions can be considered disruptive innovations (as defined by Christensen 1997).

A systemic disruptive eco-innovation 4.1 A disruptive innovation

Ecomobility or "smart mobility" as an eco-innovation consists of both responding to the needs of consumers and encouraging their support ("demand-pull innovation"), while questioning which business ecosystem is most likely to provide effective solutions to these needs ("technology-push innovation") in an evolving institutional context. In this perspective, the characterisation of smart mobility according to different types of innovation is important because it will determine the conditions of its implementation.

In an ecomobility system, new needs appear and user behaviour changes in a complex coevolution with regulatory frameworks (Dantan et al. 2015). In order to characterise the dynamics of ecomobility innovations, we will use the typology of eco-innovation proposed by [START_REF] Faucheux | IT for Green and Green IT: A Proposed Typology of Eco-Innovation[END_REF] in association with the different trends in the innovation economy (Uterback and Abernathy 1975[START_REF] Rennings | Redefining innovation -eco-innovation research and the contribution from ecological economics[END_REF][START_REF] Hellström | Dimensions of environmentally sustainable innovation: the structure of eco-innovation concepts[END_REF], Carrillo-Hermosilla et al. 2009).

The analysis of industrial regimes leads us to consider first the dynamics of innovation through technological trajectories as defined by [START_REF] Dosi | Technological Paradigms and Technological Trajectories. A Suggested Interpretation of the Determinants and Directions of Technical Change[END_REF]. In this perspective, according to [START_REF] Utterback | Mastering the Dynamics of Innovation[END_REF] innovation is identified as "dominant design". In the context of smart mobility, this means, for the automotive industry, imagining technologies that meet the needs identified, such as those of the ownership and use of vehicles with increasing constraints or opportunities: reduction in environmental impacts, innovative economic models with, for example, the electric vehicle, taking into account recent legislation, and acceptance of the economy of functionality, which provides a mobility service instead of the acquisition of a vehicle.

We observe that over time these innovations result from a creative process that occurs in a market and production context (technology push) as well as from an adaptive response to a new demand expressed by the market (demand pull).

However, these changes go beyond the issue of a technological trajectory to promote new paradigms of smart mobility (as defined by [START_REF] Dosi | Technological Paradigms and Technological Trajectories. A Suggested Interpretation of the Determinants and Directions of Technical Change[END_REF]). Thus, radical innovations (as defined by [START_REF] Dewar | The Adoption of Radical and Incremental Innovations: An Empirical Analysis[END_REF]) support technological disruptions and changes in user behaviour. This destabilisation, which was dealt with in terms of internal disruptive techniques for the overall product structure by Henderson and Clark (1990, with modular/architectural innovations), challenges the current skills of companies. The production of autonomous vehicles, as well as the development and sale of new mobility services (NMS) requires different skills for the automotive industry. However, if, like [START_REF] Markides | Disruptive Innovation: in Need of a Better Theory[END_REF], we focus on the "technological" innovation, which aims to attack new segments in an existing market8 (a different business model in a current market), we observe that a "strategic" innovation must also be envisaged in smart mobility, which involves a different value distribution in the value chain of the automotive industry. For example, car-pooling proposes a collective use of an individual mode of transport while, conversely, car-sharing describes the individual use of a collective mode of transport. These technological innovations are applied to different uses of a traditional transport mode and transform the economic model. How can companies now promote a systemic vision of technological change, characterised by a complex structure of interactions between the socio-economic environment (markets, governments), the skills of the firm linked to technological changes, and the modifications of uses? How can they promote these systemic eco-innovations (as defined by Chesbrough and Teece ( 2002)) which require new internal skills and solicit new markets with few customers, sometimes in competition with their own existing market?

Smart mobility as an eco-innovation that simultaneously destabilises the supply system and the demand system (needs of users) is considered a disruptive innovation according to the definition of Mackay and Meltcafe (2002): "A disruptive innovation represents new technologies with characteristics that are initially unfamiliar to producers and consumers, which may also require a major evolution of institutions and infrastructures, and which have the potential to disrupt market structures and lead to changes in behaviours". As well as in smart mobility, we observe the following evolutions:

