# Singular perturbations of complex polynomials

@article{Devaney2013SingularPO, title={Singular perturbations of complex polynomials}, author={Robert L. Devaney}, journal={Bulletin of the American Mathematical Society}, year={2013}, volume={50}, pages={391-429} }

. In this paper we describe the dynamics of singularly perturbed complex polynomials. That is, we start with a complex polynomial whose dynamics are well understood. Then we perturb this map by adding a pole, i.e., by adding in a term of the form λ/ ( z − a ) d where the parameter λ is complex. This changes the polynomial into a rational map of higher degree and, as we shall see, the dynamical behavior explodes. One aim of this paper is to give a survey of the many diﬀerent topological…

## Figures from this paper

## 34 Citations

On the dynamics of n-circle inversion

- MathematicsNonlinearity
- 2019

The article deals with singular perturbation of polynomial maps where is a complex parameter and n is the degree, which is a particular case of the family of rational maps known as McMullen maps. Our…

Investigating the Behavior of Perturbed Complex Quadratic Maps: Moving the singularity and the critical point∗

- Mathematics
- 2019

The study of discrete dynamical systems has substantially grown in popularity over the past 30 years with increases in computing power. The use of computer technology to simulate the behavior of…

Parameter Planes for Complex Analytic Maps

- Mathematics
- 2014

In this paper we describe the structure of the parameter planes for certain families of complex analytic functions. These families include the quadratic polynomials z 2 + c, the exponentials λ…

On McMullen-like mappings

- Mathematics
- 2014

We introduce a generalization of the McMullen family f (z) = z n +=z d . In 1988 C. McMullen showed that the Julia set of f is a Cantor set of circles if and only if 1=n+1=d < 1 and the simple…

LINEAR COMBINATION INTERPOLATION, CUNTZ RELATIONS AND INFINITE PRODUCTS

- Mathematics
- 2014

We introduce the following linear combination interpolation problem (LCI): Given N distinct numbers w1,...wN and N + 1 complex numbers a1,...,aN and c, find all functions f(z) analytic in a simply…

Interactions of the Julia Set with Critical and (Un)Stable Sets in an Angle-Doubling Map on ℂ\{0}

- MathematicsInt. J. Bifurc. Chaos
- 2015

A nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map is studied, finding the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set.

Quasisymmetric geometry of the Julia sets of McMullen maps

- MathematicsScience China Mathematics
- 2018

We study the quasisymmetric geometry of the Julia sets of McMullen maps fλ(z) = zm + λ/zℓ, where λ ∈ ℂ {0} and ℓ and m are positive integers satisfying 1/ℓ+1/m < 1. If the free critical points of fλ…

Existence of the Mandelbrot set in parameter planes for some generalized McMullen maps

- Mathematics
- 2022

In this paper we study rational functions of the form Rn,a,c(z) = z n + a zn + c and hold either a or c fixed while the other varies. We show that for certain ranges of a, the c-parameter plane…

Mandelpinski spokes in the parameter planes of rational maps

- Mathematics
- 2016

In this paper we describe a new structure that arises in the parameter plane of the family of maps where is even but is odd. We call these structures Mandelbrot–Sierpiński spokes (or, for short,…

## References

SHOWING 1-10 OF 74 REFERENCES

Singular perturbations in the quadratic family with multiple poles

- Mathematics
- 2013

We consider the quadratic family of complex maps given by , where c is the centre of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding…

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

- MathematicsErgodic Theory and Dynamical Systems
- 2005

In this paper we consider the family of rational maps of the complex plane given by \[z^2+\frac{\lambda}{z^2}\] where $\lambda$ is a complex parameter. We regard this family as a singular…

The escape trichotomy for singularly perturbed rational maps

- Mathematics
- 2005

In this paper we consider the dynamical behavior of the family of complex rational maps given by where n ≥ 2, d ≥ 1. Despite the high degree of these maps, there is only one free critical orbit up to…

Complex Dynamics and Renormalization

- Mathematics
- 1994

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid…

Limiting Julia Sets for singularly perturbed Rational Maps

- MathematicsInt. J. Bifurc. Chaos
- 2008

It is shown 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.

Checkerboard Julia Sets for Rational Maps

- MathematicsInt. J. Bifurc. Chaos
- 2013

A dynamical invariant determines which of these maps are conjugate on their Julia sets, and it is obtained an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

CONNECTIVITY OF JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

- Mathematics
- 2012

In this paper we consider the family of rational maps of the form F λ (z) = z n + λ/z n where n ≥ 2. It is known that there are two cases where the Julia sets of these maps are not connected. If the…

Cantor sets of circles of Sierpiński curve Julia sets

- MathematicsErgodic Theory and Dynamical Systems
- 2007

Abstract Our goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of…

Singular perturbations in the quadratic family

- Mathematics
- 2008

In this paper, we study the dynamics of the family of complex maps given by , where c and λ are complex parameters such that c lies in the interior of a hyperbolic component of the Mandelbrot set but…

Iteration at the boundary of the space of rational maps

- Mathematics
- 2004

Let Ratd denote the space of holomorphic self-maps of P of degree d ≥ 2, and let μf be the measure of maximal entropy for f ∈ Ratd . The map of measures f → μf is known to be continuous on Ratd , and…