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Xlim J P Cances

Performances of Chaos Coded Modulation concatenated with Space-Time Coding extended version

. The strength of our study consists in the fact that we do not only study the case of block fading channels but we propose new insights for the case of time selective channels. Our n

closed form expressions are based on the recent work of Jun He and Pooi Yuen Kam [START_REF] He | Bit Error Performance of Orthogonal Space-Time Block Codes over Time-Selective Channel[END_REF] who have studied the behaviour of the Alamouti's scheme in highly time selective channels. In all cases, we derive accurate BER bounds and we show that the use of classical Maximum Ratio Combining (MRC) equalizer makes the receiver lose all diversity gain at high SNR's. To save some diversity gain, we propose a Zero Forcing (ZF) scheme [START_REF] He | Bit Error Performance of Orthogonal Space-Time Block Codes over Time-Selective Channel[END_REF] and we quantify its performances for various Doppler effects. The contributions of this technical note are thus the following ones: -Detailed study of the distance spectra of the chaos based encoders and characterization of their distribution.

-Derivation of accurate BER bounds for quasi-static block fading channels and time selective fading channels. The rest of the note is organized as follows. In Section II, we give new insights for the chaos coded modulation schemes proposed by S. Kozic. We propose to approximate the distance distribution with some usual laws such as the Rayleigh one. In section III, we study the performances of the concatenation of the Chaos Coded Modulation (CCM) together with the Alamouti's STBC code for quasi static block fading channels. We do the same in Section IV for the case of time-selective channels considering MRC and ZF equalization. The concluding remarks are eventually given in Section V.

II. Chaos Coded Modulation Scheme, Distance Spectrum Study:

Chaotic coder structure:

We consider the Chaos-Coded modulation scheme of Fig. 1. This scheme was originally given by S. Kozic in his PhD works [START_REF] Kozic | Channel coding and modulation based on chaotic systems[END_REF]. The scheme of Fig. 1 2 . mod(1)

i Q i   

A

where A is some matrix which optimizes the distance spectrum of the code. This mapping, due to the modulus operation, is a highly non-linear operation and serves as a chaos generator. Henceforth, we have a system which combines a convolutional coder with a high dimensional mapping in the same way as Multi-level Trellis Coded Modulation (M-TCM). The corresponding convolutional coder is classically described by: , ,

... . ()

Q i i i Q i Q i D h D t t D t D bD       (1)
a a a Q Q Qi ii k i i i i Q DD           xA vv (2)
Before transmitting x k on the channel propagation medium, we modulate each of its components in NRZ-BPSK in order to obtain a average zero mean value to better interface with a zero mean additive noise such as for AWGN channel i.e:

1 . 2   k k x x
. One can see that the optimization of such multi-dimensional Chaos Coded Modulation is very complicated due to the number of parameters: matrices , ij T , matrix A, parameters Q, Q a … Rather than a global optimization algorithm which should optimize the convolutional coder together with the mapping process, we choose to fix a convolutional coder structure and then we optimize the mapping process by using a particular form of matrix A. We found that the choice , i j shift  TT for i = j and , i j tent  TT for i ≠ j enables to obtain a large set of performing non-linear mapping with A. For example, in the case n = 2, using this choice for matrices T, we are looking for matrices A with the following structure: 21 [START_REF] Chen | Analog Error Correcting Codes based on Chaotical Dynamical Systems[END_REF] 1 a

    

A

and we optimize the choice of 21 a using the distance spectrum. In the case, n = 3, we use matrix structure

A

. The choice of the remaining parameters , ij a is done using the distance spectrum of the code.

. 2 A  Q 1 2 . 2   Q A T 0Q T 00 T 01 T 1Q T 11 T 10 T QQ T Q1 T Q0 + x k Q A . 2 1  b k b k-n+1 b k-Q.n b k-(Q+1).n+1 v 0 v 1 v Q
The state of the coder is defined by vector S k :

T n Q k n Q k n k k k b b b b ] ,..., ,..., ,..., [ 1 ). 1 ( . 1        S
(3) Concerning the choice of Q, it's clear that the Viterbi decoding algorithm is rapidly limited by the complexity in the number of states which is equal to . ( 1) 2 nQ  . Practically, the number . ( 1) nQ  should not exceed 12 which correspond to 4096 states. For n = 2, this gives a maximum value of Q equal to 5, and for n = 3, this gives a maximum value of Q equal to 3. The choice of Q a , is related to the chaotic behaviour of the coder. Although it is beyond the scope of this paper to characterize the chaotic behaviour of the coder, we can illustrate it on simple examples with the phase trajectories of the encoder outputs as functions of the encoder inputs. In fact, generally speaking and depending on the choice of matrix A, varying parameter Q a reduces or increases the chaotic behaviour of the encoder. For example, we illustrate the case n = 2, with matrix

        1 8 1 1 A
. The output of the encoder

         ) 2 ( ) 1 ( k k k x x
x is a vector of two samples. To illustrate the chaotic behaviour of the decoder, we plot on Fig.

)] 1 ( ). 1 ( [ 1 k k x x E  , )] 2 ( ). 1 ( [ 1 k k x x E  , )] 1 ( ). 2 ( [ 1 k k x x E  and )] 2 ( ). 2 ( [ 1 k k x x E 
and the chaotic behaviour of the coder is characterized by the parameter:

] ) )] 2 ( ). 2 ( [ ( ) )] 1 ( ). 2 ( [ ( ) )] 2 ( ). 1 ( [ ( ) )] 1 ( ). 1 ( [ .[( 4 1 2 1 2 1 2 1 2 1 k k k k k k k k x x E x x E x x E x x E C         (4)
The smaller is C and the more chaotic the system behaves since the perfect chaotic system is a white noise with C = 0. On Fig

Spectrum distance analysis and performances over AWGN channels:

In order to optimize the coders, we study their distance spectrum. To do this, we have to determine the trajectories in the trellis which start with a common state

S S  not

necessarily equal to i S . This kind of trajectory in the trellis defines a loop and the loop is characterized by its initial state i S , its final state k S and its length L. The distance of corresponding codewords belonging to the two competing paths in the loop is: The problem of the computation of ( 5) is that, unlike linear codes when we can choose a reference path equal to a all zero sequence, due to the non-linear mapping, we have to test all the possible transmitted sequence for a given loop length together with all the possible starting states. Hence, the distance spectrum computation problem is of non polynomial complexity and in straightforward manner requires the inspection of all possible initial conditions and all possible controlled trajectories. For example, there are

2 1 1 * 2 , ,      L m m m S S L k i d x x (5)
nL Q n 2 . 2 ) 1 (  different controlled trajectories of length L.
In order to compute the distance spectrum with a reasonable complexity while keeping a sufficient accuracy, we form all the possible pair of sequences starting from a given state and both converging towards an other state after L steps with L belonging to the interval [Qn+1, .( ) n Q m  ], i.e. the length of the loop varies from Qn+1 (the constraint length of the code plus one) to .( ) n Q m  (we limit practically the search to m = 2 or 3 in our case due to the computation burden). We have partitioned the distance spectrum into subsets by distinguishing error events which entail one error bit, error events which entail two error bits, error events which entail three error bits and so on... In practice, we limit our search to error events which entail five maximum error bits since simulation results evidenced that it was sufficient to obtain accurate upper bounds for the BER on nonselective Rayleigh fading channels.

We obtain for example with matrices:

         1 0 0 1 , j i T for i = j and          1 1 0 1 i,j T (i.e. n = 2) for j i  and matrix           1 8 1 1 A , Q = Q a = 3
, the distance spectrum illustrated on Fig. 3.

Fig. 3: Distance spectrum of the chaos coded modulation

In fact, we found that, in a majority of cases, the shape of the distance spectrum is close to a Rayleigh distribution with the following probability density function: . The comparison between the Rayleigh distribution and the obtained normalized distance spectrum is illustrated on Fig. 4, it is clear that the two curves feet well. However, for some cases with small free distances, we obtain a much more complicated shape for the distance distribution. For example, in the case In these cases, we model the probability density function by a mixture of Gaussian or Rayleigh laws [START_REF] Lu | Performance analysis and design optimization of LDPCcoded MIMO OFDM systems[END_REF][36].

j C j m x j j C m x x f m x e m x x f j j        , 0 ) ( , . ) ( ) ( 2 2 . 2 / ) ( 2   (
         1 0 0 1 , j i T for i = j and          1 1 0 1 i,j T for j i  and matrix           1 5 1 1 A , Q = Q a = 3,
-Gaussian mixture:

) , ( . . . . 2 1 . ) ( 2 1 . 2 / ) ( 1 2 2 n n J n n m x n J n n C m e x f n n                (7)
In the following, unlike [START_REF] Lu | Performance analysis and design optimization of LDPCcoded MIMO OFDM systems[END_REF], we will not assume that 2

1 : 1 1 j j J j i j n i m p J j j J j j                       
The EM algorithm is an iterative procedure for solving this ML estimation problem. In the EM algorithm, the observation  is termed as incomplete data. Starting from some initial estimate ) 0 (  , the EM algorithm solves the ML estimation problem (8) by the following iterative procedure:

-E-step: Compute    ) ( log ) ( ) ( ) ( X     p E Q i i  (9)
-M-step: Solve

) ( max arg ) ( ) 1 ( i i Q       (10) Define the following hidden data   n i z i ,..., 1 ,   Z where i z is a J-dimensional indicator vector such that:      otherwise 0 ) , ( if , 1 2 , j j i j i m z    (11)
The complete data is then ) , ( Z X  

, we have :

       n i J j z j j i j j i m p 1 1 2 , , ; ( . ) , (      Z  (12)
The log-likelihood function of the complete data is then given by:

C m z z p n i J j j j i j j i n i J j j j i               1 1 2 2 2 , 1 1 , ] . 2 ) ( ) log( . 2 1 .[ log . ) , ( log      Z  (13)
Where C is some constant. The E-step can then be calculated as follows:

    ) , ( log ) ' ( ' Z     p E Q  2 , 2 11
() ( ')
.[log log( ) ] 2.

nJ ij i j j j ij j m Q z C            (14)
Where:

                 J l l l l i j j j i J l l i l i i j i j i i i j i j i j i m m z P z p z P z p z P z E z 1 ' 2 ' ' ' 2 ' ' 1 , ' , ' , ' , ' , ' ' , ' , ). , ; ( ). , ; ( ) 1
( ). 1 ( ) 1 ( ). 1 ( 1 , ˆ                   (15)
In addition, the M-step is calculated as follows. To obtain   j  , we have:

J j z n Q n i j i j j ,..., 1 , . 1 0 ) , ( 1 , '             (16)
To obtain   j m , we have:

' , 2 1 2.( ) ( , ) 0 .[ ] 0, 1,..., 2. n ij ij i jj m Q z j J m             , , 22 11
. ˆ, 1,...,

nn j i i j ij ii jj mz z j J        , 1 , 1 . , 1,..., n i i j i j n ij i z m j J z         (17)
Once,   j m have been obtained, we can obtain j  as:

n ij ij i j j j nn j i j i j i j ii m Q z j J z z m j J                        2 , 2 1 , 1 ˆ.( ) , 1,..., n i j i j i j n ij i zm jJ z         (18)
Finally, the EM algorithm for calculating the Gaussian mixture parameters for the pdf of the distance spectrum is summarized as follows.

