Random Products of Standard Maps
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Communications in
Mathematical Physics
Random Products of Standard Maps Pablo D. Carrasco ICEx-UFMG, Avda. Presidente Antônio Carlos 6627, Belo Horizonte, MG 31270-901, Brazil. E-mail: [email protected] Received: 11 August 2018 / Accepted: 31 March 2020 Published online: 29 May 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics is given by (a continuum of parameters in) the Froeschlé family. These types of coupled systems appear as some induced maps in models for the study of Arnold diffusion. Consequently, we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics. The methods are also suitable for studying cocycles over shift spaces, and do not demand any low dimensionality condition on the fiber. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Products of conservative systems with some hyperbolicity . . . . . . . 2.2 Partially hyperbolic skew products . . . . . . . . . . . . . . . . . . . 2.3 Coupled families . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Existence of physical measures and surface maps . . . . . . . . . . . 3.2 Examples: Random products of standard maps . . . . . . . . . . . . . 3.2.1 The Standard Map. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Higher dimensional examples: random products of (coupled) standard maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cocycles over the shift . . . . . . . . . . . . . . . . . . . . . . . . . Admissible Curves and Adapted Fields . . . . . . . . . . . . . . . . . . . 4.1 Partial Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Curves tangent to E u . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3 Adapted fields . . . . . . . . Positivity of the Center Exponents 5.1 Study of the integral . . . . . 5.2 Good and bad vector fields .
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1. Introduction Let f be a C 2 diffeomorphism of a compact Riemannian manifold M preserving a smooth probability measure μ. A central tool to detect chaotic behavior in the system is by means of its Lyapunov exponents. Definition 1.1. For p ∈ M, v ∈ T p M\{0}, the Lyapunov exponent of v is log d p f n (v) . χ (
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