1. The emergence of a new technology such as electromobility (or previous automobile technologies and ICT assembled differently), which underperforms compared to the "dominant design" (individual combustion vehicle) and the expectations of consumers. However, this new technology follows a faster technological evolution than the established one. The new mobility services (NMS) are also disruptive compared to the current use of individual and owned vehicles. 2. The technology of NMS is provided by actors at the competitive fringe, or even outside the market9 . For example, Bolloré, a competitor outside the automobile sector, has been one of the pioneers in car-sharing practices. The success of this innovation will move the strategic value of the old paradigm toward the new, which will significantly change the industrial structure, with the constitution of a new ecosystem. 3. The new technology requires new skills and a learning process by the providers to gain them. It also introduces new uses and/or considerably modifies the initial use of the technology. New practices in the economy of services related to the acquisition or maintenance of the car and the spread of connected objects in the vehicle are transformations in the uses, and thus in the supply, of products. 4. The innovations linked to ecomobility are addressed at new users. For example, the autonomous vehicle will target the visionary "early adopters" who are sensitive to the new technology required for its own sake. The diffusion of this innovation will follow the classic sinusoidal curve of innovation, which associates each phase of technological development with a particular category of consumers. 5. Consumers are categorised according to the time they choose to acquire the innovation in question: "early adopter" and "early majority". The success or failure of the "ecomobility" innovation will be judged when the early majority (who prefigure the transition from a niche to a mass market) adopt ecomobility with a greater importance accorded to the traditional economic determinants (price, reliability, flexibility of choice, interoperability). A new business will thus arise if the management of the transition between consumer categories is assured.

Whether the innovations in ecomobility are "demand-pull" or "technology-push", they can dramatically disrupt usages and change market boundaries. They thus involve new reflexions about the business model10 that companies in the automotive industry need to develop and the evolution of the structure of the automobile ecosystem. In this sense, [START_REF] Christensen | The Innovator's Dilemma: When New Technologies Cause Great Firms to Fail[END_REF] speaks about a disruptive innovation.

Many conditions need to be fulfilled to implement these disruptive eco-innovations:

have a design-user approach, i.e. identify and deal with what could be the uses of tomorrow in order to design technologies and products according to those needs (what are the different autonomous car models proposed by Tesla on one hand, and by existing manufacturers on the other hand?).

agree to costly investments to explain to future users what innovation is and the value it will bring them. Given that most people are rather resistant to change and prefer to keep their habits, convincing them to adopt a new product or service that requires them to change their practices will not be a simple and rapid process. It is up to the innovative companies to identify both the facilitating and the obstructing factors to ensure a wider and faster adoption of the innovation. invest heavily in R&D (facilities, expertise, patents) in order to stay ahead in terms of skills in the new market identified. The company must also equip itself with the new skills indispensable for developing this innovation. becoming the leader in a new market also involves rapid growth, i.e. continuing to invest heavily once the innovation is launched. lastly, the sales cycles of a disruptive innovation can be very long, certainly longer than classic innovative products. This is an additional risk factor.

The disruptive nature of smart mobility imposes a methodology of putting the user at the heart of the strategic reflexion and choosing the business model most likely to provide effective solutions to their needs in a perspective of co-conception and co-innovation. The implementation conditions are thus crucial in both the diffusion of these innovations and the success of the new business model.

Implementation conditions

Different variables will impact whether or not the disruptive eco-innovation is successfully adopted. The first concerns the choice of developing new skills internally or not. [START_REF] Chesbrough | When Is Virtual Virtuous?[END_REF] envisage two organisational models to implement the innovation process: a decentralised (or virtual) approach for "autonomous"11 innovations and an internal development of skills for "systemic" innovations. This "organisational design" is one of the accompanying variables of disruptive eco-innovations. In an institutionalist approach, it consists of identifying the most suitable institutional determinants and socio-economic organisation modes for decision-making: how do collaborative practices, known as "open innovation", promote disruptive innovation? What are the main obstacles to overcome for the public and private actors?

The second variable affects the institutional level. How can we ensure the conditions for disruptive innovations to be accepted? For example, should we preserve the acquired positions of the organisation of mobility with new actors such as Uber? How can we choose between the protection of private life and big data? In fact, the major pitfall for a new mobility solution is achieving the required critical mass of users; many of these solutions emerge and then just as frequently decline only one or two years after their launch. For example, this is the case of the solution Djump launched in 2013 but abandoned in 2015, which came close to a dynamic car-pooling. This critical mass, i.e. the transition from the adoption of the solution by the early adopters toward the early majority, is often easier to reach in dense housing areas, but this is not usually enough.

The third and last variable for a successful adoption of the offer concerns the support of public policies, with a focus on financing and territorial questions. What are the favourable and unfavourable factors for the birth and development of the disruptive innovation? Is the financing from private and public funds sufficient and well adapted to the specific needs of the disruptive innovation, in the territories concerned? How does the territorial dimension affect the disruptive innovation? Are there "best practices" that can be transposed from one territory to another?