-Given the initial set of values For i = 1, …, I, do the following:

- [START_REF] Barbulescu | Chaotic turbo codes[END_REF][START_REF] Zhou | Chaotic turbo codes in secure communications[END_REF]. Set:

Let ) 1 ( '   i   ,
   ) (i .
In the above EM algorithm, the number of mixture components J is fixed. Note that when J increases, ) ( log

  p increases, or ) ( log   p 
decreases. The minimum description length (MDL) principle can be used to determine J.

         ) log( . 2 ) ( log min arg ˆ, n J p J J J MDL   (19)
Where, in the MDL criterion, a penalty term ) log( 2 / n J is introduced. Hence, we can first set an upper bound of the number of mixture components, max J . In addition, for each max J J  , we run the above EM algorithm and calculate the corresponding MDL value. Finally, we choose the optimal J with the minimum MDL. For example, for the distribution of Fig. 5, we find a mixture of two Gaussian laws with parameters, 

1  0.
n J n J ij i j j i j i j j i j i j j m p z z m C                     Z (21)
We denote here 2 ( ; , ) x    as the pdf of a 2 ( , )  R random variable. We obtain:

    ) , ( log ) ' ( ' Z     p E Q  2 , 2 11
() ( ')
.[log log( ) 2.log( ) ] 2.

nJ ij i j j i j j ij j m Q z m C                (22)
With:

    ' ' ' , , , ' '2 ' ' '2 ' 1 ˆ,1 ( ; , ).
( ; , ).

i j i j i j i i j j j J i l l l l z E z P z m m                  (23)
The M-step is calculated as follows. To obtain   j  , we have:

J j z n Q n i j i j j ,..., 1 , . 1 0 ) , ( 1 , '
            (24)
To obtain   j m , we have:

' , 2 1 () ( , ) 1 0 .[ ] 0, 1,..., n ij ij i j i j j m Q z j J mm               22 , 22 1 () ˆ.[ ] 0, 1,..., .(
)

n j i j ij i j i j m z j J m          2 2 2 ,, 11 2 2 2 , , , , 1 1 1 1 ˆ
nn i j j i j i j i j ii n n n n i j i j i j i j i j i j j i i i i z z m m j J z m z m z z j J                         (25)
To obtain j  we have the set of equations: In the case n = 3, we have optimized the non-linear mapping with matrix A in the same way, using , i j shift  TT for i = j and , i j tent  TT for i ≠ j. We obtain the following results.

n ij ij i j j j nn j i j i j i j ii m Q z j J z z m j J                        (26)
With             1 8 2 1 1 2 1 1 1 A
, we have the following distance spectrum :

Fig. 7: Distance spectrum for             1 8 2 1 1 2 1 1 1 A , n = 3
The best results we found in our optimization is given on Fig. 8 with matrix

            1 2 6 1 1 4 1 1 1

A

, we obtain the following spectrum :

Fig. 8: Distance spectrum for             1 2 6 1 1 4 1 1 1 A , n = 3
The minimum distance in this case is nearly equal to: 13 free  d . To end this part, we give some BER results on AWGN channels, using the optimization obtained by the distance spectrum computation. Due to a lack of place we only give simulation results. The use of accurate upper bounds with the approximate pdf of the distance spectrum will be illustrated in the next part when the chaotic coder will be serially concatenated with a Space Time Block code (STBC). For n = 2, we obtain the following result: The chaotic coder outperforms uncoded BPSK at high SNR's due to good asymptotic properties with a moderate high free distance. The weakness of this kind of code is their poor coding rate. There are several solutions to improve this. The first is to make input bits enter the coder by groups of k bits. In this case, the coding rate becomes equal to:

) 1 .(  Q n k .
However, this considerably reduces the correlation degree between consecutive states and renders the trellis non-binary. We found that the penalty encountered by this method too much important (using k = 2 results in 4 dB losses compared to k = 1) so we prefer using puncturing to increase the coding rate of our proposed coders. We added the case of punctured codes on Fig. 9 with the best puncturing patterns we found for rate 8/7 and 4/3. The performances of punctured codes are improved at low or medium SNR's and they outperform the optimized 1/8 chaotic coder whilst at high SNR's they exhibit a smaller free distance and their diversity gain (i.e. the slope of the BER curb) is worse than those of the optimized 1/8 chaotic coder.

As predicted by the spectrum distance, the performances of the code with mapping matrix

        1 5 1 1 A are degraded when compared to the case         1 8 1 1 A .
The conclusions are nearly the same, considering the case n = 3 as it is illustrated on Fig. 10. The number of states for n = 3 and Q = 2 is 2 9 = 512. Due to a better free distance than for the case n = 2, the BER performances are improved and the chaotic coder outperforms uncoded BPSK at a threshold SNR level equal to 8.5 dB. The best punctured codes with puncturing rates 9/7 and 3/2 are unable to outperform the uncoded BPSK even at high SNR's due to a rapid decrease of the free distance value.

The best performances of chaotic coders for n = 3, Q = 2 are illustrated on Fig. 11. This time, with a free distance equal to 13 for the optimized code with rate 1/9, the punctured codes (9/7) are able to outperform uncoded BPSK at high SNR's. To complete this overview of BER performances over AWGN channels, it is important to say that using the approximate pdf's of the distance spectrum, we are able to accurately predict the BER at high's SNR's. To complete the results, we give on Fig. 12 the best performances we found with n = 3, Q = 3 (i.e. the number of states is 4096). In fact, as it is expected, increasing the quantization level for a given dimensionality n, entails some losses. Compared to Fig. 11, the loss in terms of SNR for a BER of 10 -4 , 10 -5 is approximately 1 dB and, once again, punctured codes are nable to outperform uncoded BPSK. .

III. Chaos Coded Modulation Scheme concatenated with STBC, performances over quasi-static block fading channels:

The purpose of this report is to study the behaviour of the chaotic coder on a more general and complex context, when it is concatenated with a STBC (Space-Time Block Code) for a MIMO system. This coder is concatenated with a (Space Time Block Code) STBC as the well known Alamouti's scheme as illustrated on Fig. 13.

Fig. 13: Concatenation of the chaos coded modulation encoder and a STBC code

We consider to simplify the case where n = 2; i.e k x is a vector of two transmitted analog symbols:

         ) ( ) ( 2 1 k x k x k x
. These two symbols are transmitted using the well known Alamouti's

scheme           ) ( ) ( ) ( ) ( * 1 2 * 2 1 k x k x k x k x S
or, in the case of real transmitted symbols,

          ) ( ) ( ) ( ) ( 1 2 2 1 k x k x k x k x S .

3.1-Computation of the Pairwise Error Probability (PEP):

In the case of quasi-static block fading channels, channel remains constant over the duration of a transmitted packet but changes from packet to packet and there is no need to use an interleaver between the chaos-based channel encoder and the STBC. The received signal within two consecutive time slots can be written in the following way:

) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( 2 * 1 21 * 2 11 2 1 1 2 21 1 11 1 n n n x h n x h n y n y n n n x h n x h n y          (29)
11 h and 21 h are two complex random variables with mean zero and variance 2.

n n h n n h n x h h n Z n n h n n h n x h h n Z         (30)
Remark: Z , there will be an error event when having sent the sequence x

= )) 1 ( ), 1 ( ),..., ( ), ( ( 2 1 2 1     L m x L m x m x m x
, each time the decoder chooses x x  ' , both sequences starting in the same state and merging again in possibly other state after L steps, when the decision metric implying x' will be inferior to those with x. Assuming

P h h Z n h h x n Z n h h x n Z n h h x n Z n h h x n                    x x x
Transforming (31), we obtain the following result:

       ) ) ( ) ( )). ( Re( ) ( ) ( )). ( Re( . . 2 ] ) ( ) ( ) ( ) ( .[ ( P roba ' 2 2 2 ' 1 1 1 1 2 ' 2 2 2 1 2 ' 1 2 1 2 n x n x n Z n x n x n Z B n x n x n x n x B m L m n m L m n                with: ) ( ) ( . )) ( Re( ) ( ) ( . )) ( Re( 2 2 2 1 1 1 n N n x B n Z n N n x B n Z R R     and )) ( . ) ( . Re( )) ( Re( ) ( )) ( . ) ( . Re( )) ( Re( ) ( * 2 11 1 * 21 2 2 * 2 21 1 * 11 1 1 n n h n n h n N n N n n h n n h n N n N R R       (32)
We obtain the following rule:

) ) )) ( ) ( ( )) ( ) ( ( .( )) ( ) ( ).( ( )) ( ) ( ).( ( . P roba(2. 2 ' 2 2 1 2 ' 1 1 2 ' 2 2 2 ' 1 1 1 1 n x n x n x n x B n x n x n N n x n x n N B m L m n R m L m n R                (33) Denoting: ) ( . ) ( ) ( ) ( . ) ( ) (
) ( ) ( ) ( ) ( h I I R R h E h E h E h E     
(Note that, for the moment, we consider 11 h and 21 h as constant i.e. not as random variables). We suppose also that: [START_REF] He | Bit Error Performance of Orthogonal Space-Time Block Codes over Time-Selective Channel[END_REF] and:

