Page 1 Next

Displaying 1 – 20 of 43

Showing per page

A dynamical invariant for Sierpiński cardioid Julia sets

Paul Blanchard, Daniel Cuzzocreo, Robert L. Devaney, Elizabeth Fitzgibbon, Stefano Silvestri (2014)

Fundamenta Mathematicae

For the family of rational maps zⁿ + λ/zⁿ where n ≥ 3, it is known that there are infinitely many small copies of the Mandelbrot set that are buried in the parameter plane, i.e., they do not extend to the outer boundary of this set. For parameters lying in the main cardioids of these Mandelbrot sets, the corresponding Julia sets are always Sierpiński curves, and so they are all homeomorphic to one another. However, it is known that only those cardioids that are symmetrically located in the parameter...

A family of critically finite maps with symmetry.

Scott Crass (2005)

Publicacions Matemàtiques

The symmetric group Sn acts as a reflection group on CPn-2 (for n>=3).Associated with each of the (n2) transpositions in Sn is an involution on CPn-2 that pointwise fixes a hyperplane -the mirrors of the action. For each such action, there is a unique Sn-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations...

A parabolic Pommerenke-Levin-Yoccoz inequality

Xavier Buff, Adam L. Epstein (2002)

Fundamenta Mathematicae

In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...

Borel summation and splitting of separatrices for the Hénon map

Vassili Gelfreich, David Sauzin (2001)

Annales de l’institut Fourier

We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation...

Combinatorics of distance doubling maps

Karsten Keller, Steffen Winter (2005)

Fundamenta Mathematicae

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̅(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates f n of a distance doubling map f exhibit “distance doubling behavior”. The results include well known statements for h related to the structure of the Mandelbrot set M. For h̅ they suggest some analogies to the structure of...

Conformal measures and matings between Kleinian groups and quadratic polynomials

Marianne Freiberger (2007)

Fundamenta Mathematicae

Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of C₂ ∗ C₃ and quadratic polynomials carries δ-conformal measures, and that if the correspondence is geometrically finite then the real number δ is equal to the Hausdorff dimension of the limit set. Moreover, when f is the limit of a pinching deformation f t 0 t < 1 we give sufficient conditions for the dynamical convergence of f t .

Contractive curves.

Bonifant, Araceli, Dabija, Marius (2002)

International Journal of Mathematics and Mathematical Sciences

Déformation J-équivalente de polynômes géometriquement finis

Peter Haïssinsky (2000)

Fundamenta Mathematicae

Any geometrically finite polynomial f of degree d ≥ 2 with connected Julia set is accessible by structurally stable sub-hyperbolic polynomials of the same degree. Moreover, they are topologically conjugate to f on their Julia sets.

Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

Julian Ławrynowicz, Tatsuro Ogata, Osamu Suzuki (2009)

Banach Center Publications

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III)....

Diffusion to infinity for periodic orbits in meromorphic dynamics

Janina Kotus, Grzegorz Świątek (2002)

Fundamenta Mathematicae

A small perturbation of a rational function causes only a small perturbation of its periodic orbits. We show that the situation is different for transcendental maps. Namely, orbits may escape to infinity under small perturbations of parameters. We show examples where this "diffusion to infinity" occurs and prove certain conditions under which it does not.

Dynamic classification of escape time Sierpiński curve Julia sets

Robert L. Devaney, Kevin M. Pilgrim (2009)

Fundamenta Mathematicae

For n ≥ 2, the family of rational maps F λ ( z ) = z + λ / z contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ ≥ 3. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of n and κ.

Dynamics of symmetric holomorphic maps on projective spaces.

Kohei Ueno (2007)

Publicacions Matemàtiques

We consider complex dynamics of a critically finite holomorphic map from Pk to Pk, which has symmetries associated with the symmetric group Sk+2 acting on Pk, for each k ≥1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.

From Newton’s method to exotic basins Part I: The parameter space

Krzysztof Barański (1998)

Fundamenta Mathematicae

This is the first part of the work studying the family 𝔉 of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of 𝔉 and give a detailed study of the subfamily 2 consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in 2 from Newton maps to maps with so-called exotic basins.

From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

Krzysztof Barański (2001)

Fundamenta Mathematicae

This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a fixed point of multiplier 1. Then it bifurcates into two paths of Mandelbrot-like...

High-order phase transitions in the quadratic family

Daniel Coronel, Juan Rivera-Letelier (2015)

Journal of the European Mathematical Society

We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as x exp ( x 2 ) near x = 0 , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets J 0 , σ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of J 0 , σ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS ϕ σ , α n , k is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of J 0 , σ . The parameter...

Currently displaying 1 – 20 of 43

Page 1 Next