Search

We establish a version of the Pommerenke–Levin–Yoccoz inequality for the modulus of a polynomial-like (PL) restriction of a polynomial and give two applications. First we show that if the modulus of a PL restriction of a polynomial is bounded from below then this restricts the combinatorics of the polynomial. The second application concerns parameter slices of cubic polynomials given by the non-repelling multiplier of a fixed point. Namely, the intersection of the so-called Main Cubioid and the multiplier slice lies in the closure of the principal hyperbolic domain, with possible exception of queer components. [ABSTRACT FROM AUTHOR]

To investigate the degree 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 \sigma _d-invariant laminations , where \sigma _d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f(z) = z^2 +c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials 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_sCL together with the induced factor space \mathbb {S}/C_sCL of the unit circle \mathbb {S}. As will be verified in the third paper of the series, \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. [ABSTRACT FROM AUTHOR]

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the main cubioid in this parameter space. The main cubioid is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials z^2+c for c in the filled main cardioid. [ABSTRACT FROM AUTHOR]

We prove that there is a sequence of purely imaginary parameter values λ n converging to 0 such that the Julia set for z ↦ λ n (z + 1 / z) is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate. [ABSTRACT FROM AUTHOR]

For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by R c (z) = c (z d − (d − 1)) / z with c ∈ R ∖ { 0 }. Also, when c ∈ C ∖ { 0 } we show that the geometric limit of the Julia set and filled Julia set of the family R c exists and is the unit circle. [ABSTRACT FROM AUTHOR]

Let n and k be positive integers. For n > k ≥ 2 , we consider the connectedness locus M n , k of the family of polynomials P c (z) = z n + c z k , where c is a complex parameter. We show that the geometric limit of the connectedness locus sets M n , k , when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of M n , k .When parameter c belongs to the open unit disk, we show that the geometric limit of the Julia sets J c , when n tends to infinity, exists and is the unit circle. Finally, we have established some properties on the hyperbolic components of this family. [ABSTRACT FROM AUTHOR]

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semiconjugate 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2. [ABSTRACT FROM AUTHOR]

We study the closure of the cubic Principal Hyperbolic Domain and its intersection Pλ with the slice Fλ of the space of all cubic polynomials with fixed point 0 defined by the multiplier λ at 0. We show that any bounded domain W of Fλ\Pλ consists of J-stable polynomials f with connected Julia sets J(f) and is either of Siegel capture type (then f ∈ W has an invariant Siegel domain U around 0 and another Fatou domain V such that f|V is two-to-one and fk(V) = U for some k > 0) or of queer type (then a specially chosen critical point of f ∈ W belongs to J(f), the set J(f) has positive Lebesgue measure, and it carries an invariant line field). [ABSTRACT FROM AUTHOR]

Let P be a polynomial of degree d with Julia set J P . Let N ˜ be the number of non-repelling cycles of P . By the famous Fatou–Shishikura inequality N ˜ ≤ d − 1 . The goal of the paper is to improve this bound. The new count includes wandering collections of non-(pre)critical branch continua , i.e., collections of continua or points Q i ⊂ J P all of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval ( Q i ) ≥ 3 external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense. A weak version of the inequality reads N ˜ + N irr + χ + ∑ i ( eval ( Q i ) − 2 ) ≤ d − 1 where N irr counts repelling cycles with no periodic rays landing at points in the cycle, { Q i } form a wandering collection B C of non-(pre)critical branch continua, χ = 1 if B C is non-empty, and χ = 0 otherwise. [ABSTRACT FROM AUTHOR]

Abstract: Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map φ onto a locally connected continuum , i.e. a monotone map such that for any other monotone map there exists a monotone map h with . Then we extend φ onto the complex plane (keeping the same notation) and show that φ monotonically semiconjugates to a topological polynomial . If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi''s fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map φ not to collapse all of J into a point. [Copyright &y& Elsevier]

An introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions. [ABSTRACT FROM AUTHOR]

The dynamics of the map[image omitted] is investigated for [image omitted] . This map is derived from multiple geometric circle inversions. Conditions are given for Cantor set and Sierpinski curve Julia sets for this family. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings of the invariant set, where each consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given. [Copyright &y& Elsevier]

In this paper, we consider the family of rational maps given by \[ F_\lambda(z) = z^n + \frac{\lambda}{z} \] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere. [ABSTRACT FROM AUTHOR]

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components. [ABSTRACT FROM AUTHOR]

Abstract: The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets. [Copyright &y& Elsevier]

Abstract: We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let be a homeomorphism of a plane domain U and let be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles are K-equivalent if for some angles the impressions of and are non-disjoint, ; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense -set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent. [Copyright &y& Elsevier]

Abstract: We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors. We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable. [Copyright &y& Elsevier]