Connected Julia Sets Research Articles

Local connectivity of boundaries of tame Fatou components of meromorphic functions

We prove the local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class B, and the boundaries of the basins (possibly unbounded) are allowed to contain an infinite number of post-singular values, as well as the essential singularity at infinity. A basic assumption is that the unbounded parts of the basins are contained in regions which we call ‘repelling petals at infinity’, where the map exhibits a kind of ‘parabolic’ behaviour. In particular, our results apply to a wide class of Newton’s methods for transcendental entire maps. As an application, we prove the local connectivity of the Julia set of Newton’s method for sinz, providing the first non-trivial example of a locally connected Julia set of a transcendental map outside class B, with an infinite number of unbounded Fatou components.

Hausdorff limits of external rays: The topological picture

AbstractWe study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree with connected Julia sets.

Symmetric cubic laminations

To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Cantor and connected Julia sets of the parameterized Dixon elliptic functions

ABSTRACT We discuss connectivity properties of Julia sets of the parameterized Dixon elliptic functions. Our main result is that the connectivity locus of the parameterized Dixon sine function is the exterior of the open unit disc, and the Julia set is Cantor in the open unit disc minus the origin. We prove the parameterized Dixon cosine function also exhibits a fundamental dichotomy in the connectivity of the Julia set. The Julia and Fatou sets exhibit a variety of rotational symmetries, and no Herman rings exist for any function in either family.

On hyperbolic rational maps with finitely connected Fatou sets

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show that any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence of rational maps of a fixed degree with infinitely many non-monomial rescaling limits.

Unicritical laminations

Thurston introduced invariant (quadratic) laminations in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map $\sigma _2

On Hausdorff dimension of polynomial not totally disconnected Julia sets

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.

Criterion for rays landing together

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set.

A structure theorem for semi-parabolic Hénon maps

Consider the parameter space Pλ⊂C2 of complex Hénon mapsHc,a(x,y)=(x2+c+ay,ax),a≠0, which have a semi-parabolic fixed point with one eigenvalue λ=e2πip/q. In this paper we provide a complete topological description of the Julia set and forward Julia set within a family of semi-parabolic Hénon maps, i.e. those Hénon maps from the curve Pλ which are perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in Pλ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J+. The forward Julia set J+ has a nice local description: inside a certain bidisk it is a trivial fiber bundle over Jp, the Julia set of the polynomial p, with fibers biholomorphic to a disk. The Julia set J is homeomorphic to the quotient of a solenoid by an equivalence relation and J=J⁎, the closure of the saddle periodic points.

Models for spaces of dendritic polynomials

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the pinched disk model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.

A core decomposition of compact sets in the plane

A Peano continuum means a locally connected continuum. A compact metric space is called a Peano compactum if all its components are Peano continua and if for any constant C>0 all but finitely many of its components are of diameter less than C. Given a compact set K⊂C, there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the core decomposition of K with Peano quotient. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into C. We also construct a concrete continuum K⊂R3 such that the core decomposition of K with Peano quotient does not exist. For specific choices of K⊂C, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map f:Cˆ→Cˆ restricted to its Julia set.

Combinatorial models for spaces of cubic polynomials

W. Thurston constructed a combinatorial model of the Mandelbrot set M2 such that there is a continuous and monotone projection of M2 to this model. We propose the following related model for the space MD3 of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3, then every point z in the Julia set of the polynomial P defines a unique maximal finite set Az of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let G(z) denote the convex hull of Az. The convex sets G(z) partition the closed unit disk. For (P,c1,c2)∈MD3 let c1⁎ be the co-critical point ofc1. We tag the marked dendritic polynomial (P,c1,c2) with the set G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by MD3comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3 to MD3comb so that MD3comb serves as a model for MD3.

A PRELIMINARY STUDY ON THE FRACTAL PHENOMENON: “DISCONNECTED+ DISCONNECTED=CONNECTED”

The well-known Parrondo’s paradox: “losing+losing=winning” [G. P. Harmer and D. Abbott, Parrondo’s paradox, Stat. Sci. 14 (2009) 206–213.] indicated that two games with negative gains can generate a new game with positive gain. By extending the Parrondo’s philosophy into chaos research, it was shown that the periodic alteration of two chaotic dynamics results in an ordered dynamics, that is the phenomenon: “chaos+chaos=order” [J. Almeida, D. Peralta-Salas and M. Romera, Can two chaotic systems give rise to order, Physica D 200 124–132 (2005)]. This paper further extends these researches into fractal research by proposing that two disconnected Julia sets can originate a new connected Julia set via alternating order. This new parrondian paradoxical phenomenon can be stated in the Parrondo’s terms as “disconnected+disconnected=connected”.

Complex Hénon maps and discrete groups

Consider the standard family of complex Hénon maps H(x,y)=(p(x)−ay,x), where p is a quadratic polynomial and a is a complex parameter. Let U+ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set U+ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of (C−D‾)×C by a discrete group of automorphisms Γ isomorphic to Z[1/2]/Z. On the other hand, the boundary J+ of U+ is a complicated fractal object on which the Hénon map behaves chaotically. We show how to extend the group action to S1×C, in order to represent the set J+ as a quotient of S1×C/Γ by an equivalence relation. We analyze this extension for Hénon maps that are perturbations of hyperbolic polynomials with connected Julia set or polynomials with a parabolic fixed point.

Non-archimedean connected Julia sets with branching

We construct the first examples of rational functions defined over a non-archimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.

Repelling periodic points and landing of rays for post-singularly bounded exponential maps

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most nitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.

Spiders’ webs and locally connected Julia sets of transcendental entire functions

AbstractWe show that if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider’s web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider’s web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider’s web.

Dynamics of McMullen maps

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to the question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.

Asymptotic porosity of planar harmonic measure

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set \(\mathcal{A}\) in the boundary of the Mandelbrot set such that for every \(c\in \mathcal{A}\), β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λn for n from a set with positive density amongst natural numbers.

Julia and John revisited

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov).