On the dynamics near infinity of some polynomial mappings in

Journal of the American Mathematical Society, 1991

Fundamenta Mathematicae, 2005

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 Fr Ω, with positive Lyapunov exponent, there is an invariant subset of Fr Ω 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 (1994) by the author and Anna Zdunik, where the density of periodic orbits in Fr Ω was proved. 1. Introduction. Let Ω be a simply connected domain in C and f be a holomorphic map defined on a neighbourhood W of Fr Ω to C. Assume f (W ∩ Ω) ⊂ Ω, f (Fr Ω) ⊂ Fr Ω and Fr Ω repells to the side of Ω, that is, ∞ n=0 f −n (W ∩ Ω) = Fr Ω. An important special case is where Ω is an immediate basin of attraction of an attracting fixed point for a rational function. This covers also the case of a component of the immediate basin of attraction to a periodic attracting orbit, as one can consider an iterate of f mapping the component to itself. Distances and derivatives are considered in the Riemann spherical metric on C. Let R : D → Ω be a Riemann mapping from the unit disc onto Ω and let g be a holomorphic extension of R −1 • f • R to a neighbourhood of the unit circle ∂D. It exists and it is expanding on ∂D (see [P2, Section 7]). We prove the following Theorem A. Let ν be an ergodic g-invariant probability measure on ∂D such that for ν-a.e. ζ ∈ ∂D the radial limit R(ζ) := lim rր1 R(rζ) exists. Assume that the measure µ := R * (ν) has positive Lyapunov exponent χ µ (f).

American Journal of Mathematics, 2005

2003

1991

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$ When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory

Annals of Mathematics, 2016

We show that there exist polynomial endomorphisms of C 2 , possessing a wandering Fatou component. These mappings are polynomial skewproducts, and can be chosen to extend holomorphically of P 2 (C). We also find real examples with wandering domains in R 2. The proof is based on parabolic implosion techniques.

2012

We consider a family of maps which are similar to quadratic maps in that they are degree two branched covers of the Riemann sphere, but not in general conformal. In particular, for α real and fixed, we study maps fc which are given in polar coordinates by fc(re iθ) = r 2α e 2iθ + c.

Mathematical Proceedings of the Cambridge …, 2005

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions

Transactions of the American Mathematical Society, 1998

We prove that for every rational map on the Riemann sphere f : C ¯ → C ¯ f:\overline {\mathbb {C}} \to \overline {\mathbb {C}} , if for every f f -critical point c ∈ J c\in J whose forward trajectory does not contain any other critical point, the growth of | ( f n ) ′ ( f ( c ) ) | |(f^{n})’(f(c))| is at least of order exp ⁡ Q n \exp Q \sqrt n for an appropriate constant Q Q as n → ∞ n\to \infty , then HD ess ⁡ ( J ) = α 0 = HD ⁡ ( J ) \operatorname {HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname {HD} (J) . Here HD ess ⁡ ( J ) \operatorname {HD}_{\operatorname {ess}} (J) is the so-called essential, dynamical or hyperbolic dimension, HD ⁡ ( J ) \operatorname {HD} (J) is Hausdorff dimension of J J and α 0 \alpha _{0} is the minimal exponent for conformal measures on J J . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of J J also coincides with HD ⁡ ( J ) \operatorname {HD}(J)...

2020

Let h : V −→ V be a Cohomological Expanding Mapping 1 of a smooth complex compact homogeneous manifold with dim C (V) = k ≥ 1 and Kodaira Dimension ≤ 0. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit O h (x) = {h n (x), n ∈ N or Z} of a generic point. Using pluripotential methods, we have constructed in our previous paper [1] a natural invariant canonical probability measure of maximal Cohomological Entropy ν h such that χ −m 2l (h m) * Ω → ν h as m → ∞ for each smooth probability measure Ω in V. We have also studied the main stochastic properties of ν h and have shown that ν h is a smooth equilibrium measure , ergodic, mixing, K-mixing, exponential-mixing. In this paper we are interested on equidistribution problems and we show in particular that ν h reflects a property of equidistribution of periodic points by setting out the Third and Fourth Main Results in our study. Finally we conjecture that 1 D(x, E γ) β/2 ζ C β γ −βm/2 .

Advances in Mathematics, 2008

We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C k , under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps, to regular polynomial automorphisms of C k and to their small pertubations.

Ann. Acad. Sci. Fenn. Math, 1998

The paper examines some properties of the dynamics of entire functions which extend to general meromorphic functions and also some properties which do not. For a transcendental meromorphic function f (z) whose Fatou set F (f ) has a component of connectivity at least three, it is shown that singleton components are dense in the Julia set J(f ) . Some problems remain open if all components are simply or doubly connected. Let I(f ) denote the set of points whose forward orbits tend to ∞ but never land at ∞ . For a transcendental meromorphic function f (z) we have J(f ) = ∂I(f ) , I(f ) ∩ J(f ) = ∅ . However in contrast to the entire case, the components of I(f ) need not be unbounded, even if f (z) has only one pole. If f (z) has finitely many poles then, as in the entire case, F (f ) has at most one completely invariant component.

Bulletin of the Polish Academy of Sciences Mathematics, 2006

We 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 proved by A. Zdunik and relies on the theory elaborated by M. Urbański, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in ∂Ω constructed in [P2]. To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Hölder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Hölder potentials on Julia sets.

arXiv (Cornell University), 2020

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.