Dynamics Of Polynomials Research Articles

Dynamics of quadratic polynomials and rational points on a curve of genus 4

Let f t ( z ) = z 2 + t f_t(z)=z^2+t . For any z ∈ Q z\in \mathbb {Q} , let S z S_z be the collection of t ∈ Q t\in \mathbb {Q} such that z z is preperiodic for f t f_t . In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S z S_z over z ∈ Q z\in \mathbb {Q} . In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C C of genus 4 4 defined over Q \mathbb {Q} . We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian J J of C C . We give two proofs that the rank is 1 1 : an analytic proof, which is conditional on the BSD rank conjecture for J J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 12 and degree 24 24 , respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J J .

Combining rational maps and Kleinian groups via orbit equivalence

AbstractWe develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion‐free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are orbit equivalent to Fuchsian punctured sphere groups. We call these higher Bowen–Series maps. The existence of this class ensures that the Teichmüller space of matings has one component corresponding to Bowen–Series maps and one corresponding to higher Bowen–Series maps. We also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers boundary groups that are mateable in our framework.

Roots and Dynamics of Octonion Polynomials

This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed, especially over fields of characteristic not 2. We then turn to study the dynamics of monic quadratic real octonion polynomials, classifying the fixed points into attracting, repelling and ambivalent, and concluding with a discussion on the behavior of pseudo-periodic points.

Interval Uncertainty Quantification for the Dynamics of Multibody Systems Combing Bivariate Chebyshev Polynomials with Local Mean Decomposition

Interval quantification for multibody systems can provide an accurate dynamic prediction and a robust reliability design. In order to achieve a robust numerical model, multiple interval uncertain parameters should be considered in the uncertainty propagation of multibody systems. The response bounds obtained by the bivariate Chebyshev method (BCM) present an intensive deterioration with the increase of time history in the interval dynamic analysis. To circumvent this problem, a novel method that combines the bivariate Chebyshev polynomial and local mean decomposition (BC-LMD) is proposed in this paper. First, the multicomponent response of the system was decomposed into the sum of several mono-component responses and a residual response, and the corresponding amplitude and phase of the mono-component were obtained. Then, the bivariate function decomposition was performed on the multi-dimensional amplitude, phase, and residual to transform a high-dimensional problem into several one-dimensional and two-dimensional problems. Subsequently, a low order Chebyshev polynomial can be used to construct surrogate models for the multi-dimensional amplitude, phase, and residual responses. Then, the entire coupling surrogate model of the system can be established, and the response bounds of the system can be enveloped. Illustrative examples of a slider-crank mechanism and a double pendulum are presented to demonstrate the effectiveness of the proposed method. The numerical results indicate that, compared to the BCM, BC-LMD can present a tight envelope in the long time-dependent dynamic analysis under multiple interval parameters.

Near Parabolic Renormalization for Unicritical Holomorphic Maps

Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.

Markov processes and some PCF quadratic polynomials

For any $$n\ge 1$$ , let $$T_n$$ be the complete binary rooted tree of height n, and $$f(x)=(x+a)^2-a-1$$ such that $$a\ne \pm b^2$$ for any $$b\in \mathbb {Z}$$ . In [5], Jones and Boston empirically observed that iteratively applying a certain Markov process on the factorization types of f gives rise to certain permutation groups $$M_n(f)\le \text {Aut}(T_n)$$ for $$n\le 5$$ . We prove a refined version of this phenomenon for all n, and for all the irreducible post-critically finite quadratic polynomials with integer coefficients, except for certain conjugates of $$x^2-2$$ . We do this by constructing these groups explicitly. Although there have already been some conjectures relating the Markov processes to the dynamics of quadratic polynomials, our results are the first to prove such a connection. If $$f(x)\in \mathbb {Z}[x]$$ is a post-critically finite quadratic polynomial, and $$G_n(f)$$ is the Galois group of $$f^n$$ over $$\mathbb {Q}(i)$$ , then we conjecture that for all $$n\ge 1$$ , $$M_n(f)$$ contains a subgroup which is conjugate in $$Aut(T_n)$$ to $$G_n(f)$$ , analogous to the role of Mumford-Tate groups in classical arithmetic geometry. We provide evidence that this is implied by a purely group theoretical statement.

Conformal equivalence of measures and dynamics of orthogonal polynomials

ABSTRACTWe introduce a notion of asymptotically orthonormal polynomials for a Borel measure μ with compact non-polar support in . Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials, see also Christiansen et al. [Julia sets of orthogonal polynomials, Potential Anal. 50(3) (2019), pp. 401–413]. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study the dynamics of sequences of asymptotically orthonormal polynomials for the pullback measure under affine mappings φ. We prove that the sequences of Julia sets and filled Julia sets of affine deformations of sequences of asymptotically orthonormal polynomials for μ also have the same convergence properties as the Julia sets and filled Julia sets of the orthonormal polynomials. This leads to theorems on the convergence properties of affine deformations of the family of iterates of any fixed monic centred polynomial and, in the case the polynomial is hyperbolic, on the corresponding family of affine parameter spaces.

