Misiurewicz Points Research Articles

A note on Misiurewicz polynomials

Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz points $c_0,c_1\in \bar{\mathbb{Q}}$ are such that the polynomials $f_{c_0,d}$ and $f_{c_1,d}$ have the same orbit type. One classical question is whether $c_0$ and $c_1$ need to be Galois conjugates or not. Recently there has been a partial progress on this question by several authors. In this note, we prove some new results when $d$ is a prime. All the results known so far were in the cases of period size at most $3$. In particular, our work is the first to say something provable in the cases of period size greater than $3$.

The connectedness locus of the family of polynomials P c (z) = z n –cz

For the study of the two-dimensional space of cubic polynomials, Milnor [J. Milnor, Cubic polynomials with periodic critical orbit, Part I, in Complex Dy-namics Families and Friends, D. Scheleicher and A.K. Peters, eds., 2009, pp. 333–411] considers the complex one-dimensional slice 𝒮 n of the cubic polynomials which have a super-attracting orbit of period n. These polynomials are described – modulo affine conjugacy – by the family f c (z) = z 3 + cz 2. Milnor gives a detailed description of 𝒮 n and a number of questions were left open which were resolved recently by Roesch [P. Roesch Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Scientifiques de L' Ecole Normal Sup. 40 (2007)] and Faught [D. Faught, Local connectivity in a family of cubic polynomials, Thesis Cornell University, 1992]. Roesch generalizes these results for the families of polynomials f c (z) = z d + cz d−1; d ≥ 3. In the present note, we consider for n ≥ 3, the family of polynomials P c (z) = z n − cz, where c is a complex parameter. It is proved that the connectedness locus ℳ n of the family of polynomials P c (z) contains the unitary disc {c; |c| ≤ 1}, and it is a subset of the disc . Thus, we obtain that lim n→∞ℳ n = {c; |c| ≤ 1}. Also, we prove that the sets of centred polynomials of the hyperbolic components of ℳ n and the Misiurewicz points are dense in the boundary of ℳ n . We observe that the family of polynomials z m (z n − c) has been considered by Rabii [M. Rabii, Local connectedness of the Julia set of the family z m (z n + b), Ergodic Theory Dyn. Syst. 18 (2) (1998), pp. 457–470]. She studied the local connectedness of the Julia set of these families.

FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

Calculation of the Structure of a Shrub in the Mandelbrot Set

We calculate the external arguments of the structure of any shrub in the Mandelbrot set. Before calculating, we revise, expand, and clarify some tools useful for this paper: harmonics, pseudoharmonics, the concept of structure, the structure of a shrub, and the ancestral route. Finally we present the main contribution of this paper, a three‐step algorithm which allows us to calculate the structure of the shrub. In the first step, we use pseudoharmonics that were previously introduced by us, in order to calculate the first and last external arguments of a structural node. In the second step, starting from two general properties of the Misiurewicz points external arguments introduced here by us, we present a new method to calculate the intermediate external arguments. In the last step we introduce a third property that allows us to calculate the external arguments of the representatives of the branches emerging from the structural nodes.

A general view of pseudoharmonics and pseudoantiharmonics to calculate external arguments of Douady and Hubbard

Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmonics. Pseudoharmonics turn out to be a more powerful tool than harmonics since they allow the calculation of external arguments of the Mandelbrot set in many more cases. Likewise, unlike antiharmonics, pseudoantiharmonics turn out to be a powerful tool to calculate external arguments of the Mandelbrot set in some cases.

Distribution of rational maps with a preperiodic critical point

Let $\{f_\lambda\}$ be a family of rational maps of a fixed degree, with a marked critical point $c(\lambda)$. Under a natural assumption, we first prove that the hypersurfaces of parameters for which $c(\lambda)$ is periodic converge as a sequence of positive closed $(1,1)$ currents to the bifurcation current attached to $c$ and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree $d$. By intersecting the $d-1$ currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure $\mu_{\rm bif}$ of finite mass which is supported on the connectedness locus. They showed that its support is included in the closure of the set of parameters admitting $d-1$ neutral cycles. We show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials (i.e., of Misiurewicz points).

ALGORITHM FOR EXTERNAL ARGUMENTS CALCULATION OF THE NODES OF A SHRUB IN THE MANDELBROT SET

Misiurewicz points are essential points of the Mandelbrot set. External arguments are the most useful tool in order to analyze these points; therefore, it is important to go deeper in the study of the external arguments' calculation of these points. In this paper we introduce an algorithm to calculate the external arguments of the branch points, or nodes, of a shrub, since such nodes are the most outstanding Misiurewicz points of the chaotic region of the Mandelbrot set. Devaney and Moreno-Rocha had already presented a paper where they calculated the external arguments of one of these Misiurewicz points, the main node of a shrub. Our algorithm widens the study of Devaney and Moreno-Rocha because it allows one to calculate the external arguments of any node of the shrub.

