Moscow Mathematical Journal

Volume 19, Issue 2, April–June 2019 pp. 189–216.

Quasi-Periodic Kicking of Circle Diffeomorphisms Having Unique Fixed Points

Authors: Kristian Bjerklöv (1)
Author institution:(1) Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden

Summary:

We investigate the dynamics of certain homeomorphisms F:𝕋²→𝕋² of the form F(x,y)=(x+ω,h(x)+f(y)), where ω∈ℝ/ℚ, f: 𝕋→𝕋 is a circle diffeomorphism with a unique (and thus neutral) fixed point and h: 𝕋→𝕋 is a function which is zero outside a small interval. We show that such a map can display a non-uniformly hyperbolic behavior: (small) negative fibred Lyapunov exponents for a.e. (x,y) and an attracting non-continuous invariant graph. We apply this result to (projective) SL(2,ℝ)-cocycles G: (x,u)↦(x+ω,A(x)u) with A(x)=R_φ(x)B, where R_θ is a rotation matrix and B is a parabolic matrix, to get examples of non-uniformly hyperbolic cocycles (homotopic to the identity) with perturbatively small Lyapunov exponents.

2010 Math. Subj. Class. 37C60, 37C70, 37D25, 37E30.

Keywords: Lyapunov exponents, quasi-periodic forcing, nonuniform hyperbolicity, cocycles.

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