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Simple Lyapunov spectrum for certain linear cocycles over partially hyperbolic maps. (English) Zbl 1500.37027

Summary: Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by H. Furstenberg [Trans. Am. Math. Soc. 108, 377–428 (1963; Zbl 0203.19102)], Y. Guivarc’h and A. Raugi [Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 31–54 (1986; Zbl 0592.60015)], I. Ya. Gol’dshejd and G. A. Margulis [Russ. Math. Surv. 44, No. 5, 11–71 (1989; Zbl 0705.60012); translation from Usp. Mat. Nauk 44, No. 5(269), 13–60 (1989)], and, more recently, C. Bonatti and M. Viana [Ergodic Theory Dyn. Syst. 24, No. 5, 1295–1330 (2004; Zbl 1087.37017)] and A. Avila and M. Viana [Invent. Math. 181, No. 1, 115–178 (2010; Zbl 1196.37054)]. In all the cases, the authors consider cocycles over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms.
In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of \(u\)-states.
Our main results are stated for certain partially hyperbolic skew-products whose iterates have bounded derivatives along center leaves. They allow us, in particular, to exhibit non-trivial examples of stable simplicity in the partially hyperbolic setting.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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References:

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