Abstract
We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.
Highlights
A quadratic polynomial Qc(x) = c − x2(1 < c ≤ 2) induces a unimodal map on the interval [c − c2, c]
When considered as a holomorphic map defined on the Riemann Sphere, the complex dynamics generated by Qc is exhaustedly studied
Since the base map h is uniformly expanding, we have proved that F has two positive Lyapunov exponents
Summary
In Viana [11], Misiurewicz-Thurston quadratic polynomials were used to construct nonuniformly expanding maps in dimension greater than one The main effort is to control recurrence of typical orbits to the critical line y = 0, see Proposition 4.3 This proposition, together with the “Building Expansion” Lemmas in [11], which are summarized in Lemma 2.7, implies that F is nonuniformly expanding in the sense of [3], and the results proved therein complete the proof of our Main Theorem. Lemma 2.6, serving as an intermediate step, is proved by combining subhyperbolicity of Qa with the super attracting behavior of Qa near infinity It remains an interesting open problem whether the statement of our Main Theorem holds for (a, b) chosen from a positive subset of [1, 2] × [1, 2)
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