Dynamics of quadratic polynomials, III Parapuzzle and SBR measures

Dynamics of quadratic polynomials, III Parapuzzle and SBR measures

Dynamics in the Eremenko-Lyubich class

The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials – in other words transcendental entire functions – is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics.In this paper we survey the dynamics of functions in the Eremenko-Lyubich class,B\mathcal {B}. Among transcendental entire functions, those in this class have properties that make their dynamics most “similar” to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of classB\mathcal {B}functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases.We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.

Statistical properties of quadratic polynomials with a neutral fixed point

We describe the statistical properties of the dynamics of the quadratic polynomials P_\alpha(z)=e^{2\pi \alpha \mathbf{i}} z+ z^2 on the complex plane, with \alpha of high type. In particular, we show that these maps are uniquely ergodic on their measure-theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behaviour of typical orbits in the Julia set. This confirms a conjecture of Pérez–Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.

The classification of polynomial basins of infinity

We consider the problem of classifying the dynamics of complex polynomials f : C ! C restricted to the basins of innity X(f). We synthesize existing combinatorial tools | tableaux, trees, and laminations | into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to innity, we obtain a complete description of the set of topological conjugacy classes with given pictograph. For arbitrary polynomials, we compute the total number of topological conjugacy classes of basins (f;X(f)) with a given pictograph. We also dene abstract pictographs and prove that every abstract pictograph is realized by a polynomial. Extra details are given in degree 3, and we provide examples that show the pictograph is ner invariant than both

A local–global principle in the dynamics of quadratic polynomials

Let [Formula: see text] be a number field, [Formula: see text] a quadratic polynomial, and [Formula: see text]. We show that if [Formula: see text] has a point of period [Formula: see text] in every non-archimedean completion of [Formula: see text], then [Formula: see text] has a point of period [Formula: see text] in [Formula: see text]. For [Formula: see text] we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over [Formula: see text] for which this local–global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn–Poonen–Schaefer in the case [Formula: see text]. More precisely, we show that for every quadratic polynomial [Formula: see text] there exist infinitely many primes [Formula: see text] such that [Formula: see text] does not have a point of period four in the [Formula: see text]-adic field [Formula: see text]. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period five.

Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods.

Complex dynamics with focus on the real part

We consider the dynamics of holomorphic polynomials in$\mathbb{C}$. We show that the ergodic properties of the map can be seen already from the real parts of the orbits.

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

A transcendental entire function fa(z) = z +e z +a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials Pa,d(z) = (1+a/d)z +(1+z/d) d+1 +a, which converges uniformly on compact sets to fa as d → ∞. We show its dynamical convergence under a certain assumption, even though fa has a Baker domain or a wandering domain. We also investigate the parameter spaces of fa and Pa,d.

Real cubic polynomials with a fixed point of multiplicity two

The aim of this paper is to study the dynamics of the real cubic polynomials that have a fixed point of multiplicity two. Such polynomials are conjugate to fa(x)=ax2(x−1)+x, a≠0. We will show that when a>0 and x≠1, then |fan(x)| converges to 0 or ∞ and, if a<0 and a belongs to a special subset of the parameter space, then there is a closed invariant subset Λa of R on which fa is chaotic.

English

The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z n z + c = 0 for n 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. In order to solve this function by Jungck –type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = z n + c and Sz = z. Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated Relative Mandelbrot sets and Relative Julia sets.

Wandering Fatou Components and Algebraic Julia Sets

We study the dynamics of polynomials with coefficients in a non-Archimedean field $K,$ where $K$ is a field containing a dense subset of algebraic elements over a discrete valued field $k.$ We prove that every wandering Fatou component is contained in the basin of a periodic orbit. We obtain a complete description of the new Julia set points that appear when passing from $K$ to the Berkovich line over $K$. We give a dynamical characterization of polynomials having algebraic Julia sets. More precisely, we establish that a polynomial with algebraic coefficients has algebraic Julia set if every critical element is nonrecurrent.

The size of quadratic p-adic linearization disks

We find the exact radius of linearization disks at indifferent fixed points of quadratic maps in Cp. We also show that the radius is invariant under power series perturbations. Localizing all periodic orbits of these quadratic-like maps we then show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics of polynomials over Cp where the boundary of the linearization disk does not contain any periodic point.

Cooperation principle, stability and bifurcation in random complex dynamics

We investigate the random dynamics of rational maps and the dynamics of semigroups of rational maps on the Riemann sphere Cˆ. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in Cˆ, due to the automatic cooperation of the generators. We investigate the iteration and spectral properties of transition operators acting on the space of (Hölder) continuous functions on Cˆ. We also investigate the stability and bifurcation of random complex dynamics. We show that the set of stable systems is open and dense in the space of random dynamical systems of polynomials. Moreover, we prove that for a stable system, there exist only finitely many minimal sets, each minimal set is attracting, and the orbit of a Hölder continuous function on Cˆ under the transition operator tends exponentially fast to the finite-dimensional space U of finite linear combinations of unitary eigenvectors of the transition operator. Combining this with the perturbation theory for linear operators, we obtain that for a stable system constructed by a finite family of rational maps, the projection to the space U depends real-analytically on the probability parameters. By taking a partial derivative of the function of probability of tending to a minimal set with respect to a probability parameter, we introduce a complex analogue of the Takagi function, which is a new concept.