0 )) ( ). ( ( )) ( ). ( ( )) ( ). ( (
)) ( ). ( ( )) ( ). ( ( )) ( ). ( ( 2 2 2 1 2 1 2 1 2 1 1 1       n n n n E n n n n E n n n n E n n n n E n n n n E n n n n E I R I I R I I R R R I R
2 2 2 2 2 1 1 1 1 )) ( ). ( ( )) ( ). ( ( ... ... )) ( ). ( ( )) ( ). ( (       n n n n E n n n n E n n n n E n n n n E R R I I R R I I
We deduce:

. ) ( )) ( ( . )) ( ( . ) ) ( (   B h h n n E h n n E h n N E R      (35) 2 2 1 2 2 . ) ) ( ( ) ) ( (  B n N E n N E R R   The random variable     ) ( ) ( ). ( ) ( ) ( ). ( . . 2 ' 2 2 2 ' 1 1 1 1 n x n x n N n x n x n N B D R m L m n R         for given sequences x, '
x is a Gaussian random variable as linear combination of random variables with mean zero and variance:

    2 ' 2 2 1 2 ' 1 1 2 2 ) ( ) ( ) ( ) ( . . . 4 n x n x n x n x B B m L m n           (36) 2 2 3 . . 4  C B   with     2 ' 2 2 1 2 ' 1 1 2 ) ( ) ( ) ( ) ( n x n x n x n x C m L m n        
We can now compute the PEP Pairwise Error Probability of detecting x' instead of x. (41) To derive e P , the complete probability of error event, we need to average (41) over the distribution of ) ( 2

( ) , , ' ( 21 11 P h h P e   x x x D > A) (37) with     2 2 2 ' 2 2 1 2 ' 1 1 2 . ) ( ) ( ) ( ) ( . C B n x n x n x n x B A m L m n          This yields to: dx e C B h h P C B C B x e . . . . . 2 2
    ) . 2 . 2 ) ( ) ( ) ( ) ( . ) ( ( . 2 1 ) , , ' ( 2 / 1 2 ' 2 2 1 2 ' 1 1 2 / 1 2 21 2 11 21 11                    n x n x n x n x h h erfc h h P m L m n e x x x
        x x x ( 43
)
Where we used the equality:

)! 1 ( ) (    n n .
To

                                                          (
                                 h h u h h C
                                  h h u h h C du e u C J ( 46
)
Using the well known equalities for  function: 1) sin( . )

            dt e t z t z . . ) ( 0 1 ( 1) . ( ) ( ). (
z z z zz z           (47)
We obtain:

   ) 2 / 1 ( and 2 / ) 2 / 3 (    (48)
Then, putting these results into H in (44), we find:

                 2 /
                 h h h h h h h C C C C H               2 / 1 2 2 2 4 2 / 3 2 2 2 2 2 2 4 . 4 . . 8 . . 2 . 2 2 / . 4 . . 8 . . 2 . 4                      h h h h h h h C C C C H ( 49
)
We obtain eventually the probability of error event for a given error length sequence:

                                             x x x x x x
At high SNR's, it is interesting to compute the diversity gain, we have:

C C C C C P C                                             x x x x x x x x x 1) With 2 0 1 6.( 1). .( / ) b Q n E N    , ( see
e hb P C n Q E N    x x x ( 52
)
That is, at high SNR's, we have a diversity gain of two.

To compute the probability ) ' (

x x  e P
, it is then necessary to study the distance spectrum of the channel coder, i.e, we need to average (50) over the distribution of the quantity C. As we showed in part 2.2, a Rayleigh distribution or mixtures of Rayleigh or Gaussian distributions can be used with good accuracy. However, before doing this, and in order to obtain a closed form expression of e P in terms of a series expansions of 2  , we develop ( ' ) e P  x x x using the following rules.

           n x n n x x x 1 1 1 1 ( ' ) . . 2
C n C                    x x x
We obtain:

P m C n C                        x x x 2( 1) 1 2 ( 1) 2 2( 1) 1
( 1) .2 . .( 1).( 2

                 xx ( 54
)
We have: Using the classical expansion for x < 1:

                          
1 1 . ! ) 1 ( ! . ) 1 ( ). 2 )...( 1 .( ) 1 ( ) 1 ( 1                      k n k n n k n k n n k x k n n x k n n n x
We obtain:

. . ) / ( ) 2 ) 1 ( )...( 1 ( . . ) 1 ( . 1 ) ( . 2 2 min min 2 2 . 2 / 1 ) 1 ( 1 1 1 . 2 /                          du e u
                             
And, for the second integral form, we have:

                               22 / 2. 1 1 0 ' 1 1 1! ( 1)
                      xx (56)
Remark:

1-The integral expressions involved in the computation of (54-56) can be computed recursively. We have:

. ) ( 1                                 1 1 0 1 1 ! . . ) 1 (



).

For J 0 , we have: p I the computation is straightforward using the expansion series of function

. 2 / 1 . ! . 2 ) 1 ( ) . 2 ( 1 . ! ) 1 ( 2 2                
, we obtain:

) 1 2 ( ) ( . . ! . 2 1 . ) 1 ( . . . . ! . 2 1 . ) 1 ( 1 2 2 0 0 2 2 0 '                  p
1 . ). 1 .( 6
1 ) ). 1 .( /( / 0 2 2 0 0 0 N E n Q n Q N n Q N Q n E N E b s b           (57) 3 / 1  s E 3
corresponds to the average energy per transmitted symbol. The derivation of the complete Pairwise Error Probability in (54, 56) is useful to obtain an accurate upper-bound of the Bit Error Probability (BEP) for the STBC based chaos coded modulation system.

Bit Error Probability:

We consider at first a trellis section between time m and m+1 and we assume that a particular sequence x starting from a given state S has been transmitted. We have an overall number of states equal to N S . Let ) , , w ( l d E be the event that an other incoming sequence x d , starting from the same state S, with Hamming distance w between x and x d (error weight sequence is equal to w ), with Euclidean distance between G(x) and G(x d ) equal to d (G is the encoding function) and length l is active, i.e., has the smallest path metric in the interval (m, m+1). Also, denote with is the probability of an error event produced by an incorrect path x d with Hamming distance w from x, Euclidean distance d between G(x) and G(x d ), and length l; To obtain b P we have to average formula (61) over all the possible departure states and over all the possible reference transmitted sequences. Assuming uniform density probability for each probability distribution (the probability to be in a given state and the probability to send a given reference sequence), this yields to: In the case of the studied chaotic encoder, we can replace, due to the richness of the distance spectrum, the summation over d and l by continuous integration since we found that, for a given error event length w, the probability density distribution of d follows either a Rayleigh function similar as in ( 6) or a mixture of Gaussian or Rayleigh distributions. We obtain the following result: e P  xx , has been given in (54, 56) since we suppose that, for a given value w, the distance spectrum follows either a Rayleigh function or a mixture of Gaussian or Rayleigh distributions.

Simulation results:

We checked formula (63) using the chaotic coder of Fig. 1 (a = 8, n =2, Q = 3) and the coder optimized for n 3, = 2. We compare the upper bound of (63) with simulation results on Fig. 14. One can see that the two curves are very close to each other in the two cases (n = 2 and n = 3) showing the accuracy of the formulas ( 56) and (63). The system for n = 3, Q = 2 has a high diversity gain at high SNR's since its slope is much more important that those for the case n = 2, Q = 3. This is due to a better free distance.

IV. Chaos Coded Modulation Scheme concatenated with STBC, performances over time selective fading channels:

We start from the classical Alamouti's model, but this time we consider that channel parameters may vary from sample to sample. We have the following equation:

) 1 2 ( ) 1 2 ( ). 1 2 ( ) 2 ( ). 1 ( ) 1 2 ( ) 2 ( ) 1 2 ( ). 2 ( ) 2
            n n n x n h n x n h n y n n n x n h n x n h n y ( 64
)
We can write these equations in equivalent matrix form:

                                        ) 1 2 ( ) 2 ( ) 1 2 ( ) 2 ( ) 1 2 ( ) 1 2 ( ) 2 ( ) 2 ( ) 1 2 ( ) 2 ( * * 11 * 21 21 11 * n n n n n x n x n h n h n h n h n y n y ) ( ) ( . ) ( ) ( n n n n N X H Y   (65)

Classical MRC use:

Using the classical MRC equalization is equivalent to multiplying the received vector Y by ) (n

H H that is: ) ( ). ( ) ( . ) ( ). ( ) ( ). ( n N n n n n n n H H H H X H H Y H  
(66) We have:

                                         
(

n h n h n h n h n h n h n h n h n h n h n h n h n n n h n h n h n h n h n h n h n h n n H H

H H H H

Similarly as in [START_REF] Lu | Performance analysis and design optimization of LDPCcoded MIMO OFDM systems[END_REF], we use the following notations:

) 1 2 ( ). 1 2 ( ) 2 ( ). 2 ( ) ( ) 1 2 ( ) 2 ( ) ( ) 1 2 ( ) 2 ( ) ( 21
          n h n h n h n h n n h n h n n h n h n    The off-diagonal elements of ) ( ). ( n n H H H
are non-zero and therefore they cause ISI. Consequently, the interferences are introduced into the decision metrics of x(2n) and x(2n+1).