Most new mobility solutions are deployed primarily in central Paris, and sometimes in the Parisian inner circle and the city centres of large conurbations. This is demonstrated by the use of different private hire services (Uber, le Cab, etc.), whose attractiveness decreases rapidly further from the very centre of cities. Thus, while 42% of Parisian households used a private hire service at least once in 2016, only 11% did in the urban centres of more than 100,000 inhabitants outside the Ile-de-France, and only 4% in the rural municipalities (source: Chronos/ L'ObSoCo study 2016).

This extremely contrasting situation makes support for the development of the solution by the local authority indispensable. Even solutions that are relatively inexpensive to implement for the territories and whose success seems guaranteed, like bike-sharing schemes, remain dependent on characteristics linked to the population density, amongst others. The recent failures of this type of service in several provincial cities, some of which have more than 100,000 inhabitants (Perpignan, Angers, and Dijon), are due to various factors: too high an operational cost for the local authority compared to the revenue recovered, and underutilisation of the system. In both cases, the lack of infrastructure or development dedicated to cyclists is highlighted; in other words, although there was sufficient public and private financing, it was not adapted to the territory.

Car-pooling is another example of a solution that may succeed or fail. While the long-distance car-pooling represented by BlaBlaCar has proved to be a success over the whole national territory (and internationally), the same group of solutions applied to shorter distances, like WayzUp, is difficult to deploy in less dense areas, despite this solution's position in the market since 2013. For a car-pooling solution to operate in more extensive territories, it often needs to be part of a local experiment: for example, the dynamic car-pooling of Covivo in Isère in 2010, or RézoPouce in the Montauban region (France). However, this experimental nature makes it difficult to identify the reproducibility of the solution, including in territories with similar socio-demographic characteristics. The local "best practices" cannot always be transposed.

To try out innovative offers, there is regular cooperation between start-ups and historic actors of mobility (car manufacturers and associated services, public transport operators). Nevertheless, these partnerships remain limited over time and are generally subject to cautious investment, thus restricted. For example, the RATP (Parisian public transport provider) and the start-up Sharette set up a partnership in summer 2015 during works on Line A of the Express Regional Network. Despite its success announced at the time, the partnership was not renewed during a new series of works in summer 2016, due to the bankruptcy of Sharette in the intervening period.

Promoting the socio-economic conditions for the birth and development of disruptive innovations remains insufficient. In the presence of negative externalities associated with transport, public regulatory actions are indispensable to restore the well-being of the population suffering from the negative effects to its original state. Individual mobility (solo driving) presents many advantages for the user: in a car, there are often more opportunities for travel. One can travel to destinations further away and generally in a shorter time than on public transport. The car provides "seamless mobility", door-to-door and without a break, especially in less dense areas. It is practical, offering the possibility of transporting heavy objects. Solo driving remains synonymous with independence and flexibility: the user depends on no one to make his/her journey. Lastly, it provides the driver with a space completely isolated from the outside world, if he/she wants it.

On the other hand, the use of the car leads to many negative externalities. It causes noise, air pollution, greenhouse gas emissions, traffic congestion, road accidents, etc. The role of public actors is to limit all these externalities and preserve an acceptable quality of life for the inhabitants. However, public mobility policies are often expensive to implement, even though they provide significant and long-term effects [START_REF] Quinet | La pratique de l'analyse coût-bénéfice dans les transports: le cas de la France[END_REF]. Converting a city thoroughfare into an urban boulevard with more space dedicated to public transport, pedestrians and bicycles represents an ambitious development for a local authority, with a certain irreversibility of choice.

In this respect, mobility policies must be assessed as thoroughly and robustly as possible, before or in progress [START_REF] Rousval | De l'aide multicritère à la décision à l'aide multicritère à l'évaluation[END_REF]. The implementation of a policy represents a number of potential projects or actions between which the decision-maker must choose with regard to various considerations. Will the project envisaged meet its objectives of correcting the negative externalities? Will it be too expensive? Is it assessed using the correct criteria? Is it preferable to another project? In other words, the cost of the opportunity of a project must be assessed.

Assessment methods

In the last fifteen years, less structuring mobility policies have increasingly entered the evaluation field; for example, a car-pooling centre facilitated by an average-sized conurbation (source: Ministère de l'Ecologie, du Développement sustainable et de l'Energie, 2015).

