J. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Science, vol.98, issue.2, pp.163-197, 1992.
DOI : 10.1016/0304-3975(92)90001-V

J. Allouche and J. Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, Sequences and their applications, pp.1-16, 1998.
DOI : 10.1007/978-1-4471-0551-0_1

J. Allouche and J. Shallit, Automatic sequences, 2003.
DOI : 10.1017/cbo9780511546563

R. Béjian and H. Faure, Discrépance de la suite de van der Corput, C. R. Acad. Sci. Paris Sér. A-B, vol.285, issue.5, pp.313-316, 1977.

R. Béjian and H. Faure, Discrépance de la suite de Van der Corput, Séminaire Delange-Pisot-Poitou Théorie des nombres, p.14, 1977.

J. Berstel and C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science, vol.12, 1988.
DOI : 10.1007/978-3-642-73235-5
URL : https://hal.archives-ouvertes.fr/hal-00619791

P. Billingsley, Probability and measure Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, 1995.

D. Vincent, R. Fabian, and . Wirth, Special issue on the joint spectral radius: Theory, methods and applications, Linear Algebra and its Applications, vol.428, issue.10, pp.2259-2404, 2008.

D. Vincent, Y. Blondel, J. Nesterov, and . Theys, On the accuracy of the ellipsoid norm approximation of the joint spectral radius, Linear Algebra Appl, vol.394, pp.91-107, 2005.

D. W. Boyd, J. Cook, and P. Morton, On sequences of ±1's defined by binary patterns, Dissertationes Math. (Rozprawy Mat, vol.283, p.64, 1989.

J. Brillhart and L. Carlitz, Note on the Shapiro polynomials, Proceedings of the American Mathematical Society, vol.25, issue.1, pp.114-118, 1970.
DOI : 10.1090/S0002-9939-1970-0260955-6

J. Brillhart, P. Erd?, and P. Morton, On sums of Rudin-Shapiro coefficients. II, Pacific Journal of Mathematics, vol.107, issue.1, pp.39-69, 1983.
DOI : 10.2140/pjm.1983.107.39

V. Canterini and A. Siegel, Automate des pr??fixes-suffixes associ?? ?? une substitution primitive, Journal de Th??orie des Nombres de Bordeaux, vol.13, issue.2, pp.353-369, 2001.
DOI : 10.5802/jtnb.327

J. Cassaigne, Counting overlap-free binary words, STACS 93, pp.216-225, 1993.
DOI : 10.1007/3-540-56503-5_24
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.8340

F. George, B. Clements, and . Lindström, A sequence of (±1)?determinants with large values, Proc. Amer, pp.548-550, 1965.

J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae, vol.32, issue.2, pp.107-115, 1983.
DOI : 10.1007/BF01393827

I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.61, 1992.

I. Daubechies and J. C. Lagarias, Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions, SIAM Journal on Mathematical Analysis, vol.22, issue.5, pp.1388-1410, 1991.
DOI : 10.1137/0522089

I. Daubechies and J. C. Lagarias, Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals, SIAM Journal on Mathematical Analysis, vol.23, issue.4, pp.1031-1079, 1992.
DOI : 10.1137/0523059

H. Delange, Sur la fonction sommatoire de la fonction " somme des chiffres, Enseignement Math, vol.21, issue.21, pp.31-47, 1975.

B. Thomas, D. P. Dinsenbacher, and . Hardin, Multivariate nonhomogeneous refinement equations, J. Fourier Anal. Appl, vol.5, issue.6, pp.589-597, 1999.

M. Drmota and J. Gajdosik, The distribution of the sum-of-digits function, Journal de Th??orie des Nombres de Bordeaux, vol.10, issue.1, pp.17-32, 1998.
DOI : 10.5802/jtnb.216

P. Dumas, Algebraic aspects of B-regular series, Automata, Languages and Programming Proceedings of the 20th International Colloquium ICALP 93, pp.457-468, 1993.
DOI : 10.1007/3-540-56939-1_94
URL : https://hal.archives-ouvertes.fr/inria-00074743

P. Dumas, H. Lipmaa, and J. Wallén, Asymptotic behaviour of a noncommutative rational series with a nonnegative linear representation, Discrete Mathematics & Theoretical Computer Science, vol.9, issue.1, pp.247-274478, 1836.
URL : https://hal.archives-ouvertes.fr/hal-00966501

J. Dumont, Summation Formulae for Substitutions on a Finite Alphabet, Number theory and physics, pp.185-194, 1989.
DOI : 10.1007/978-3-642-75405-0_19

J. Dumont, N. Sidorov, and A. Thomas, Number of representations related to a linear recurrent basis, Acta Arith, vol.88, issue.4, pp.371-396, 1999.

