Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-13T23:32:54.090Z Has data issue: false hasContentIssue false
  • English
  • Français

A semi-invertible operator Oseledets theorem

Published online by Cambridge University Press:  11 March 2013

CECILIA GONZÁLEZ-TOKMAN  and
ANTHONY QUAS
CECILIA GONZÁLEZ-TOKMAN
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada, V8W 3R4 email ceciliag@uvic.caaquas@uvic.ca
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada, V8W 3R4 email ceciliag@uvic.caaquas@uvic.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1230 - 1272
Copyright
Copyright ©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baladi, V.. Anisotropic Sobolev spaces and dynamical transfer operators: ${C}^{\infty } $ foliations. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005, pp. 123–135.CrossRefGoogle Scholar
Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (4) (2009), 1453–1481.CrossRefGoogle Scholar
Baladi, V., Isola, S. and Schmitt, B.. Transfer operator for piecewise affine approximations of interval maps. Ann. Inst. H. Poincaré Phys. Théor. 62 (3) (1995), 251–265.Google Scholar
Baladi, V., Kondah, A. and Schmitt, B.. Random correlations for small perturbations of expanding maps. Random Comput. Dynam. 4 (2–3) (1996), 179–204.Google Scholar
Barreira, L. and Silva, C.. Lyapunov exponents for continuous transformations and dimension theory. Discrete Contin. Dyn. Syst. 13 (2) (2005), 469–490.CrossRefGoogle Scholar
Buzzi, J.. Exponential decay of correlations for random Lasota–Yorke maps. Comm. Math. Phys. 208 (1) (1999), 25–54.CrossRefGoogle Scholar
Buzzi, J.. Absolutely continuous S.R.B. measures for random Lasota–Yorke maps. Trans. Amer. Math. Soc. 352 (7) (2000), 3289–3303.Google Scholar
Buzzi, J.. No or infinitely many a.c.i.p. for piecewise expanding ${C}^{r} $ maps in higher dimensions. Commun. Math. Phys. 222 (2001), 495–501.CrossRefGoogle Scholar
Cowieson, W.. Absolutely continuous invariant measures for most piecewise smooth expanding maps. Ergod. Th. & Dynam. Sys. 22 (2002), 1061–1078.Google Scholar
Dellnitz, M., Froyland, G., Horenkamp, C., Padberg-Gehle, K. and Sen Gupta, A.. Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators. Nonlinear Process. Geophys. 16 (6) (2009), 655–663.CrossRefGoogle Scholar
Dellnitz, M., Froyland, G. and Sertl, S.. On the isolated spectrum of the Perron–Frobenius operator. Nonlinearity 13 (4) (2000), 1171–1188.Google Scholar
Dellnitz, M. and Junge, O.. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (2) (1999), 491–515.Google Scholar
Doan, T. S.. Lyapunov exponents for random dynamical systems. PhD Thesis, Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2009.Google Scholar
Froyland, G.. Approximating physical invariant measures of mixing dynamical systems in higher dimensions. Nonlinear Anal. 32 (7) (1998), 831–860.Google Scholar
Froyland, G.. Ulam’s method for random interval maps. Nonlinearity 12 (4) (1999), 1029–1052.Google Scholar
Froyland, G.. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete Contin. Dyn. Syst. 17 (3) (2007), 671–689 (electronic).Google Scholar
Froyland, G.. Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps. Physica D 237 (6) (2008), 840–853.Google Scholar
Froyland, G., Lloyd, S. and Quas, A.. Coherent structures and isolated spectrum for Perron-Frobenius cocycles. Ergod. Th. & Dynam. Sys. 30 (3) (2010), 729–756.Google Scholar
Froyland, G., Lloyd, S. and Quas, A.. A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., to appear.Google Scholar
Froyland, G., Lloyd, S. and Santitissadeekorn, N.. Coherent sets for nonautonomous dynamical systems. Physica D: Nonlinear Phenomena 239 (16) (2010), 1527–1541.Google Scholar
Froyland, G., Padberg, K., England, M. and Treguier, A.-M.. Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98 (22) (2007), 224503.Google Scholar
Giusti, E.. Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel, 1984.Google Scholar
González-Tokman, C., Hunt, B. and Wright, P.. Approximating invariant densities of metastable systems. Ergod. Th. & Dynam. Sys. 31 (2011), 1345–1361.Google Scholar
Hennion, H.. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118 (2) (1993), 627–634.Google Scholar
Kaĭmanovich, V.. Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), 29–46, 196–197.Google Scholar
Karlsson, A. and Ledrappier, F.. On laws of large numbers for random walks. Ann. Probab. 34 (5) (2006), 1693–1706.Google Scholar
Kato, T.. Perturbation Theory for Linear Operators (Classics in Mathematics). Springer, Berlin, 1995 (reprint of the 1980 edition).Google Scholar
Keane, M., Murray, R. and Young, L.-S.. Computing invariant measures for expanding circle maps. Nonlinearity 11 (1) (1998), 27–46.Google Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super Pisa Cl. Sci.(4) 28 (1) (1999), 141–152.Google Scholar
Keller, G. and Rugh, H.-H.. Eigenfunctions for smooth expanding circle maps. Nonlinearity 17 (5) (2004), 1723–1730.CrossRefGoogle Scholar
Kifer, Y.. Equilibrium states for random expanding transformations. Random Comput. Dynam. 1 (1) (1992/93), 1–31.Google Scholar
Kifer, Y.. Thermodynamic formalism for random transformations revisited. Stoch. Dyn. 8 (1) (2008), 77–102.CrossRefGoogle Scholar
Kingman, J.. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 (1973), 499–510.Google Scholar
Lian, Z. and Lu, K.. Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space. Mem. Amer. Math. Soc. 206 (967) (2010).Google Scholar
Lyons, R., Pemantle, R. and Peres, Y.. Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (3) (1995), 1125–1138.Google Scholar
Mañé, R.. Lyapounov exponents and stable manifolds for compact transformations. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 522–577.Google Scholar
Oseledec, V. I.. A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems. Tr. Mosk. Mat. Obš. 19 (1968), 179–210.Google Scholar
Raghunathan, M.. A proof of Oseledec’s multiplicative ergodic theorem. Israel J. Math. 32 (4) (1979), 356–362.Google Scholar
Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27–58.Google Scholar
Sickel, W.. Pointwise multipliers of Lizorkin–Triebel spaces. The Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998) (Operator Theory Advances and Applications, 110). Birkhäuser, Basel, 1999, pp. 295–321.CrossRefGoogle Scholar
Strichartz, R.. Multipliers on fractional Sobolev spaces. J. Math. Mech. 16 (1967), 1031–1060.Google Scholar
Thieullen, P.. Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987), 49–97.Google Scholar
Thomine, D.. A spectral gap for transfer operators of piecewise expanding maps. Discrete Contin. Dyn. Sys. 30 (3) (2011), 917–944.Google Scholar
Triebel, H.. Theory of Function Spaces. II (Monographs in Mathematics, 84). Birkhäuser, Basel, 1992.Google Scholar
Tsujii, M.. Piecewise expanding maps on the plane with singular ergodic properties. Ergod. Th. & Dynam. Sys. 20 (2000), 1851–1857.Google Scholar