                            ) 1 2 ( ). 1 2 ( ) 2 ( ). 2 ( ) 2 ( ). ( ) 1 2 ( . ) 1 2 ( ) 2 ( ) 1 2 ( ) 1 2 ( ). 1 2 ( ) 2 ( ). 2 ( ) 1 2 ( ). ( ) 2 ( . ) 1 2 ( ) 2 ( ) 2 ( 11
) 1 ( 1 h R  (see 77-79) ) ) 1 ( 1 ), 2 ). 1 ( ( ) 1 2 ( 2 11 ) 2 ( 11 11 h h n h R n h R n h    
Similarly, one could demonstrate that:

) ) 1 ( 1 ), 1 2 ( ). 1 ( ( ) 2 ( 2 21 ) 1 2 ( 21 21 h h n h R n h R n h     
Using these two characteristics, it is easy to demonstrate that the interference term ) (n



is a conditional complex Gaussian random variable, which is given by:

)) ) 1 ( 1 ).( ) 1 2 ( ) 2 ( ( , 0 ( ) ( 2 2 21 2 11 ) 1 2 ( ), 2 ( 21 11 h n h n h R n h n h n        (67) Conditioning on the channel parameters ) 1 ) 1 ( 2 ( ),..., 1 2 ( ), 2 ( 21 11 11
    m L h m h m h
, one can compute the Pairwise Error Probability:

2 2 1 2 1 2 2 1 2 1 1 : ), ( ) 1 2 ( ' ). ( ) 1 2 ( ) 2 ( ' ). ( ) 2 ( ) 1 2 ( ). ( ) 1 2 ( ) 2 ( ). ( ) 2 ( Proba ) , ' ( 
                             n x n n Z n x n n Z n x n n Z n x n n Z P m L m n m L m n m L m n n e     x x x H
We then replace:

) ( ) 1 2 ( ). ( ) 1 2 ( ) ( ) 2 ( ). ( ) 2 ( 2 2 1 1 n N n x n n Z n N n x n n Z         With: ) 1 2 ( ). 1 2 ( ) 2 ( ). 2 ( ) 2 ( ). ( ) ( ) 1 2 ( ). 1 2 ( ) 2 ( ). 2 ( ) 1 2 ( ). ( ) ( 11
           n n n h n n n h n x n n N n n n h n n n h n x n n N  
We obtain the PEP:

1 * ( ), : 1 1 1 * 22 1 2 2 2 2 12 ( ' , ) Proba 2.Re( ( ).[ ( ).( (2 )
'( 2))] ...

2.Re( ( ).[ ( ).( (2 1) '

(2 1))] ( ) '(2 ) (2 ) ( ) '(2 1) (2 1) (68) Lm e n n m L m nm Lm nm P n N n x n x n n N n x n x n n x n x n n x n x n                                         H x x x
The random variable:

1 ** 1 1 2 2

2.Re[ ( ).[ ( ).( (2 ) '(2 ))] ( ).[ ( ).( (2 1) '(2 1))]]

Lm nm X n N n x n x n n N n x n x n           X,
) mL h X h n h n nm h n x x n n R x n n n x n x n n R x n n                         
Then, the probability (68) is equal to:

( ), : 1

1 22 22
1 2 2 2 1 1/ 2 1 22 2 2 2 2 2 2 2 2 1 2 1 1 2 2 1 1 2 ( ' , ) ( ) . ( ) . 1 . ( ) 2 2. 2.

( ). .[ ( ).(1 (1) ). (2 1) . ( )] ( ). .[ ( ).(1 (1) ). (2 ) . ( )]

e n n m L m mL nn nm mL n h n h nm P n C n C erfc n C n R x n n n C n R x n n                                 H x x x
With:

2 2 2 )) 2 ( ' ) 2 ( ( n x n x C n   and 2 2 1 2 )) 1 2 ( ' ) 1 2 ( (      n x n x C n
We use the following approximations: ]

            L m m n m h m h n h n h n h n h i.e.: 1 2 1 ( ) ( ) ( ), [ , 1] n
n m n m m L          , we obtain : 11 21
(2 1), (2 )

1 1/ 2 2 2 1 2 2 1 1/ 2 1 2 2 2 2 2 2 2 2 2 2 1 ( ' , ) ( ) ( ) 1 .
(

) 2 2. 2. .[(1 (1) ). ( 2 1)
] .

[(1 (1) ). ( 2) ]

e h m h m mL n n nm mL n h n h nm P m C C erfc C R x n C R x n                            x x x (70)
To complete, the PEP computation, we have to average (70) over the distribution of 1 () m

 . 1 () m  follows a chi-squared distribution: 1 () 42 1 . .exp( ) 4. . (2) 2. m hh x px     . 11 21 2
(2 1), ( 2)

1 1/ 2 2 2 2 2 1 / 2. 4 1/ 2 1 22 0 2 2 2 2 2 2 2 2 1 ( ' , ) () 1 (
P x C C erfc x e dx C R x n C R x n                            x x x
Setting:

mL n h n h nm C R x n C R x n              which is independent of 2 
and thus the expression (72) shows that we obtain an irreductible error floor at high SNR's, that is the MRC equalizer completely loses diversity gain. Therefore, the time selectivity of the channel makes the conventional (MRC) linear equalizer lose all the diversity. It is necessary to suppress the interfering terms in the expressions of ( 2), (2 1) Z n Z n  to restore some diversity gain. To obtain the BER, we still have to average (69) over the distance spectrum distribution. Once again, using a Rayleigh distribution approximation, we are able to derive a closed form expression of a lower bound in a form of a series expansion in terms of where our hypothesis which consists in considering channel parameters constant over the length of the diverging sequences is too much optimistic. The difference between simulation results and the lower bound in this case is 2 dB whilst it is less than 1 dB for the other Doppler rates.

ZF equalization:

Since the irreductible error floor of the conventional MRC equalizer is caused by ISI arising from the off-diagonal elements () n  , it is of interest to construct a decision metric that mitigates () n  . In order to suppress ISI due to the temporal variations of the channel coefficients, one has to diagonalize ) (n H . A possible matrix which diagonalizes ) (n H is given by [START_REF] Lu | Performance analysis and design optimization of LDPCcoded MIMO OFDM systems[END_REF]:

            ) 2 ( ) 1 2 ( ) 2 ( ) 1 2 ( ) ( 11 * 21 21 * 11 n h n h n h n h Q (73)
Multiplying the two sides of (65) by ) (n Q , we obtain:

N Q H Q Y Q ). ( ). ( ). ( ). ( n X n n n   (74)
The matrix

) ( ). ( n n H Q is equal to:                ) 1 2 ( ). 2 ( ) 1 2 ( ). 2 ( 0 0 ) 1 2 ( ). 2 ( ) 2
h h h h h h h h h h h h h h R R R R R R R h n h E R n h n h E R n h n h E n h R n h n h R n h E                       (79)
h h n h R n h R n h     (80)
Similarly, one could demonstrate that:

) ) 1 ( 1 ), 1 2 ( ). 1 ( ( ) 2 ( 2 21 ) 1 2 ( 21 21 h h n h R n h R n h      (81)
Using the equivalent model (74), we obtain, after diagonalization, the equivalent mathematical model:

) 1 2 ( ). 2 ( ) 2 ( ). 1 2 ( ) 1 2 ( ). ( ) 1 2 ( ) 1 2 ( ). 2 ( ) 2 ( ). 1 2 ( ) 2 ( ). ( ) 2 (
            n n n h n n n h n x n n Z n n n h n n n h n x n n Z   (82)
                              n x n n Z n x n n Z n x n n Z n x n n Z P m L m n m L m n m L m n n e     x x x H
We eventually obtain:

P n x n x n x n n N n x n x n n x n x n x n n N n x n x n n x n x n x n                                 H x x x 1 22 1) (2 1) (83) Lm nm xn            With: ) ( ) 1 2 ( ). ( ) 1 2 ( ) ( ) 2 ( ). ( ) 2 ( 2 1 n N n x n n Z n N n x n n Z         and: ) 1 2 ( ). 2 ( ) 2 ( ). 1 2 ( ) ( ) 1 2 ( ). 2 ( ) 2 ( ).
P n N n x n x n N n x n x n n x n x n x n x n                                  H x x x
To further develop the computation of the error probability, one has to characterize the random variable:

)

)] 1 2 ( ' ) 1 2 ( ).( ( )) 2 ( ' ) 2 ( ).( ( ).[ ( Re( . 2 * 2 1 * 1           n x n x n N n x n x n N n X m L m n



Conditioning on the channel parameters, X is a Gaussian random variable with mean zero and variance 2 X  .

 

)

) 1 2 ( ' ) 1 2 ( )].( 1
                 n x n x n n n h n n n h n n x n x n n n h n n n h n X m L m n   (
             n n j n n n n n h j n h n h n n j n n n n n h j n h n h n j n n I R I R I R I R I R    (86)
Reporting these expressions into (85), we obtain eventually:

                                 n x n x n n j n n n h j n h n n j n n n h j n h n j n n x n x n n j n n n h j n h n n j n n n h j n h n j n X I R I R I R I R I R I R I R m L m n I R I R I R     (87)

2.(

Lm R R R I I R R I I nm I I R R I R I I R X x n x n n h n n n h n n n h n n n h n n n n h n n n h n n n h n n n h n n n x                         21
] R R R I I R R I I I I R R I I R R I n x n n h n n n h n n n h n n n h n n n n h n n n h n n n h n n n h n n n                     
Using this last expression, we obtain:

                  (88)
To obtain (88), we have used the fact that

(2 ), ( 2), (2 1), (2 1)...