There are many economic methods to estimate the effects of a public policy. These aim to calculate the individual's willingness to pay, "that is, how much the individuals would be prepared to pay to benefit from an increase in the supply of a non-market commodity" (Meunier and Marsden 2009, p. 6). Two groups of methods are traditionally contrasted: the stated preference and the revealed preference methods [START_REF] Mahieu | Non-market valuation in France. An overview of the research activity: Introduction[END_REF]. In these two groups, there are applications specific to transport projects, or to transport externalities for the local authority.

Stated preference methods are often applied to transport projects [START_REF] Bristow | International meta-analysis of stated preference studies of transportation noise nuisance[END_REF]. Among these, contingent valuation "estimates by survey techniques how much individuals would be ready to pay to enjoy the benefits of a project" (Meunier and Marsden 2009, p. 7).

The assessment of transport projects by contingent valuation has been the subject of many applications: reduction of noise and air pollution in Navarre (Lera-Lopez et al. 2012), estimation of parking charges in Greece [START_REF] Anastasiadou | Determining the parking fee using the contingent valuation methodology[END_REF], improvement in the provision of public transport in Dubai [START_REF] Worku | Demand for improved public transport services in the UAE: A contingent valuation study in Dubai[END_REF], and rail services in Korea [START_REF] Chang | Estimation of option and non-use values for intercity passenger rail services[END_REF].

In terms of revealed preferences, the best known method applied to transport projects is that of hedonic pricing. This approach is based mainly on the estimation of property prices, which depend partly on the proximity to transport networks and to sources of amenities and nuisances. Thus, the property market indirectly provides a monetary value of these attributes through the difference observed between the values of two identical properties, with the exception of one of the characteristics studied. This difference in value is explained by the gain or loss of well-being attributed by the buyers to these proximities (Rosen 1974, Le Boennec and[START_REF] Boennec | Nouvelles centralités, choix modal et politiques de déplacements: le cas nantais[END_REF]. Besides the property's own characteristics and the sociodemographic variables of the neighbourhood, the variables of accessibility (to a public or road transport network) constitute another group of accommodation attributes considered in a local market. Many applications of the hedonic pricing method continue to characterise the submarkets: in Paris (Bureau and Glachant 2010), Ireland [START_REF] Mayor | A Hedonic Analysis of the Value of Rail Transport in the Greater Dublin Area[END_REF], and the United States [START_REF] Duncan | The Impact of Transit-Oriented Development on Housing Prices in San Diego, CA[END_REF][START_REF] Bajari | A rational expectations approach to hedonic price regressions with time-varying unobserved product attributes: The price of pollution[END_REF].

In these approaches, the calculation of willingness to pay may provide a basis to fuel decision-making methods of the cost-benefits analysis (CBA) type. The CBA enables the public decision-maker to choose a transport project in a context in which the decision criteria can be valued in monetary terms. In this way, even if recent evolutions have expanded the perspectives of the cost-benefits analysis, "the quantified economic assessment constitutes the core of the evaluation", as Quinet (2010) points out.

The CBA first clarifies the different elements to take into account in the analysis and provides a structured framework for the public debate [START_REF] Meunier | Analyse coût-bénéfices: guide méthodologique[END_REF]. However, several limitations of the method are generally highlighted. Some impacts are not taken into account in the assessment of projects, which risks biasing the selection; this situation arises because it is sometimes difficult to choose between an inaccurate costing of an impact, an essential condition for monetary evaluation, and no costing at all, in which case the impact will not be taken into account (Forum International des Transports 2011[START_REF] Bueno | Sustainability assessment of transport infrastructure projects: A review of existing tools and methods[END_REF].

Finally, although a cost-benefits analysis undeniably improves the transparency and reliability of the analysis by objectifying it, a better appreciation of the diverse effects of a mobility project may be sought by using alternative or complementary tools, in all those situations in which the analytical criteria may not easily be valued in monetary terms [START_REF] Joubert | Fynbos (fine bush) vegetation and the supply of water: a comparison of multi-criteria decision analysis and cost-benefit analysis[END_REF][START_REF] Tudela | Comparing the output of cost benefit and multicriteria analysis: An application to urban transport investments[END_REF]. Moreover, in practice, the CBA described above continues to favour large infrastructure projects [START_REF] Beria | Multicriteria versus Cost Benefit Analysis: A comparative perspective in the assessment of sustainable mobility[END_REF][START_REF] Hüging | Need for a holistic assessment of urban mobility measures -Review of existing methods and design of a simplified approach[END_REF]. For a mobility project without a very heavy infrastructure (such as bike-sharing or car-sharing), a multi-criteria analysis may be preferred by the decision-maker. Finally, it is generally agreed that a multi-criteria analysis "takes more impacts into account (…), responds more directly to the concerns of decisionmakers and is open to different assessments of the weight given to different impacts" (Forum International des Transports 2011, p. 13). In many cases, this greater flexibility can make the multi-criteria analysis preferable to a cost-benefits analysis in the eyes of decision-makers [START_REF] Bueno | Sustainability assessment of transport infrastructure projects: A review of existing tools and methods[END_REF].

