, Let V be a p-dimensional subset of E and W a closed complement of E. Let h 1 ,. .. , h p?1 be independent vectors of V. Then there exists B > 0 such that: ?y ? W

. Proof, We prove this result for the norm. ?l,p , which is enough by Lemma 2.2.12. Let l 1 ,. .. , l p?1 be linear forms such that l i (h i ) = 1, l i (h j ) = 0 if i = j

A. Avila, J. Bochi, and J. Yoccoz, Uniformly hyperbolic finite-valued SL(2, R)-cocycles, Comment. Math. Helv, vol.85, issue.4, pp.813-884, 2010.

V. Baladi, Decay of correlations, AMS Summer Institute on Smooth ergodic theory and applications, vol.69, pp.297-325, 1999.
URL : https://hal.archives-ouvertes.fr/hal-01162368

V. Baladi, Positive Transfer Operators and Decay of Correlations, Book Advanced Series in Nonlinear Dynamics, vol.16, 2000.

E. Berkson, Some metrics on the subspaces of a Banach space, Pacific J. Math, vol.13, pp.7-22, 1963.

G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc, vol.85, pp.219-227, 1957.

J. Bochi and N. Gourmelon, Some characterizations of domination, Mathematische Zeitschrift, vol.263, issue.1, pp.221-231, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00489039

J. Bochi and I. D. Gourmelon, Continuity properties of the lower spectral radius, Proc. London Math. Soc, vol.110, pp.477-509, 2015.

A. , A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst. A, vol.36, issue.5, pp.2377-2403, 2016.

A. Blumenthal and I. Morris, Characterization of dominated splittings for operator cocycles acting on Banach spaces

J. Bochi and I. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc, vol.110, issue.3, pp.477-509, 2015.

N. Bourbaki, Éléments de mathématique, Algèbre. Chapitre, vol.1, issue.3, 2007.

N. Bourbaki, Éléments de mathématique. Variétés différentielles et analytiques, 2007.

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2010.

J. D. Biggins and A. R. Sani, Convergence results on multitype, multivariate branching random walks, Adv. in Appl. Probab, vol.37, issue.3, pp.681-705, 2005.

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math, vol.161, issue.2, pp.1423-1485

L. Carleson and T. W. Gamelin, Universitext: Tracts in Math, Complex Dynamics, 1993.

L. Dubois, Projective metrics and contraction principles for complex cones, J. Lond. Math. Soc, vol.79, pp.719-737, 2009.

F. Froyland, C. Gonzalez-tokman, and A. Quas, Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles, Comm. Pure Appl. Math, vol.68, pp.2052-2081, 2015.

F. Froyland, C. Gonzalez-tokman, and A. Quas, Hilbert Space Lyapunov Exponent stability, Trans. Amer. Math. Soc

G. Froyland, S. Lloyd, and A. Quas, A semi-invertible oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst, p.33, 2013.

G. Frobenius, Über Matrizen aus nicht negativen Elementen, S.-B. Preuss. Akad. Wiss, pp.456-477, 1912.

A. Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Annales de l'institut Fourier, vol.4, pp.73-112, 1952.

C. Gonzalez-tokman and A. Quas, A semi-invertible operator oseledets theorem, Ergodic Theory Dynam. Systems, vol.34, issue.04, pp.1230-1272, 2014.

C. Gonzalez-tokman and A. Quas, A concise proof of the Multiplicative Ergodic Theorem on Banach Spaces, J. Mod. Dyn, vol.9, pp.237-255, 2015.

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math, vol.164, pp.279-315, 2006.

J. Harris, Algebraic Geometry: A First Course, 1992.

T. Kato, Perturbation Theory for Linear Operators, 1995.

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, 1995.

B. Lemmens and R. Nussbaum, Birkhoff 's version of Hilbert's metric and its applications in analysis, Handbook of Hilbert Geometry, IRMA Lectures in Mathematics and Theoretical Physics, issue.22, p.10, 2014.

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics, pp.522-577, 1983.

I. D. Morris, A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory, Adv. Math, vol.225, pp.3425-3445, 2010.

P. Ney and E. Numellin, Markov additive processes, I and II, Ann. Probab, vol.15, pp.561-592, 1987.

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc, vol.19, pp.179-210, 1968.

O. Perron, Zur Theorie der Matrices, Math. Ann, vol.64, pp.248-263, 1907.

E. R. Pujals and M. Sambarino, On the dynamics of dominated splitting, Ann. of Math, vol.169, issue.2, pp.675-739, 2009.

M. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math, vol.32, issue.4, pp.356-362, 1979.

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. Math, vol.32, pp.68-80, 1979.

D. Ruelle, Characteristic Exponents and Invariant Manifolds in Hilbert Space, Ann. of Math, vol.115, issue.2, pp.243-290, 1982.

H. H. Rugh, Coupled maps and analytic function spaces, Ann. Sci. École Norm. Sup, vol.35, pp.489-535, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00096228

H. H. Rugh, Cones and gauges in complex spaces: Spectral gaps and complex PerronFrobenius theory, Ann. of Math, vol.171, pp.1707-1752, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00157201

P. Thieullen, Entropie, Fibrés dynamiques asymptotiquement compacts exposants de lyapounov, vol.4, pp.49-97, 1987.

P. Walters, An Introduction to Ergodic Theory, 1982.