R I R I n n n n n n n n 

are white Gaussian noise samples, i.e:

l k l n k n E I R    , 0 )] ( ). ( [ and l k l n k n E l n k n E I I R R     , )] ( ). ( [ )] ( ). ( [ 2 2 2 ] )) ( [( ] )) ( [(    k n E k n E I R
Hence, we obtain:

                 n x n x n h n h n x n x n h n h n m L m n X    (89)
Using the Gaussian distribution of X, we compute the Pairwise Error Probability:

) )), 1 ( 2 ( ), 1 ) 1 ( 2
                                    L m m n n n L m m n n n e C n h n h C n h n h n C C n erfc L m h L m h m h m h m P     x x x (90) With : 2 2 2 )) 2 ( ' ) 2 ( ( n x n x C 0 21 11 21 11                                       L m m n n n L m m n n n e C n h n h n h n h n d C C n L m h L m h m h m h m P         x x
P m h m h m m C C d m h n h n C h n h n C m C C d                                x x x     1 1 2 2



remains constant over the block length L. This corresponds to a rather optimist case (as Fig. 15 showed it) and, to be more accurate, one has to choose a better approximation of

V. Conclusion:

Based on the former work of Kozic & al, we have proposed new insights for the stud of performing Chaos Coded Modulation (CCM) schemes. We have studied in detail the distance spectrum of some CCM schemes and we have approximated their pdf's with some well known laws. Using these approximations, we have been able to study the performances of these CCM schemes when they are concatenated with a Space Time Block Code (STBC). We obtained accurate bounds even in the case of time selective channels. In the case of time selective channels, we showed that the MRC classical equalizer makes the lose all the diversity gain at high SNR's exhibiting BER floor while the zero-forcing equalizer structure loses only one order of diversity. The provided simulation results, together with the computation of the BER bounds, illustrate the accuracy of our closed form expressions except in the case of highly time selective channels.

  can be represented by means of a convolutional coder of rate 1/( .(1)) nQ   , where at each time step k, one bit

S

  choice of the matrix A, we can write the transmitted vector at the output of the modulator:

Fig 1 :

 1 Fig 1: Trellis chaos coded modulation encoder

Fig. 2 :

 2 Fig. 2: Phase trajectory of the chaotic encoder for different values of Q a .

6 )and 2 j

 62 For example, with the distance spectrum plotted on Fig 3, we calculate parameters j m to obtain the best fitting between the pdf of the distance spectrum and )

Fig. 4 :Fig. 5 :

 45 Fig. 4: Distance spectrum v.s. Rayleigh distribution

-

  Given the initial set of values j  -Given the distance values   i  , the number of mixture components J, and the total number of EM iterations I, starting from the initial parameters ) 0 (  .

  equations[START_REF] Escribano | Iteratively decoding chaos encoded binary signals[END_REF][START_REF] Kozic | Controlled One-and Multidimensional Modulations Using Chaotic Maps[END_REF] is a set of coupled non-linear equations and we use the optimization toolbox with the function fsolve to solve[START_REF] Escribano | Iteratively decoding chaos encoded binary signals[END_REF][START_REF] Kozic | Controlled One-and Multidimensional Modulations Using Chaotic Maps[END_REF] at each maximization step.

Fig. 6 :

 6 Fig. 6: Normalized Probability Density Distribution (pdf) approximation with Gaussian mixture for Fig. 5.

Fig. 9 :

 9 Fig. 9: Performances of Trellis Chaos-Coded Modulation over AWGN channels for n = 2, Q = 3

Fig. 10 :

 10 Fig. 10: Performances of Trellis Chaos-Coded Modulation over AWGN channels for n = 3, Q = 2

Fig. 11 :

 11 Fig. 11: Performances of optimized Trellis Chaos-Coded Modulation over AWGN channels for n = 3, Q = 2

Fig. 12 :

 12 Fig. 12: Performances of optimized Trellis Chaos-Coded Modulation over AWGN channels for n = 3, Q = 3 (4096 states)

  .

  .

  We suppose that:

2 - 2 

 22 The expressions of the Pairwise Error Probability (54, 56) are serial expansions in terms of , the variance of the additive white Gaussian noise on the link. It can also expressed in terms of E b

  the multiplicity of such incorrect paths. The inequality in (58) relies on the fact that, to be active in the interval (m, m+1), the error event ) l must have started at a node between m-l+1 and m.The bit error probability conditioned on x and S, is in error in the interval (m, m+1), is given by: fact w is the number of information digits which differ between x and x d and l is the overall number of transmitted digits in x x d (incoming bits are encoded one after one). Substituting (60) into (59), we obtain:

  Pairwise Error Probability (PEP) for pair of sequences which having w error length weight. ,w ( ')

Fig. 14 :

 14 Fig. 14: Performance comparison between simulation results and the Upper-Bound results

  results clearly highlight the BER floor phenomenon at high SNR's as it is plotted on Fig. 15. Depending on the Doppler rate, the simulation results and the lower bounds feet quite well except for the case . 0.01 ds fT

Fig. 15 :

 15 Fig. 15: MRC Simulation results and lower bounds for various Doppler effects

  Applying the above lemma, we can obtain:

  shows a diversity order of one. Therefore, the time selectivity of the channel makes the conventional linear decoder lose one order of diversity, compared with the full diversity order of two in the quasi-static fading channel case. To obtain the BER bounds, we have to average (98) over the distance spectrum distribution. We obtain the curves of Fig.16. It is clear from Fig. 16 that the BER floor has been avoided. The BER performances have been considerably improved for the high rates Doppler . 0.01 ds fT , since we obtain a BER of 4 7.10  at E b /N 0 = 20 dB whilst the BER saturates at 3 1.7.10  on Fig. 15.

Fig. 16 :

 16 Fig. 16: ZF Simulation results and lower bounds for various Doppler effects

  2.

								nn m  	, i.e. that the variance is twice
	the mean. Finding J,
	Fortunately, the EM (Expectation Maximization) algorithm enables to reduce considerably the
	complexity. For a fixed number of mixtures J, based on the observations		  	i	i , 	 n ,..., 1	,
	the parameters		  j j , , , j m  j	1,...,	J		can be estimated using the EM (Expectation
	Maximization) algorithm.
	The problem can be formulated as follows. Denote	;  x (		,		2	)	as the pdf of an		(		,		2	)
	random variable. Then, the maximum likelihood (ML) estimate of the parameters  is given
	by:
			, ˆ2 1 ; ( . log max arg ) ( log max arg 1 1 :	)	) 8 (

n  , m n and n  is a complicated multidimensional optimization problem.

  2

																					variables
	) 1 n ( Z	and	Z	) 2 n (	:
					1	(	)	(	11	2	21	2	).	1	(	)	. 11 *	1	(	)	. 21	* 2	(	)
					2	(	)	(	11	2	21	2	).	2	(	)	. 21 *	1	(	)	. 11	* 2	(	)

h  . 1 n and 2 n are two additive complex white Gaussian noise samples with mean zero and variance 2. 2  .

At the output of the Maximum Ratio Combiner (MRC) we form the two decision

  Using the definitions of B and C, we obtain the conditional probability of error:

															P e	(	x		x	'	11 h	,	21 h	,	x	)		2 1	( erfc .	. 2 B 1	.  C 2 2 /	)	(40)
																												2
	(	x		x	'	11	,	21	,	x	)									1 			3	/	2	2   	2		. 8	3	.	2	. 	2	(38)
	Setting	u 	x	. 2 /(	. 2 B	3	/	2	.  C .	)	, we eventually obtain :
															P e	(	x		x	'	h 11 ,	h 21 ,	x	)		1 	.	e C  B 2 / 1 		u	2	du .	(39)
																												. 2	2	. 

  conditioning on the channel parameters, is a Gaussian random variable with mean zero and variance:

	2	11	21 (2 ), (2 1)	1	2 1 4. ( ).( (2 )	2 '(2 )) .[ ( ).(1 1	2 (1) ). (2 1) 2	2	1 . ( )] ...
			2 2 4. ( ).( (2 1)	2 '(2 1)) .[ ( ).(1 1	2 (1) ). (2 ) 2	2	2 . ( )] (69

  x

	To	further	proceed	the	above	expression,	we	use	the	hypothesis
	] ( ' ( ), (2 1), (2 ), )
	e
						mL
					( ) .(	[	] ) .
	1						nn
		. exp(				nm					)
			mL
		2. . ( ) (	[	(2 1)	(2 ) .	(2 1)		(2 ) .	]).sin
							nn
			nm
					( ) .(	[	] ) .
	1						nn
		. exp(

  Furthermore (if we consider the case of moderate Doppler effects), we suppose: Lemma 1: if x is a real Gaussian random variable with mean

	(					'		(	2			), 1		(	2	),	)
	1	.			( [exp	. 2	.( Re(	( (	2	))	) 1 .(				[		(	2	)	sin ). ]	)	)].	[exp(	.( Im( . 2	(	2 )) (	) 1 .(	[	21	(	2	)	2	sin ). ). ] 2	)]
							2 nn 2 2 2 2 21 11 2 1 (2 1) (2 ) . h n h n C  2 2 11 21 2. .( [ (2 1) (2 ) . ]).sin  mL   nm  mL nm h n h n C        	)
	2 ( 21 n h		) 1	2		) n 2 ( 11 h	2		2 ( 11 n h		) 1	2		) n 2 ( 21 h	2		2 ( 11 m h		) 1	2		) m 2 ( 21 h	2	 n		, m [	m		L		] 1
	Using the following lemma:
																																m	x	x and variance 2  , we have:
																																2
																			. w [exp( E	x	2	)]		2 x  . x w . . . 2 . 2 w m w    1 1 exp(	2 x	)	(91)
	we obtain [34]:
													P e 	. 2 (   ( exp ' 2 m  / 0  . x x 1 (   	sin ). ). ] 1 2  2 d  ) n 2 ) 2 m C 2 ), ( 21 h 2 2 2 C ( 21 [ 2 1 m h ) 1 ), 1 x L m .( m 2 2 2 ( ) ( 11 11 ( .( ), 2  h m m h  m n n       		)	(92)
	To average (92) over the distribution of		(m	)	, we can see that conditioned on
	(	2			), 1			(	2	)
		m	 			R h			h			m				h			m		R h	h	m			h	m
	Since		)) Re( m ( 	is independent of		)) Im( m ( 	we can average the conditional probability of
	error separately as:


Hervé Gaussier
email: herve.haussier@univ-grenoble-alpes.frharish@math.iisc.ernet.in

Harish Seshadri

HOLOMORPHIC MOTIONS AND COMPLEX GEOMETRY HERV É GAUSSIER AND HARISH SESHADRI

Introduction and Main Result

Let B be a connected complex (n -1)-manifold with a basepoint z 0 ∈ B. A holomorphic motion of the unit disc ∆ ⊂ C parametrized by B is a continuous map f : B × ∆ → CP 1 = C ∪ {∞} satisfying the following conditions:

(1) f (z 0 , w) = w for all w ∈ ∆, (2) the map f (z, .) : ∆ → CP 1 is injective for each z ∈ B, (3) the map f (., w) : B → CP 1 is holomorphic for each w ∈ ∆.

