Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere.     

Julia sets that are full of holes

Conclusion  Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Although there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura’s surprising result (1998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, provides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean e-porosity, both found in Koskela and Rohde (1997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Julia sets are porous and which are not. In fact, forJ _R, little is known about its dimension or measure. There is much left to explore.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Quatre applications du lemme de Zalcman à la dynamique complexe

We give four applications of Zalcman’s lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets; similarity between the Mandelbrot set and the Julia sets; a construction of the Lyubich-Minsky lamination and its variant; and a unified characterization of conical points by Lyubich and Minsky and those by Martin and Mayer.     

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible. Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”     

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials in terms of the parameter c. Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001     

Parabolic Implosion and Julia-Lavaurs Sets in the Exponential Family

We deal with all the maps from the exponential family f _ε(z) = (e ⁻¹ + ε)exp(z), with ε ≥ 0. Let h _ε = HD(J _r,ε) be the Hausdorff dimension of the radial Julia sets J _r,ε. Observing the phenomenon of parabolic implosion, it is shown that the function ε ↦ h _ε is not continuous from the right. The research of the first author was supported in part by the NSF Grant DMS 0100078.     

Transversality family of expanding rational semigroups

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d$d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2$2$, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2$2$. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈C$a\in \mathbb{C}$ with |a|≠0,1$\left|a\right|\ne 0,1$, the family of small perturbations of the semigroup generated by {z²,az²}$\{z^2,a{z}^2\}$ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.     

A characterization of hyperbolic rational maps

We give a topological characterization of rational maps with disconnected Julia sets. Our results extend Thurston’s characterization of postcritically finite rational maps. In place of iteration on Teichmüller space, we use quasiconformal surgery and Thurston’s original result.     

Julia sets and periodic kneading sequences

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form f _c (z) = z ² + c, having an associated periodic kneading sequence of the form [`(a*)]{\overline{\alpha\ast}} which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial f _c , with similar dynamical properties.     

McMullen's root-finding algorithm for cubic polynomials

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

     

Julia set of λ exp(<Emphasis Type="Italic">z</Emphasis>)/<Emphasis Type="Italic">z</Emphasis> with real parameters λ

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.     

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

Wandering sets for a class of borel isomorphisms of [0,1)

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity, the existence of a Borel wandering set is equivalent to rationality of the Poincaré rotation number. We compute the rotation numbers for a special class of such functions. The main result provides a concrete method of connecting certain pairs of wavelet sets.     

Geometry and ergodic theory of non-recurrent elliptic functions

We explore the class of elliptic functions whose critical points all contained in the Julia set are non-recurrent and whose ω-limit sets form compact subsets of the complex plane. In particular, this class comprises hyperbolic, subhyperbolic and parabolic elliptic maps. Leth be the Hausdorff dimension of the Julia set of such an elliptic functionf. We construct an atomlessh-conformal measurem and show that theh-dimensional Hausdorff measure of the Julia set off vanishes unless the Julia set is equal to the entire complex plane ℂ. Theh-dimensional packing measure is positive and is finite if and only if there are no rationally indifferent periodic points. Furthermore, we prove the existence of a (unique up to a multiplicative constant) σ-finitef-invariant measure μ equivalent tom. The measure μ is shown to be ergodic and conservative, and we identify the set of points whose open neighborhoods all have infinite measure μ. In particular, we show that ∞ is not among them. The research of the first author was supported in part by the Foundation for Polish Science, the Polish KBN Grant No 2 PO3A 034 25 and TUW Grant no 503G 112000442200. She also wishes to thank the University of North Texas where this research was conducted. The research of the second author was supported in part by the NSF Grant DMS 0100078. Both authors were supported in part by the NSF/PAN grant INT-0306004.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space . (Received 11 August 2000; in revised form 18 April 2001)     

Dynamics of a Discrete Brusselator Model: Escape to Infinity and Julia Set

We consider a discrete version of the Brusselator Model of the famous Belousov-Zhabotinsky reaction in chemistry. The original model is a reaction-diffusion equation and its discrete version is a coupled map lattice. We study the dynamics of the local map, which is a smooth map of the plane. We discuss the set of trajectories that escape to infinity as well as analyze the set of bounded trajectories – the Julia set of the system. The work was partially supported by National Science Foundation grant #DMS-0088971 and U.S.-Mexico Collaborative Research grant 0104675 The article is available online on SpringerLink (www.springerlink.com) using colors instead of greyscales in the pictures. Lecture held in the Seminario Matematico e Fisico on July 1, 2003 Received: March 2005     

Collet, Eckmann and Hölder

We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical orbits, yields the H?lder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected, and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least one non-empty fully invariant Fatou component. Oblatum 22-III-1996 & 15-VII-1997     

The Adjoint Action of an Expansive Algebraic -Action

 Let , and let α be an expansive -action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup is a -action by automorphisms of . By duality, there exists a -action by automorphisms of the compact abelian group : this action is called the adjoint action of α. We prove that is again expansive and has completely positive entropy, and that α and are weakly algebraically equivalent, i.e. algebraic factors of each other. A -action α by automorphisms of a compact abelian group X is reflexive if the -action on the compact abelian group adjoint to is algebraically conjugate to α. We give an example of a non-reflexive expansive -action α with completely positive entropy, but prove that the third adjoint is always algebraically conjugate to . Furthermore, every expansive and ergodic -action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic -actions with zero entropy. Received 11 June 2001; in revised form 29 November 2001