Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log ∣f′∣, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

Ergodic properties of semi-hyperbolic functions with polynomial Schwarzian derivative

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.

Jarník and Julia; a Diophantine analysis for parabolic rational maps for Geometrically Finite Kleinian Groups with Parabolic Elements

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarník in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a "weak multifractal analysis" of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps.

The geometry of conformal measures for parabolic rational maps

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

Subshifts on an infinite alphabet

We study transfer operators over general subshifts of sequences of an infinite alphabet. We introduce a family of Banach spaces of functions satisfying a regularity condition and a decreasing condition. Under some assumptions on the transfer operator, we prove its continuity and quasi-compactness on these spaces. Under additional assumptions—existence of a conformal measure and topological mixing—we prove that its peripheral spectrum is reduced to one and that this eigenvalue is simple. We describe the consequences of these results in terms of existence and properties of invariant measures absolutely continuous with respect to the conformal measure. We also give some examples of contexts in which this setting can be used—expansive maps of the interval, statistical mechanics.

ON HAUSDORFF MEASURES AND SBR MEASURES FOR PARABOLIC RATIONAL MAPS

If J is the Julia set of a parabolic rational map having Hausdorff dimension h<1, we show that Sullivan's h-conformal measure on J is either absolutely continuous or orthogonal with respect to the Hausdorff measures defined by the function [Formula: see text], according to whether τ>τ0 or τ<τ0 for some explicitly computable τ0>0.