Thickness of Julia sets of Feigenbaum polynomials with high order critical points

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point. To cite this article: G. Levin, G. ?wi?tek, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Local connectivity of Julia sets for a family of biquadratic polynomials

The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z~2-2c~2)z~2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.     

Recurrent critical points and typical limit sets of rational maps

We consider a rational map of the Riemann sphere with normalized Lebesgue measure and show that if there is a subset of the Julia set of positive -measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then for -a.e. point and is conservative, ergodic and exact.

     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

     

Multiplicities of fixed points of holomorphic maps in several complex variables

Let Δ^υ be the unit ball in ℂ^υ with center 0 (the origin of υ) and let F:Δ^υ→ℂ^υbe a holomorphic map withF(0) = 0. This paper is to study the fixed point multiplicities at the origin 0 of the iteratesF ⁱ =F∘⋯∘F (i times),i = 1,2,.... This problem is easy when υ = 1, but it is very complicated when υ > 1. We will study this problem generally.     

Building blocks for Quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

     

Mandelbrot-like sets in dynamical systems with no critical points

We report the discovery of an infinite quantity of Mandelbrot-like sets in the real parameter space of the Hénon map, a bidimensional diffeomorphism not obeying the Cauchy–Riemann conditions and having no critical points. For practical applications, this result shows to be possible to stabilize infinitely many complex phases by tuning real parameters only. To cite this article: A. Endler, J.A.C. Gallas, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings

Let △n be the ball |x| 1 in the complex vector space C n , let f :△n→ C n be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, . . . , 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1 :△n→ C n sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f M 1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point. According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0)≥1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true. In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0)≥2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0)≥2 for all holomorphic mappings f :△2 → C 2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M .     

Continuity of Julia sets

A sufficient and necessary condition is given for the continuity of Julia sets in the space of all rational maps with degreek>1. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing

In this paper we study a semi-Kolmogorov type of population model, arising from a predator–prey system with indirect effects. In particular we are interested in investigating the population dynamics when the indirect effects are time dependent and periodic. We first prove the existence of a global pullback attractor. We then estimate the fractal dimension of the attractor, which is done for a subclass by using Leonov’s theorem and constructing a proper Lyapunov function. To have more insights about the dynamical behavior of the system we also study the coexistence of the three species. Numerical examples are provided to illustrate all the theoretical results.     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

Ruelle operators with rational weights for Julia sets

LetR be a rational function with nonempty set of normality that consists of basins of attraction only and let

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors theorem.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points I

It is known that the Julia set of the Newton method of a non-constantpolynomial is connected (Mitsuhiro Shishikura, Preprint, 1990,M/90/37, Inst. Hautes Études Sci.). This is, in fact,a consequence of a much more general result that establishesthe relationship between simple connectivity of Fatou componentsof rational maps and fixed points which are repelling or parabolicwith multiplier 1. In this paper we study Fatou components oftranscendental meromorphic functions; that is, we show the existenceof such fixed points, provided that immediate attractive basinsor preperiodic components are multiply connected.     

On Julia sets concerning phase transitions

The sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family of rational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures of these sets are described completely.     

On the possibility of globally stabilizing invariant sets

In this paper we study the possibility of globally stabilizinginvariant sets of autonomous continuous nonlinear systems bythe state (output) feedback control law. We show that the topologicalstructure of invariant sets can give rise to obstruction tothe existence of a continuous control law for global stabilizationof invariant sets.     

Not all Julia sets are quasi-self-similar

We show that there exist rational functions, whose Julia set fails to be quasi-self-similar.