Holomorphic motions were introduced by R. Mãne, P. Sad and D. Sullivan [START_REF] Mãne | On the dynamics of rational maps[END_REF] and have been intensively studied since then (see, for instance, [START_REF] Slodkowski | Holomorphic motions and polynomial hulls[END_REF][START_REF] Chirka | Holomorphic motions and uniformization of holomorphic families of Riemann surfaces[END_REF][START_REF] Douady | Prolongements de mouvements holomorphes[END_REF][START_REF] Astala | Papers on analysis: a volume dedicated to Olli Martio on the occasion of his 60th birthday[END_REF]). In this note we study the complex-analytic structure of the graph D of f :

D = {(z, f (z, w)), z ∈ B, w ∈ ∆} ⊂ B × CP 1 .
Our main result is the following Theorem 1.1. The graph D of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n .

Denoting the unit ball in C n by B n , Theorem 1.1 will be a consequence of the following Theorem 1.2. Let S ⊂ C be a bounded domain. If A(C n ) denotes the set of complex affine (n -1)-dimensional subspaces of C n , then there does not exist a map f : S → A(C n ) satisfying the following conditions:

(1) For t ∈ S, if W t = f (t) ∩ B n , then B n = ∪ t∈S W t .

(2) Either W t ∩ W s = φ or W s = W t for s, t ∈ S.

(3) There is a holomorphic map π :

B n → B n-1 such that π : W t → B n-1 is bijective for all t ∈ S.
To derive Theorem 1.1 from Theorem 1.2, we use a rescaling argument based on a recent result of K. T. Kim and L. Zhang [START_REF] Kim | On the uniform squeezing property of bounded convex domains in C n[END_REF].

To put these results in context, we recall the following result of K. Liu [START_REF] Liu | Geometric height inequalities[END_REF] and V. Koziarz-N. Mok [START_REF] Koziarz | Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations[END_REF] : [START_REF] Koziarz | Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations[END_REF]) Let n > m ≥ 1 and let Γ 1 ⊂ SU (n, 1), Γ 2 ⊂ SU (m, 1) be torsion-free cocompact lattices. Then there does not exist a holomorphic submersion from B n /Γ 1 to B m /Γ 2 .

Theorem 1.3. ([10],
2010 Mathematics Subject Classification. 32F45, 32Q45, 53C23.

This was proved for n = 2, m = 1 by K. Liu [START_REF] Liu | Geometric height inequalities[END_REF] and for all n > m ≥ 1 by V. Koziarz and N. Mok [START_REF] Koziarz | Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations[END_REF]. This result is natural from various points of view. In particular, it is related to the following well-known question in the study of negatively curved Riemannian manifolds:

Let f : M n → N m be a smooth fibre bundle where M and N are smooth compact manifolds of dimensions n > m ≥ 2. Can M admit a Riemannian metric with negative sectional curvature ?

If the bundle above is trivial then Preissman's theorem implies that the answer to the above question is in the negative. Also, it is essential that m ≥ 2: a theorem of W. Thurston states that certain 3-manifolds fibering over a circle admit hyperbolic metrics. Theorem 1.2 implies the Liu-Koziarz-Mok result when n = m + 1 by the Bers-Griffiths uniformization theorem as explained later in the paper. The compactness of the manifolds is essential in the question above and the result of Liu-Koziarz-Mok. In other words, cocompact group actions on universal covers are needed. Our point of view is that given the Bers-Griffiths theorem, the cocompact actions are not necessary. The proof we present involves some elementary facts about the Kobayashi metric and Riemannian geometric techniques.

Note that an equivalent formulation of Theorem 1.2 is that the graph of a holomorphic motion cannot admit a complete Kähler metric with constant negative holomorphic sectional curvature. Hence the following question is natural:

Can the graph of a holomorphic motion of the unit disc admit a complete Kähler metric with variable negative sectional curvature?

A related question, mentioned to the authors by Benoit Claudon and Pierre Py, is:

Can the graph of a holomorphic motion of the unit disc be Gromov hyperbolic with respect to the Kobayashi metric ?

The method in this paper appears to hold some promise for tackling these questions. In this connection, it is important to point out that metrics with weaker negative curvature conditions can exist on such domains: a result of S. K. Yeung [START_REF] Yeung | Geometry of domains with the uniform squeezing property[END_REF] asserts that the universal cover of a Kodaira fibration, which is necessarily the graph of a holomorphic motion by the Bers-Griffiths theorem, admits complete Kähler metrics with negative holomorphic bisectional curvature.

2. Proof of Theorem 1.2

The Kobayashi metric on D. Let

• B be a connected complex (n -1)-manifold with a basepoint z 0 ∈ B,

• f : B × ∆ → CP 1 a holomorphic motion, • D = {(z, f (z, w)) : z ∈ B, w ∈ ∆} ⊂ B × ∆, • F : B × ∆ → D be defined by F (z, w) = (z, f (z, w)), • π : D → B denote the first projection, • for p ∈ D, let S p = π -1 (π(p)),
• for w ∈ ∆, let

F w : B → D be F w (z) = f (z, w) and Σ w = F w (B). Note that π -1 (z) = F ({z} × ∆) for every z ∈ B.
Lemma 2.1. For every w ∈ ∆, the map F : B × {w} → D is a holomorphic embedding which is totally geodesic for the Kobayashi metrics on B and D.

Proof: Since

d K ∆ (z 1 , z 2 ) ≥ d K D (F (z 1 , w), F (z 2 , w)) ≥ d K ∆ (π • F (z 1 , w), π • F (z 2 , w)) = d K ∆ (z 1 , z 2 )
by the distance decreasing property of the Kobayashi metric the result follows.

2.2. Proof of Theorem 1.2. Let S ⊂ C be a bounded domain and f : S → A(C n ) be a map, where A(C n ) is the set of complex affine (n -1)-dimensional subspaces of C n . Suppose that we have

(1) a partition B n = ∪ t∈S W t where W t = f (t) ∩ B n and
(2) a holomorphic map π :

B n → B n-1 such that π| Wt : W t → B n-1 is bijective.
Before stating the next lemma, we recall that the Kobayashi metric on B n coincides with the Bergman metric and is, in particular, a C 2 Riemannian metric. Lemma 2.2. (1) For every t ∈ S and z ∈ B n-1 , W t ∩ π -1 (z) consists of a single point and the intersection is orthogonal.

(2) The fibers of π are equidistant, i.e., for any z 1 , z 2 ∈ B n-1 and p ∈ π -1 (z 1 ) we have

d K D (p, π -1 (z 2 )) = d K D (π -1 (z 1 ), π -1 (z 2 )) Proof: Let t ∈ S and z 1 ∈ B n-1 . It is clear that W t ∩ π -1 (z 1 ) is a singleton since π| Wt : W t → B n-1 is bijective. Fix p ∈ π -1 (z 1 ). Since B n = ∪ t∈S W t , p ∈ W t for some t ∈ S. Let γ : [0, 1] → D be a unit-speed geodesic with γ(0) = p and γ ′ (0) ∈ T p W t . By Lemma 2.1 we can assume that γ([0, 1]) ⊂ W t . Let γ(1) = q ∈ π -1 (z) ∩ W t .
We claim that γ is the shortest geodesic between π -1 (z 1 ) and π -1 (z). This is because

d K D (p, q)) ≥ d K B n-1 (z 0 , z) = l(γ) ≥ d K D (p, q) for any w 1 , w 2 ∈ ∆.
The equality above comes from the assumption that π| Wt : W t → B n-1 is bijective and hence an isometry. Since k is a Riemannian metric, the first variation for arc-length implies that γ meets π -1 (z 1 ) and π -1 (z) orthogonally. Hence γ ′ (0) is orthogonal to

T p π -1 (z 1 ).
In what follows we use the following notation: for any p ∈ D S p := π -1 (π(p)).

Corollary 2.3. Let γ : [0, L] → D be a geodesic with γ(0) = p, γ ′ (0) ∈ (T p S p ) ⊥ . If P s denotes the parallel transport of T p S p along γ, then

P s = T γ(s) S γ(s) .
Proof: Let {e 1 , e 2 , ..., e 2n } be an orthonormal basis of T p D with e 1 , e 2 ∈ T p S p and let E i (s) be the parallel translate of e i , i = 1, ..., 2n, along γ. By Lemma 2.1 for i ≥ 2, γ lies in W t for some t ∈ S. By Lemma (1) of 2.2,

e i ∈ T p W t for 3 ≤ i ≤ 2n. Since W t is totally geodesic, E i (s) is tangent to W t for all s ∈ [0, L] and 3 ≤ i ≤ 2n. Hence E 1 (s), E 2 (s) ∈ (T γ(s) W t ) ⊥ = T γ(s) S γ(s) . . 2.3. Distance between complex submanifolds. Fix z 0 , z 1 ∈ B n-1 . Let p 0 ∈ π -1 (z 0 ), p 1 ∈ π -1 (z 1 ) be points satisfying d K D (p 0 , p 1 ) = d K D (π -1 (z 0 ), π -1 (z 1 )
). Let γ : [0, L] → D be the unit speed geodesic with γ(0) = p 0 , γ(L) = p 1 which realizes the distance between π -1 (z 0 ) and π -1 (z 1 ). Since γ ′ (0) is orthogonal to T p π -1 (z 0 ), Lemma 2.1 implies that γ ′ (0) ∈ T p W t and γ([0, ∞)) ⊂ W t for some t ∈ S.

By considering the normal exponential map to π -1 (z 0 ), we can find a unit normal vector field X to π -1 (z 0 ) in a neighbourhood U (in π -1 (z 0 )) of p 0 such that the holds: for any q ∈ U , the geodesic γ q : [0, L] → D with γ q (0) = q, γ ′ q (0) = X q satisfies γ q (L) ∈ π -1 (z 1 ). Note that X p 0 = γ ′ (0). Let u ∈ T p 0 π -1 (z 0 ) and σ : [-a, a] → π -1 (z 0 ) ⊂ D a curve with σ(0) = p 0 and σ ′ (0) = u. Define a geodesic variation H :

[0, L] × [-a, a] → D of γ by (2.1) H(s, t) = Exp σ(t) (sX σ(t) ).
Let Y (s, t) = ∂H ∂t (s, t) be the variation vector field and, for each t ∈ [-a, a], let γ t be the geodesic given by γ t (s) = H(s, t). Let T (s, t) = γ ′ t (s) = ∂H ∂s (s, t).

