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Weakly expanding skew-products of quadratic maps. (English) Zbl 1037.37014
The authors study ergodic properties of quadratic skew-products $$F:S^1\times\mathbb R\to S^1\times\mathbb R$$ over angle-doubling of the circle $$S^1$$: $F(\theta ,x)=(d\theta \text{ mod }1,a-x^2+\alpha\sin(2\pi\theta )),$ where $$d\geq 2$$ is an integer, $$\alpha$$ is a small real number and the parameter $$a\in (1,2)$$ is such that the map $$f_a(x)=a-x^2$$ has a pre-periodic (but not periodic) critical point. M. Viana [Publ. Math., Inst. Hautes Étud. Sci. 85, 63–96 (1997; Zbl 1037.37016)] proved that, under the assumption $$d\geq 16$$, these maps admit two positive Lyapunov exponents at Lebesgue almost every point, provided $$\alpha$$ is sufficiently small. J. F. Alvès [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, 1–32 (2000; Zbl 0955.37012)] showed that these maps admit an absolutely continuous invariant measure (a.c.i.m.). In the paper, the results of M. Viana and J. F. Alvès mentioned above are extended to weakly expanding skew-products by replacing the assumption $$d\geq 16$$ by the more natural one $$d\geq 2$$. The authors obtain the following:
Theorem: For any integer $$d\geq 2$$ and for ever sufficiently small $$\alpha >0$$, $$F$$ has two positive Lyapunov exponents at Lebesgue almost every point and admits a.c.i.m. The same is true for any (not necessary a skew-product) sufficiently small $$C^{\infty}$$ perturbation of $$F$$.

##### MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37E10 Dynamical systems involving maps of the circle 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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