Disconnected Julia Sets Research Articles

Disconnected Julia set of Halley's method for exponential maps

We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.

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.

On uniformly disconnected Julia sets

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in \(\mathbb {S}^n\), for \(n\ge 2\). Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in \(\mathbb {S}^n\), then it is the Julia set of a hyperbolic UQR map \(f:\mathbb {S}^N \rightarrow \mathbb {S}^N\) where \(N=n\) if \(n=2\) and \(N=n+1\) otherwise.

On the cycles of components of disconnected Julia sets

For any integers $$d\ge 3$$ and $$n\ge 1$$ , we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its Julia set except single points and Jordan curves.

Real Analyticity of Hausdorff Dimension Function of Disconnected Julia Sets of Parabolic Polynomials fa,b(z) = z(1 – az – bz<sup>2</sup>)

Abstract not available
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 119-125

Rational maps with disconnected Julia set

Rational maps with disconnected Julia set

Iteration of the Translated Tangent

The iteration of the map $$z \mapsto \lambda + \tan z$$ on the complex plane is investigated for each complex number $$\lambda $$ . It is shown that its Fatou set always contains a completely invariant component and the Julia set is either totally disconnected or connected. A necessary and sufficient condition for totally disconnected Julia set is proved. When the Julia set is connected, it is seen that the Fatou set can consist of additional components, which can be completely invariant attracting domain, a Siegel disk, a two periodic attracting domain or parabolic domain. Examples of each case are given. Further, a necessary and sufficient condition is provided for the Julia set to be a Jordan curve passing through infinity. It is proved that the set of parameters corresponding to Jordan curve Julia sets is connected, whereas the set of those corresponding to totally disconnected Julia sets is disconnected.

Dynamics and limiting behavior of Julia sets of König's method for multiple roots

A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f, there exists at least one point in Sd converging under iterations of Nf to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton's method. We show that for all order σ≥2, there exists a complex polynomial f such that the Julia set of König's method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König's method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric.

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”.

Elliptic Functions with Disconnected Julia Sets

In this paper, we investigate elliptic functions of the form [Formula: see text], where [Formula: see text] is the Weierstrass elliptic function on a real rhombic lattice. We show that a typical function in this family has a superattracting fixed point at the origin and five other equivalence classes of critical points. We investigate conditions on the lattice which guarantee that [Formula: see text] has a double toral band, and we show that this family contains the first known examples of elliptic functions for which the Julia set is disconnected but not Cantor.

A family of rational maps with buried Julia components

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.

Totally disconnected Julia set for different classes of meromorphic functions

We study a class of functions A \textbf {A} given by Epstein in Towers of finite type complex analytic maps, ProQuest LLC, Ann Arbor, MI, 1993, called finite-type maps. We extend a result related with the Julia set given by Baker, Domínguez in Some connectedness properties of Julia sets, Complex Variable Theory Appl. 41 (2000), 371–389, and Baker, Domínguez, and Herring in Dynamics of functions meromorphic outside a small set, Ergodic Theory Dynam. Systems 21 (2001), 647–672, to functions in class A \textbf {A} .

On the ergodicity of conformal measures for rational maps with totally disconnected Julia sets

On the ergodicity of conformal measures for rational maps with totally disconnected Julia sets

On dynamical Teichmüller spaces

Following ideas from a preprint of the second author (see Automorphisms of a rational function with disconnected Julia set, Orsay, Preprint, 03 1992), we investigate relations of dynamical Teichmüller spaces with dynamical objects. We also establish some connections with the theory of deformations of inverse limits and laminations in holomorphic dynamics (see J. Diff. Geom. 47 (1997), 17–94).

Disconnected Julia sets of quartic polynomials and a new topology of the symbol space

For a certain quartic polynomial, there exists a homeomorphism between the set of all components of the filled-in Julia set with the Hausdorff metric and some subset of the corresponding symbol space with the ordinary metric. But these sets are not compact with respect to each metric. We introduce a new topology with respect to which these sets are compact.

On a certain kind of polynomials of degree 4 with disconnected Julia set

For a polynomial of degree at least two, the Julia set and thefilled-in Julia set are either connected or have uncountably manycomponents. In the case that the Julia set of a polynomial of degree 4is neither connected nor totally disconnected, there exists ahomeomorphism between the set of all components of the filled-inJulia set and some subset of the corresponding symbol space.Furthermore the polynomial is topologically conjugate to the shiftmap via the homeomorphism. Moreover there exists ahomeomorphism between the Julia sets of the polynomial and that of acertain polynomial semigroup.

Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets

Let R R be a rational map with a totally disconnected Julia set J ( R ) J(R) . If the postcritical set on J ( R ) J(R) contains a non-persistently recurrent (or conical) point, then we show that the map R R cannot be a structurally stable map.

Dynamics of polynomials with disconnected Julia sets

We study the structure of disconnected polynomial Julia sets. We consider polynomials with an arbitrary number of non-escaping critical points, of arbitrary multiplicity, which interact non-trivially. We use a combinatorial system of a tree with dynamics to give a sufficient condition for the Julia set a polynomial to be an area zero Cantor set. We show that there exist uncountably many combinatorially inequivalent polynomials, which satisfy this condition and have multiple non-escaping critical points, each of which accumulates at all the non-escaping critical points.

Dynamics of functions meromorphic outside a small set

The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.