Lemma 2.4. For any (s, t)

∈ [0, L] × [-a, a] we have (i) Y (s, t) ∈ T γt(s) S γt(s) . (ii) Y ′ (s, t) := ∇ T Y (s, t) ∈ T γt(s) S γt(s) .
Proof: (i) This follows if we can show that for each s ∈ [0, L] the curve t → H(s, t) lies in a fiber of π. To see this consider the curves, for t 1 , t 2 ∈ (-a, a), s → (π•γ t 0 )(s) and s → (π•γ t 1 )(s). These curves are unit-speed geodesics in B n-1 connecting π(z 0 ) and π(z 1 ) by Lemma 2.1. Since B n-1 has negative curvature, uniqueness of geodesics forces π(H(s, t 0 )) = π(H(s, t 1 )).

(ii) We show this for t = 0 for notational simplicity. Let {e 1 , e 2 } be an orthonormal basis of T p 0 S p 0 . Let E 1 (s), E 2 (s) be the parallel vector fields along γ with E i (0) = e i . We then have, by (i) and Corollary 2.3,

Y (s, 0) = f 1 (s)E 1 (s) + f 2 (s)E 2 (s) for some functions f 1 , f 2 : [0, L] → R. Hence Y ′ (s, 0) = f ′ 1 (s)E 1 (s) + f ′ 2 (s)E 2 (s) ∈ T γ(s) S γ(s) .
We continue to denote p = F (z 0 , w 0 ) in what follows. We recall the notation and constructions above:

• X denotes a local unit normal vector field on S p such that the geodesics starting in the direction X pass through the same fibers subsequently,

• γ : [0, ∞) → D is a geodesic with γ(0) = p, γ ′ (0) = X p ,

• for u ∈ T p S p , H u (s, t) = γ u t (s) denotes a geodesic variation of γ such that (a) ∂Hu ∂s (0, t) = X H(0,t) (b) the variation vector field Y u (s, t) = ∂Hu ∂t (s, t) satisfies Y u (0, 0) = u, • T u (s, t) = ∂Hu ∂s (s, t) denotes the tangent vector (γ u t ) ′ (s) to the variation geodesic γ u t (s). Note that T u (s, 0) = γ ′ (s) for all s and T u (0, t) = X Hu(0,t) for all t. Lemma 2.5. For any u ∈ T p S p and s ∈ [0, ∞), we have

(i) ∇ Yu Y u (s, 0), γ ′ (s) = -1 2 (|Y u | 2 ) ′ (s, 0). (ii) ∇ u X ∈ T p S p .
Proof:

∇ Yu Y u , γ ′ = Y u Y u , γ ′ -Y u , ∇ Yu γ ′ = -Y u , ∇ γ ′ Y u = - 1 2 (|Y u | 2 ) ′
where we have used T u , Y u = 0 and ∇ Yu T u = ∇ Tu Y u . This proves (i). For (ii), [Y u , T u ] = 0 and (ii) of Lemma 2.4 imply that

∇ u X = ∇ γ ′ (0) Y u ∈ T p S p .
By (ii) above, the shape operator L : T p S p → T p S p of S p along the normal vector field X is given by

L(v) = ∇ v X.
As L is a symmetric operator we can find an orthonormal basis {e 1 , e 2 } of T p S p consisting of eigenvectors of L. Since S p is a minimal submanifold (being a complex subvariety), the corresponding eigenvalues are given by α, -α for some α ≥ 0. As before we denote the parallel transports of e 1 , e 2 along γ by E 1 (s), E 2 (s).

Next we observe that if Y 1 (s) := Y e 1 (s, 0), Y 2 (s) := Y e 2 (s, 0) are Jacobi fields constructed as earlier with

Y 1 (0) = e 1 , Y 2 (0) = e 2 then Y ′ 1 (0) = ∇ γ ′ (0) Y 1 = ∇ e 1 X = L(e 1 ) = αe 1 . Similarly Y ′ 2 (0) = -αe 2 . Hence if K 1 : [0, L] → R denotes the function K 1 (s) = R(E 1 (s), γ ′ (s), γ ′ (s), E 1 (s))
and f 1 : [0, L] → is the solution to

y ′′ + K 1 y = 0, y(0) = 1, y ′ (0) = α then Y 1 = f 1 E 1 . Similarly Y 2 = f 2 E 2 where f 2 satisfies y ′′ + K 2 y = 0, y(0) = 1, y ′ (0) = -α with K 2 (s) = R(E 2 (s), γ ′ (s), γ ′ (s), E 2 (s)).
2.4. The case of CH n . In case D is biholomorphic to the unit ball in C n , the Kobayashi metric on D has constant holomorphic sectional curvature -1 and the curvature tensor has the property that R(X, Y )Y, X = -1 4 whenever {X, Y } is an orthonormal pair spanning a totally real 2-plane, i.e., whenever X, Y = X, JY = 0. Hence f 1 and f 2 satisfy y ′′ -y 4 = 0.

It follows that

f 1 (s) = cosh( s 2 ) + 2α sinh( s 2 ), f 2 (s) = cosh( s 2 ) -2α sinh( s 2 )
.

Case 1: α = 1 2 . In this case, (i) of Lemma 2.5 implies that

∇ Y i Y i , γ ′ (s) = - 1 2 (f 2 i ) ′ (
s) < 0 for i = 1, 2 and s large enough. On the other hand, Lemma 2.4 and the fact that S γ(s) is a minimal submanifold implies that 0 =

2 i=1 ∇ E i E i , γ ′ (s) = 2 i=1 f -2 i ∇ Y i Y i , γ ′ (s).
This contradiction completes the proof.

Case 2: α = 1 2 for all p ∈ D, all q ∈ S p and all T 0 ∈ (T q S p ) ⊥ . In this case, one can check that the second fundamental form of S p in every normal direction is parallel. O'Neill's formula [START_REF] O'neill | The fundamental equations of a submersion[END_REF] (Page 465, 2. of Corollary 1) for the curvature of a Riemannian submersion then shows that the sectional curvature of the 2-plane {u, v} is zero where u ∈ T p S p and v ∈ (T p S p ) ⊥ . 2.5. Holomorphic motions and Bers-Griffiths uniformization. The following fundamental theorem allows us to deduce (when n = m + 1) Theorem 1.3 from Theorem 1.2: Theorem 2.6. ( [START_REF] Bers | Uniformization, moduli and Kleinian groups[END_REF], [START_REF] Griffiths | Complex-Analytic Properties of Certain Zariski Open Sets on Algebraic Varieties[END_REF]) Let M and N be compact complex manifolds and φ : M → N a holomorphic submersion. Suppose that dim(M ) = dim(N ) + 1 and the fibers of φ are compact Riemann surfaces of genus ≥ 2. Then the universal cover of M is biholomorphic to the graph of a holomorphic motion over Ñ .

For a detailed account of holomorphic motions and uniformization we refer the reader to [START_REF] Chirka | Holomorphic motions and uniformization of holomorphic families of Riemann surfaces[END_REF].

Proof of Theorem 1.1.

We assume, to get a contradiction, that the graph D of some holomorphic motion is biholomorphic to some bounded strictly pseudoconvex domain Ω. We denote by Φ a biholomorphism from D to Ω.

For every ν ≥ 1, let F ν : B → D be the totally geodesic holomorphic embedding defined by F ν (z) := F (z, 1 -1 ν ). For every ν, the map Φ • F ν is holomorphic from B to Ω. Let z 0 ∈ B. We may assume, taking a subsequence if necessary, that lim ν→∞ z

ν := Φ • F ν (z 0 ) = p ∈ ∂Ω.
According to [8, Theorem 4.1] it holds lim z→p σ Ω (z) = 1, where σ Ω is the squeezing function of Ω (see Definition in [START_REF] Kim | On the uniform squeezing property of bounded convex domains in C n[END_REF]). This means that for every ν ≥ 1 there exists a biholomorphism ϕ ν from Ω to some strongly pseudoconvex domain Ω ν and there exists a sequence (r ν ) ν with lim ν→∞ r ν = 1 such that for every ν ≥ 1:

(3.1) ϕ ν (z ν ) = 0 and B(0, r ν ) ⊂ Ω ν ⊂ B n .
Here B(0, r ν ) denotes the ball in C n centered at the origin with radius r ν .

For every ν ≥ 1, let Σ ν := {F ν (z), z ∈ B} and let Σν 0 := (ϕ ν • Φ)(Σ ν ). Lemma 3.1. For every ν ≥ 1, the set Σν 0 is a totally geodesic complex submanifold of Ω ν . Proof of Lemma 3.1. This follows from Lemma 2.1 and the fact that biholomorphisms are isometries for the Kobayashi metric.

Moreover, we get:

Lemma 3.2. The sequence ( Σν 0 ) ν converges, for the local Hausdorff convergence of sets, to some totally geodesic complex submanifold of B n .

Proof of Lemma 3.2. Let, for every ν ≥ 1, Ψ ν := ϕ ν • Φ • F ν . Then Ψ ν is a holomorphic isometric embedding of (B, d K B ) into (Ω ν , d K Ων ) satisfying Ψ ν (z 0 ) = 0. Since for every ν ≥ 1 we have the inclusion Ω ν ⊂ B n , the sequence (Ψ ν ) ν is normal and extracting a subsequence if necessary, we may assume that (Ψ ν ) ν converges, uniformly on compact subsets of B, to some holomorphic map Ψ ∞ : B → B n satisfying Ψ ∞ (z 0 ) = 0. Finally, let 0 ∈ L ⊂⊂ B n . Since Ψ ν is an isometry for the Kobayashi distances, there exists K ⊂⊂ B such that for every ν ≥ 1 we get: L ∩ Σν 0 ⊂ Ψ ν (K). Now the uniform cnvergence of (Ψ ν ) ν on K implies that the sets Σν 0 converge to Σ∞ 0 for the Hausdorff convergence on L. Let Σ∞ 0 := Ψ ∞ (B) and let z, z ′ ∈ B. There exist q ν , q ′ ν ∈ Σν 0 , converging respectively to Ψ ∞ (z) and Ψ ∞ (z ′ ) and we have by Lemma 3.1:

d K Σ∞ 0 (Ψ ∞ (z), Ψ ∞ (z ′ )) = lim ν→∞ d K Σν 0 (Ψ ν (z), Ψ ν (z ′ )) = lim ν→∞ d K Ων (Ψ ν (z), Ψ ν (z ′ )) = d K B n (Ψ ∞ (z), Ψ ∞ (z ′ )).

