In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet–Eckmann, second Collet–Eckmann, and topological Collet–Eckmann conditions are equivalent and also invariant under topological conjugacy.

We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate $(\delta _n)$ of the critical orbit such that, for almost every non-regular parameter a, the set of n such that $\vert F^n(0;a) \vert < \delta _n$ is infinite. In particular, when $\delta _n = n^{-1}$ , this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2)161(2) (2005), 831–881].

In this paper we study rational Collet-Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, they are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree d ≥ 2. in a strong sense; they are Lebesgue density points of hyperbolic maps. We discuss the (rather weak) condition on slow recurrence more below.The Collet-Eckmann condition stems from the pioneering works by P. Collet and J.-P. Eckmann in the 1980s [7]. In the real setting, there are many works on the perturbation of such maps, see e.g. the seminal papers [4,5] by M. Benedicks and L. Carleson. M. Tsujii generalised these results for real maps in [23], see also the more recent work of B. Gao and W. Shen [9]. We are going to study perturbations of such maps in the complex setting. For the quadratic family and other unicritical families, J. Graczyk and G. Światek recently made an extensive study of perturbations of typical Collet-Eckmann maps with respect to harmonic measure, in a series of papers [15,11,13,14]. M. Benedicks and J. Graczyk also have an unpublished work on perturbations on such (quadratic or, more generally, unicritical) maps. The maps there and in the recent papers [15,11] are also slowly recurrent, and hence the results in this paper is partially a generalisation of some of those results. We will not use harmonic measure, but develop the classical Benedicks-Carleson parameter exclusion techniques and combining it with strong results on transversality, by G. Levin [18]. Technically, this paper is closely related to [1].Let f be a rational map. As usual let J (f ) and F (f ) denote the Julia and Fatou set of f respectively. Let Crit(f ) be the set of critical points of f , i.e. the set of points with vanishing spherical derivative. With Jrit(f ) we mean the the set of critical points of f contained in the Julia set, i.e. Jrit(f ) = Crit(f )∩J (f ). As is standard, we let f n denote the n-th iterate of f .In this paper we will consider perturbations of rational maps satisfying the following two properties. Recall that a rational map is called hyperbolic if it is expanding on the Julia set or, equivalently, if every critical point belongs to the Fatou set and is attracted to an attracting cycle. If a rational map is not hyperbolic, it is called non-hyperbolic. Derivatives are always with respect to the spherical metric on the Riemann sphere.Definition 1.1 (Collet-Eckmann condition). A non-hyperbolic rational map f without parabolic periodic points satisfies the Collet-Eckmann condition, if there exist C > 0 and γ > 0 such that for each critical point c in the Julia set of f , one has |Df n (f (c))| ≥ Ce γn for all n ≥ 0.We will often refer to the constant γ appearing in the above definition as the Lyapunov exponent or simply the exponent. Notice that the Co...

We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate (δn) of the critical orbit such that, for almost every nonregular parameter a, the set of n such that |F n (0; a)| < δn is infinite. In particular, when δn = n −1 , this extends an earlier result of Avila and Moreira.