Extremal exponents of random products of conservative diffeomorphisms
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Mathematische Zeitschrift
Extremal exponents of random products of conservative diffeomorphisms Pablo G. Barrientos1 · Dominique Malicet2 Received: 2 March 2019 / Accepted: 20 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that for a C 1 -open and C r -dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension d ≥ 2, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a C 1 -open and C r -dense subset of ergodic random products of independent conservative surface diffeomorphisms.
1 Introduction The notion of uniform hyperbolicity introduced by Smale in [28] was early shown to be less generic than initially thought [3,23]. In order to describe a large set of dynamical systems, Pesin theory [26] provides a weaker notion called non-uniform hyperbolicity. These systems are described in terms of non-zero Lyapunov exponents of the linear cocycle defined by the derivative transformation, the so-called differential cocycle. In contrast with the non-density of hyperbolicity, we had to wait some decades to construct the first examples of systems with robustly zero Lyapunov exponents [7,17]. Even in the conservative setting there are open sets of smooth diffeomorphisms with invariant sets of positive measure where all the Lyapunov exponents vanish identically (see [10,15,31]). However, recently in [21] it was showed that conservative diffeomorphisms without zero exponent in a set of positive volume are C 1 -dense. On the other hand, abundance of non-uniform hyperbolicity has been obtained in the general framework of linear cocycles when the base driving dynamics is fixed and the matrix group is perturbed in many different contexts [1,5,12,18,29]. However, nothing is known for random product of independent non-linear dynamics. That is, for cocycles driving by a
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Pablo G. Barrientos [email protected] Dominique Malicet [email protected]
1
Instituto de Matemática e Estatística, UFF, Rua Mário Santos Braga s/n-Campus Valonguinhos, Niterói, Brazil
2
LAMA, Université Paris-Est Marne-la-Vallée, Batiment Coppernic 5 boulevard Descartes, Champs-sur-Marne, France
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P. G. Barrientos, D. Malicet
shift map endowed with a Bernoulli probability to value in the group of diffeomorphisms of a compact manifold. To perturbe the Lyapunov exponents of these cocycles one must change the non-linear dynamics similar to the case of differential cocycles. Examples of iterated function systems (IFSs) of diffeomorphisms with robust zero extremal Lyapunov exponents with respect to some ergodic measure that not project on a Bernoulli measure were provided in [6,13]. The authors in [6] question if such examples of IFS with robust zero extremal Lyapunov exponents could be constructed by taking the dynamics conservative and the ergodic measure defined as the product measure of a shift invariant measure on the base and the volume measure on the fiber. We give
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