Conjugation of cascades

Presented in this work are some results relative to sequences found in the logistic equation bifurcation diagram, which is the unimodal quadratic map prototype. All of the different saddle-node bifurcation cascades, associated with every last appearance p-periodic orbit ( p = 3 , 4 , 5 , … ) , can also be generated from the very Feigenbaum cascade. In this way it is evidenced the relationship between both cascades. The orbits of every saddle-node bifurcation cascade, mentioned above, are located in different chaotic bands, and this determines a sequence of orbits converging to every band-merging Misiurewicz point. In turn, these accumulation points form a sequence whose accumulation point is the Myrberg–Feigenbaum point. It is also proven that the first appearance orbits in the n-chaotic band converge to the same point as the last appearance orbits of the ( n + 1)-chaotic band. The symbolic sequences of band-merging Misiurewicz points are computed for any window.

Operating with External Arguments of Douady and Hubbard

The external arguments of the external rays theory of Douady and Hubbard is a valuable tool in order to analyze the Mandelbrot set, a typical case of discrete dynamical system used to study nonlinear phenomena. We suggest here a general method for the calculation of the external arguments of external rays landing at the hyperbolic components root points of the Mandelbrot set. Likewise, we present a general method for the calculation of the external arguments of external rays landing at Misiurewicz points.

Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ``Misiurewicz points'' have invariant densities for positive measure sets of parameters.

Chaotic bands in the Mandelbrot set

Hyperbolic components and Misiurewicz points of the chaotic region of the Mandelbrot set are located in what we call shrubs. Each shrub has two well-differentiated parts. The first one, shrub 0, corresponds to the main branch of the shrub. The second one, shrub r , corresponds to all the other branches of the shrub. In this paper, we have focused on the ordering of the hyperbolic components and Misiurewicz points located at the shrub 0 of any shrub of the Mandelbrot set. We have seen that each shrub 0 is made up of infinite chaotic bands, each one corresponding to a disk of the period-doubling cascade of the hyperbolic component from where the shrub originates.

External arguments of Douady cauliflowers in the Mandelbrot set

Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays land without intersecting in its Misiurewicz points and minute Mandelbrot sets is not really well known. In this work we study the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays.

How to work with one-dimensional quadratic maps

We analyse some identifiers which can univocally identify hyperbolic components and Misiurewicz points of one-dimensional quadratic maps. After seeing the equivalence among the different identifiers and how to go from one to another, we show which are the best for some specific tasks. Likewise, we present the analytic expressions, some of them shown for the first time in this paper, to calculate these identifiers. Some experimental considerations are taken into account.

Misiurewicz point patterns generation in one-dimensional quadratic maps

In a family of one-dimensional quadratic maps, Misiurewicz points are unstable and the orbits of such a points are repulsive. On the contrary, the orbits of superstable periodic points are attractive. Here we study the patterns of the symbolic sequences of both Misiurewicz and superstable periodic points, and show that a Misiurewicz point pattern can be obtained as the limit of the sum of a superstable periodic orbit pattern plus itself or some of its heredity transmitters repeated an infinite number of times. Inversely, when a Misiurewicz point pattern is given, we also show that it is possible to find both the superstable periodic orbit pattern and the heredity transmitters that generate such a pattern.

Similarity between the Dynamic and Parameter Spaces in Cubic Mappings

We have extended the work of Lei Tan on the similarity between the Mandelbrot set and the Julia sets. We show that the fractal structures of dynamic and parameter spaces are asymtotically similar at Misiurewicz points for the cubic mappings.

A revision of the Lyapunov exponent in 1D quadratic maps

Only Misiurewicz points strictly have a Lyapunov exponent positive measure in one-dimensional (1D) quadratic maps. Hence, we represent here the Lyapunov exponent, λ, of all the Misiurewicz points which were found and inventoried by us in a former work. As a result of this representation, an infinity of alignments of Misiurewicz points are obtained, and two types of jumps in the values of λ are found.

On the calculation of Misiurewicz patterns in one-dimensional quadratic maps

In this work we give for the first time a table with all Misiurewicz points M n, p for low values of the preperiod and period (2⩽ n⩽8, 1⩽ p⩽5) in one-dimensional quadratic maps. In the particular case of M n,1 (important Misiurewicz points which are all placed in the period-1 chaotic band) the preperiod values are (2⩽ n⩽11). A brute-force algorithm to obtain all the symbolic sequences (patterns) of the M n, p and a more efficient algorithm to obtain the patterns of M n,1 are also shown.

Misiurewicz points in one-dimensional quadratic maps

Misiurewicz points are constituted by the set of unstable or repellent points, sometimes called the set of exceptional points. These points, which are preperiodic and eventually periodic, play an important role in the ordering of hyperbolic components of one-dimensional quadratic maps. In this work we use graphic tools to analyse these points, by measuring their preperiods and periods, and by ordering and classifying them.

An approach to the ordering of one-dimensional quadratic maps

In this work the antenna of the Mandelbrot set is used to order hyperbolic components of one-dimensional quadratic maps by means of graphic and experimental tools. Successive partitions of the antenna have been made to classify hyperbolic components. All the separators used in partitions are Misiurewicz points and the pre-period and period of these points have been determined.

Regularity and other properties of absolutely continuous invariant measures for the quadratic family

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.