A multi-criteria analysis is implemented in several phases [START_REF] Beria | Multicriteria versus Cost Benefit Analysis: A comparative perspective in the assessment of sustainable mobility[END_REF]. The project or the actions to be assessed must first be specified. It is then indispensable to draw up a list of criteria that will guide the researcher in evaluating the predefined actions. The foreseeable impacts of actions should be envisaged; they must be assessed with regard to each criterion, according to a weighting system. Lastly, the aggregation of assessments, criterion by criterion, should inform the decision-making.

The different stakeholders in the project or action, public or private, may help the researcher in the construction of the process [START_REF] Beria | Multicriteria versus Cost Benefit Analysis: A comparative perspective in the assessment of sustainable mobility[END_REF]). However, as described, the simple multicriteria analysis does not explicitly take into account this diversity of stakeholders, who are not involved in the implementation stages presented above. Yet, there is a wide range of actors potentially concerned by the possible impacts of a mobility project [START_REF] Macharis | The multi-actor, multi-criteria analysis methodology (MAMCA) for the evaluation of transport projects: theory and practice[END_REF]. They are primarily the users, but also the suppliers, as well as the financers (who may be partly the same but also include the public actors). This is why it may be relevant to integrate, explicitly and as soon as possible in the method, all the stakeholders in the decisionmaking process. Such a strategy makes the group of techniques revolving around multi-actor multi-criteria analyses (MAMCA) particularly timely (Macharis andBernardini 2015, Le Boennec et al. 2017).

Conclusion

There are many challenges facing national and local mobility policies. From an environmental point of view, the fight against climate warming is a priority of local policies, which should not, however, reduce the ability of individuals to travel. From a social point of view, it is therefore a question of offering mobility solutions that meet the demands of actors to travel, specific demands that depend on the territorial contexts.

The deployment of an innovative "smart mobility" ecosystem is envisaged as an opportunity to promote a better coordination of all public transport and alternative mobility services to the private car, associated with a development of car-sharing according to different modalities.

The generalisation of ICT in ecomobility facilitates the complementarity, or even the competition or substitution, of new kinds of transport. Nevertheless, these new transport modes, developed by both public and private actors, must be coordinated to meet the new needs expressed by the demand.

What could be the role of public policies in this management of disruptive eco-innovations? Moreover, the ability to invest in and exploit infrastructures is important in the implementation of a local transport policy. In the context of a rapid evolution in mobility options and systems of governance, how could a local policy on these smart mobility issues be more effectively structured than it is today? The following chapter provides an illustration of these new responsibilities.

Figure 7

 7 Figure 7.1. The Smart Cities Mandala (EU & Giffinger et al.).

An urban area in France is a group of contiguous municipalities, with no pockets of clear land, constituted by an urban centre (or urban unit) providing more than 10,000 jobs, and by rural municipalities or urban units (urban periphery) in which at least 40% of the employed resident population works in the centre or in the municipalities attracted by it (French national statistics institute INSEE definition).

The INSEE typology uses the following travel motives: home to work or studies, purchasing, personal business, accompaniment, leisure or visits.

In France, Agendas 21.

The number of inter-municipalities on 1 January 2017 was thus 1263 against 2062 previously, i.e. a fall of 39%.

"Départements" and "Régions" in France

In 2010, the contribution of users to financing the public transport network in the Paris region was only 29.7%.

The company redefines its technologies, goods and services and does not find a new market: for example Amazon, easyJet, etc.

[START_REF] Christensen | The Innovator's Dilemma: When New Technologies Cause Great Firms to Fail[END_REF] speaks of the dilemma of the innovator for whom the disruptive innovations are rarely introduced by the dominant companies in the market.

In the meaning of[START_REF] Christensen | Foundations for Growth: How to Identify and Build Disruptive New Businesses[END_REF] for whom innovation corresponds to "the creation of totally new markets and economic models" (p. 22).

The authors speak of autonomous innovations when they can be carried out independently of each other.