A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing...

Traitant la série de Poincaré d’un groupe discret d’isométries en courbure négative comme un noyau de Green, on établit une théorie du potentiel assez comparable à la théorie classique pour affirmer un parallèle entre densités conformes à la Patterson-Sullivan et densités harmoniques, et notamment définir une frontière de Martin où les densités ergodiques forment la partie minimale, et enfin l’identifier géométriquement sous hypothèse d’hyperbolicité.

Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of C₂ ∗ C₃ and quadratic polynomials carries δ-conformal measures, and that if the correspondence is geometrically finite then the real number δ is equal to the Hausdorff dimension of the limit set. Moreover, when f is the limit of a pinching deformation ${f_t}_{0\le t<1}$ we give sufficient conditions for the dynamical convergence of $f_t$.

Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.

We prove that for Ω being an immediate basin of attraction to an attracting fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure μ on the boundary FrΩ, with positive Lyapunov exponent, there is an invariant subset of FrΩ which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of μ. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145...

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0,\sigma }$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0,\sigma }$, given by Urbański and Zinsmeister. The closure of the limit set of our IFS ${\varphi }_{\sigma ,\alpha }^{n,k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of $J_{0,\sigma }$. The parameter...

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic settings....

Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if $\frac{p\left(T\right)+1}{p\left(T\right)}\delta >2$. Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.

Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $\left|detM\right|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set ${}_M$ of points in $A_M$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of ${}_M$ is positive. For this special class the full description of ${}_M$ is given as well. This paper continues our work begun...

We show that for entire maps of the form z ↦ λexp(z) such that the orbit of zero is bounded and Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.

Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{\sigma }$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,\sigma }$ is continuous at σ₀ as the function of the parameter $\sigma \in \overline{₀}$ if and only if $HD\left(J_{0,\sigma ₀}\right)\ge 4/3$. Since $HD\left(J_{0,\sigma }\right)>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD\left(J_{0,\sigma }\right)$ on an open and dense subset of ∂₀.

e prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a 2:1 factor of a Blaschke product, is larger than 1. We prove a "local version" of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials...

We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.