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.
We consider the family of transcendental entire maps given by where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing...
We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable...
We establish a Poincaré-Dulac theorem for sequences of holomorphic contractions whose differentials split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of . In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...
We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.
Rees-Shishikura’s theorem plays an important role in the study of matings of polynomials. It promotes Thurston’s combinatorial equivalence into a semi-conjugacy. In this work we restate and reprove Rees-Shishikura’s theorem in a more general form, which can then be applied to a wider class of postcritically finite branched coverings. We provide an application of the restated theorem.
Let f be a quadratic map (more generally, , d > 1) of the complex plane. We give sufficient conditions for f to have no measurable invariant linefields on its Julia set. We also prove that if the series converges absolutely, then its sum is non-zero. In the proof we use analytic tools, such as integral and transfer (Ruelle-type) operators and approximation theorems.
Let f be a polynomial of one complex variable so that its Julia set is connected. We show that the harmonic (Brolin) measure of the set of biaccessible points in J is zero except for the case when J is an interval.
We charocterize the commuting polynomial automorphisms of C2, using their meromorphic extension to P2 and looking at their dynamics on the line at infinity.
Following the attracting and preperiodic cases, in this paper we prove the existence of weakly repelling fixed points for transcendental meromorphic maps, provided that their Fatou set contains a multiply connected parabolic basin. We use quasi-conformal surgery and virtually repelling fixed point techniques.
It is shown that an infinite sequence of polynomial mappings of several complex variables, with suitable growth restrictions, determines a filled-in Julia set which is pluriregular. Such sets depend continuously and analytically on the generating sequences, in the sense of pluripotential theory and the theory of set-valued analytic functions, respectively.
We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ2↦ℂ2 under which all complete external rays from infinity for f have well defined endpoints.
We prove that for some families of entire functions whose Julia set is the complement of the basin of attraction every branch of a tree of preimages starting from this basin is convergent.
We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise sensitivity with respect to a large class of metrics. We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry. In the finite measure-preserving...