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A concise proof of the multiplicative ergodic theorem on Banach spaces

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  • We give a new proof of a multiplicative ergodic theorem for quasi-compact operators on Banach spaces with a separable dual. Our proof works by constructing the finite-codimensional `slow' subspaces (those where the growth rate is dominated by some $\lambda_i$), in contrast with earlier infinite-dimensional multiplicative ergodic theorems which work by constructing the finite-dimensional fast subspaces. As an important consequence for applications, we are able to get rid of the injectivity requirements that appear in earlier works.
    Mathematics Subject Classification: Primary: 37H15; Secondary: 37L55.

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  • [1]

    A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, arXiv:1502.06554.

    [2]

    T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, PhD thesis, Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2009.

    [3]

    G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.doi: 10.1017/S0143385709000339.

    [4]

    C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.doi: 10.1017/etds.2012.189.

    [5]

    T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

    [6]

    Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp.doi: 10.1090/S0065-9266-10-00574-0.

    [7]

    R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, 1983, 522-577.doi: 10.1007/BFb0061433.

    [8]

    V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

    [9]

    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, No. 94, Cambridge University Press, Cambridge, 1989.doi: 10.1017/CBO9780511662454.

    [10]

    M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.doi: 10.1007/BF02760464.

    [11]

    D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.doi: 10.2307/1971392.

    [12]

    P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.

    [13]

    P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, 25, Cambridge University Press, Cambridge, 1991.doi: 10.1017/CBO9780511608735.

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