In particular, since totally geodesic complex submanifolds of B n , of complex dimension (n-1), are intersections of B n with complex affine subspaces of complex dimension (n -1), we may assume that Σ∞ 0 = B n-1 × {0}. Let ζ ∈ ∆ and let q := (0, ζ) ∈ B n . Then q ∈ Ω ν for sufficiently large ν and there exists (b ν , ζ ν ) ∈ B × ∆ such that q = ϕ ν • Φ • F ζν (b ν ). We set Σν q := ϕ ν • Φ • F ζν (B). We prove, exactly as for Σν 0 , that Σν q is a totally geodesic complex submanifold of Ω ν . Lemma 3.3. The sequence (b ν ) ν is relatively compact in B.

Proof of Lemma 3.3. Since D is complete hyperbolic by assumption, it follows from Lemma 2.1 that B is complete hyperbolic. Assume to get a contradiction that (b ν ) ν is not relatively compact in B. We recall that π : D → B is holomorphic. Hence we get for every sufficiently large ν:

Consequently, extracting a subsequence if necessary, we may assume that:

for every ν we get:

This contradicts Condition (3.2).

It follows now from Lemma 3.3 that we may extract from (b ν ) ν ) a subsequence, still denoted (b ν ) ν , that converges to some point b ∞ ∈ B. Hence, extracting a subsequence if necessary, we may assume that the sequence (

and Σ∞ q is a totally geodesic complex submanifold of B n . We finally prove Proposition 3.4. For every q ∈ B n ∩ ({0 ′ } × ∆) there exists a totally geodesic complex submanifold Σ∞ q of B n passing through q. For every ν, let π ν : D → Σ ν be given by

)) = 0 according to (3.1), we have πν (0) = 0 for every ν. Hence we may extract from (π ν ) ν a subsequence, still denoted (π ν ) ν , that converges to a holomorphic map π∞ :

Moreover we have: Proposition 3.5. For every q ∈ {0} × ∆, the restriction of π∞ to Σ∞ q is a biholomorphism from Σ∞ q to B n-1 × {0}.

Proof of Proposition 3.5. By the very definition of πν , the restriction of πν to Σν q is a biholomorphism from Σν q to B n-1 × {0} for every ν. Moreover, Σν q converges to Σ∞ q for the Hausdorff distance. Finally, we have for every ν ≥ 1:

Since lim ν→∞ b ν = b ∞ ∈ B and since the sequence (ϕ ν •Φ•F ν ) ν converges, uniformly on compact subsets of B to Ψ ∞ (see the proof of Lemma 3.2) we obtain that:

Hence if g ν denotes the inverse of the restriction of πν to Σ∞ q then g ν is defined on B n-1 ×{0} and the sequence (g ν ) ν converges, uniformly on compact subsets of B n-1 × {0}, to some holomorphic map g ∞ :

Finally, let q = q ′ be two points in B n . By construction, for every ν ≥ 1, the intersection between the totally geodesics submanifolds Σν q and Σν q ′ is empty. Since ( Σν q ) ν converges to Σ∞ q and Σν q ′ converges to Σ∞ q ′ , it follows from the positivity of intersection that: Σ∞ q ∩ Σ∞ q ′ = ∅. Now Proposition 3.4 and Proposition 3.5 give a contradiction, according to Theorem 1.2.

We end the note by studying some metric properties of D. We assume that B is complete hyperbolic and that B admits an exhaustion (B k ) k∈N : B k ⊂⊂ B k+1 for every k and B = sup k∈N B k , such that B k is complete (Kobayashi) hyperbolic for every k.

Proposition 3.6. The domain D is complete (Kobayashi) hyperbolic.

Proof of Proposition 3.6. It is proved in [START_REF] Chen | Holomorphic motion and invariant metrics (Analytic Geometry of the Bergman Kernel and Related Topics[END_REF] that for every k, D k := F (B k × ∆) is complete hyperbolic.

Let Z 0 = (z 0 , w 0 ), Z ν = (z ν , w ν ) ∈ B × ∆ be such that lim ν→∞ d K B (z 0 , z ν ) = ∞. Then:

Assume, to get a contradiction, that there exists c > 0 such that d K D (F (Z 0 ), F (Z ν )) ≤ c for every k ∈ N (extracting a subsequence if necessary). There exists k 1 ≥ k 0 such that the set {y ∈ D/ d K D (y, F (Z 0 )) < c + 1} is contained in D k 1 according to (3.3). Moreover, it follows from Lamma 5.1 in [START_REF] Kim | Characterization of the Hilbert ball by its automorphisms[END_REF] that:

This contradicts the fact that D k 1 is complete hyperbolic.


Martin Tanguy

A multi-agent system to assess the vulnerability on the maritime supply chain of energy

Keywords: maritime network, graph theory, agentbased modeling

The maritime supply chain of energy concerns all trips done between ports on the maritime space. The use of this space has increase since 1970. This increase is due to the globalization, a strong increase of the demand of energy and freight, containerization of goods and economies of scale [START_REF] Rodrigue | The geography of transport systems[END_REF]. Due to the development of maritime transport, a territorialization process appears on maritime space [START_REF] Parrain | La haute mer : un espace aux frontières de la recherche géographique[END_REF]. For Di Meo ( 2001) "Territorialize space is for a society to multiply places to install networks in both concrete and symbolic". These locations exist on the maritime space, and can be physical (e.g. weather, reefs) socio-economical (E.g use for navigation, exploitation) or political/legal (e.g. TSS, EEZ). All these locations formalize a spatial heterogeneity and a spatial dynamic, due to the permanent evolution of maritime locations. These maritime territories can be risky for maritime transport, taking into account the environment, human activities or deliberate actions. Risks can product disruptions and affect the capacity of a spatial mediator (maritime space) to link the different parts within this mediator (Gleyze, 2005). The main goal of this research is to propose a modeling approach of the maritime network; we use an agent-based system to simulate vessel trajectories. This approach will be able to measure, by the spatio-temporal features of disruption, the vulnerability of the maritime network and especially the maritime supply chain of energy.

Most of the studies related to the modeling of maritime network use the graph theory. A graph is defined as a set of nodes and links. Links connect nodes each other. [START_REF] Joly | Structuration des lignes maritimes régulières de navires porte-conteneurs[END_REF] was the first to model the maritime network using the graph theory. Mains researches include now works of [START_REF] Veenstra | Analysing container flows in the Caribbean[END_REF], Hu and Zhu (2009), [START_REF] Kaluza | The complex network of global cargo ship movements[END_REF], [START_REF] Zavitsas | The Vulnerability of The Petroleum Supply Chain[END_REF], [START_REF] Ducruet | Network diversity and maritime flows[END_REF] and Xu et al (2014). These works focus on topological properties, clusters identification and maritime network configuration (smallworld property, free scale network). But these works do not consider the features of maritime space; « There are no physical links between airports and between seaports that have the characteristic of a line ». [START_REF] Veenstra | Analysing container flows in the Caribbean[END_REF].

In this paper, we propose in a first step, the formalization of the features of the maritime space and we focus on the relationship between this maritime space and the vulnerability. The spatial features can be divided between possibilities and restrictions. On the one hand, the network is totally connected (each port can be reach by another port) and the connectivity is theoretically infinite (each trip can follow an infinity of paths). On the other hand, the locations within the maritime space formalize barriers. Rodrigue(2013) distinguishes absolute, relative and arbitrary barriers. An absolute barrier prevents every movement and corresponds to sea/land interface. A relative barrier produces a friction to a movement, for example weather, straits or channels. An arbitrary barrier corresponds to mandatory areas (TSS, EEZ) on the maritime space. The barriers can be static (e.g. reefs) or dynamic (e.g. icebergs) and these barriers affect the global accessibility of the maritime space. This accessibility will be used to measure the vulnerability.

Accessibility is the capacity of a location to be reached from another one [START_REF] Rodrigue | The geography of transport systems[END_REF]. Disruptions are brutal changes of the spatial structure that affect the accessibility of this space due to a risk. Vulnerability can be assessed by the measure of disruption on the maritime network. These measures are the values of spatial distance and the spatiotemporal distance. The change of the value of the spatial distance is due to the decrease of accessibility within this space, and can be measured for example by the structural vulnerability (Gleyze,2005). The change of the spatiotemporal value is due to the change of duration to reach the destination port due to the dynamic of maritime space.

Considering previous elements, in a second step, we propose a conceptual model (Fig. 1) for building the maritime network. This network is built by the individual goal of a vessel (a port to reach) and this network is formed by the whole trips done on the maritime space. Indeed, trips are built by a spatial behavior depending of the individual goal of each vessel (supply and demand port locations -economic features) and their given spatial possibilities (accessibility of maritime space -geographic features). These trips formalize the maritime network. This approach allows the study of the relationship between topological reasoning (relationship between ports) and geometric reasoning (relationship with the spatial structure and dynamics).

To measure the vulnerability of a network by accessibility and time, a multi-agent system can be used in a third step. An agent is an entity which has its own goals and capacities in a shared environment [START_REF] Ferber | Les systèmes multi-agents : un aperçu général[END_REF]. In our case, this environment, the maritime space, is a dynamic spatial environment. Langlois(2013) proposes a paradigm Agent/Organization/Behavior which allows to simulate the relationship between social agents (vessels) and spatial agents (maritime space configurations). With simulations, it is possible to measure the spatial behavior of agents according to the dynamic of the spatial structure. These behaviors built trajectories depending of the accessibility on maritime space. This loss of accessibility and time is due to the spatial dynamics and can be compared to the shortest path on a given link of the network. An accessibility rate can be calculated for each maritime path. In case of disruption, this accessibility rate can be used to measure and quantify the disruption and its impact on the maritime network. With this method, an assessment of the maritime network vulnerability can be performed. This agent-based system can be used later for optimization of maritime network to avoid or decrease effects of disruptions. This system could be further used to support fleet management systems.

Figure 1 .

 1 Figure 1. Conceptual model