# 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*:𝕋^{2}→𝕋^{2} 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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