J. Dumont and A. Thomas, Systemes de numeration et fonctions fractales relatifs aux substitutions, Theoretical Computer Science, vol.65, issue.2, pp.153-169, 1989.
DOI : 10.1016/0304-3975(89)90041-8

N. Dyn and D. Levin, Subdivision schemes in geometric modelling, Acta Numer, vol.11, pp.73-144, 2002.

P. Flajolet and M. Golin, Mellin transforms and asymptotics, Acta Informatica, vol.32, issue.4, pp.673-696, 1994.
DOI : 10.1007/BF01177551
URL : https://hal.archives-ouvertes.fr/inria-00074307

P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, Mellin transforms and asymptotics: digital sums, Theoretical Computer Science, vol.123, issue.2, pp.291-314, 1994.
DOI : 10.1016/0304-3975(92)00065-Y
URL : https://hal.archives-ouvertes.fr/inria-00075064

P. Flajolet and L. Ramshaw, A Note on Gray Code and Odd-Even Merge, SIAM Journal on Computing, vol.9, issue.1, pp.142-158, 1980.
DOI : 10.1137/0209014

P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics, 2002.

M. Goze and Y. Khakimdjanov, Nilpotent and solvable lie algebras, pp.615-663, 2000.
DOI : 10.1016/S1570-7954(00)80040-6

C. Heil and D. Colella, Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl, vol.2, issue.4, pp.363-377, 1996.

H. Lipmaa, J. Wallén, and P. Dumas, On the Additive Differential Probability of Exclusive-Or, Fast Software Encryption, pp.317-331, 2004.
DOI : 10.1007/978-3-540-25937-4_20

M. and D. Mcilroy, The Number of 1???s in Binary Integers: Bounds and Extremal Properties, SIAM Journal on Computing, vol.3, issue.4, pp.255-261, 1974.
DOI : 10.1137/0203020

F. Morain and J. Olivos, Speeding up the computations on an elliptic curve using addition-subtraction chains, RAIRO - Theoretical Informatics and Applications, vol.24, issue.6, pp.531-543, 1990.
DOI : 10.1051/ita/1990240605311
URL : https://hal.archives-ouvertes.fr/inria-00075576

B. Mossé, Puissance de mots et reconnaissabilit?? des points fixes d'une substitution, Theoretical Computer Science, vol.99, issue.2, pp.327-334, 1992.
DOI : 10.1016/0304-3975(92)90357-L

B. Mossé, Reconnaissabilit?? des substitutions et complexit?? des suites automatiques, Bulletin de la Société mathématique de France, vol.124, issue.2, pp.329-346, 1996.
DOI : 10.24033/bsmf.2283

D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer, pp.719-721, 1969.
DOI : 10.1090/S0002-9939-1969-0244149-8

H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.63, 1992.
DOI : 10.1137/1.9781611970081

H. Andrew, P. Osbaldestin, and . Shiu, A correlated digital sum problem associated with sums of three squares, Bull. London Math. Soc, vol.21, issue.4, pp.369-374, 1989.

O. Rioul, Simple Regularity Criteria for Subdivision Schemes, SIAM Journal on Mathematical Analysis, vol.23, issue.6, pp.1544-1576, 1992.
DOI : 10.1137/0523086

G. Rota and G. Strang, A note on the joint spectral radius, Proc. Ser. A 63 = Indag, pp.379-381, 1960.

W. Rudin, Some theorems on Fourier coefficients, Proc. Amer, pp.855-859, 1959.
DOI : 10.1090/S0002-9939-1959-0116184-5

J. Sakarovitch, Elements of Automata Theory, 2005.
DOI : 10.1017/CBO9781139195218

S. Harold and . Shapiro, Extremal problems for polynomials and power series. Master's thesis, Massachusets Institute of Technology, 1951.

K. B. Stolarsky, Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity, SIAM Journal on Applied Mathematics, vol.32, issue.4, pp.717-730, 1977.
DOI : 10.1137/0132060

G. Strang and D. Zhou, Inhomogeneous refinement equations, The Journal of Fourier Analysis and Applications, vol.44, issue.6, pp.733-747, 1998.
DOI : 10.1007/BF02479677

J. Kenneth, E. M. Supowit, and . Reingold, Divide and conquer heuristics for minimum weighted Euclidean matching, SIAM J. Comput, vol.12, issue.1, pp.118-143, 1983.

J. R. Trollope, An Explicit Expression for Binary Digital Sums, Mathematics Magazine, vol.41, issue.1, pp.21-25, 1968.
DOI : 10.2307/2687954

N. John, V. D. Tsitsiklis, and . Blondel, The Lyapunov exponent and joint spectral radius of pairs of matrices are hard?when not impossible?to compute and to approximate, Math. Control Signals Systems, vol.10, issue.1, pp.31-40, 1997.

A. Project, 78153 Le Chesnay Cedex, France E-mail address