Robust Non-Hyperbolic Fibred Quadratic Pplynomial Dynamics

Abstract

We consider the dynamics of fibred quadratic polynomials over an irrational rotation of the circle. We construct a simple mechanism, so-called critical connection, that implies non-hyperbolicity. To exhibit that critical connection is a robust property in the parameter space of fibred quadratic polynomials over a fixed irrational rotation of the circle. Since there exist fibred quadratic polynomials that have such phenomena, we conclude that hyperbolicity is non-dense in this sense. In order to obtain our main results, we prove that, in the hyperbolic case, the filled-in Julia set varies continuously with the fibres. Even though the existence of robust mechanisms that give rise to non-hyperbolicity is a technique that has become standard, the novelty of this work has to do with the fact that we were able to establish this mechanism in a polynomial family of degree 2 of the complex plane, fibred over a nonchaotic map.