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A study of a meromorphic perturbation of the sine family

Domínguez Patricia, Vázquez Josué

Demonstratio Mathematica

2022

Vol. 55, nr 1

8--27

We study the dynamics of a meromorphic perturbation of the family λsinz by adding a pole at zero and a parameter μ , that is, fλ,μ(z)=λsinz+μ/z , where λ,μ∈C⧹{0} . We study some geometrical properties of fλ,μ and prove that the imaginary axis is invariant under fn and belongs to the Julia set when ∣λ∣≥1 . We give a set of parameters (λ,μ) , such that the Fatou set of fλ,μ has two super-attracting domains. If λ=1 and μ∈(0,2) , the Fatou set of f1,μ has two attracting domains. Also, we give parameters λ,μ such that ±π/2 are fixed points of fλ,μ and the Fatou set of fλ,μ contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of fλ,μ , where the Fatou set contains two types of domains, for λ,μ given.

Rays to renormalizations

Levin Genadi

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 2

133--149

Let KP be the filled Julia set of a polynomial P, and Kf the filled Julia set of a renormalization f of P. We show, loosely speaking, that there is a finite-to-one function λ from the set of P-external rays having limit points in Kf onto the set of f-external rays to Kf such that R and λ(R) share the same limit set. In particular, if a point of the Julia set Jf = δKf of a renormalization is accessible from C \ Kf then it is accessible through an external ray of P (the converse is obvious). Another interesting corollary is that a component of KP \ Kf can meet Kf only in a single (pre-)periodic point. We also study a correspondence induced by λ on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set Kp ) of some results of Levin and Przytycki (1996), Blokh et al. (2016) and Petersen and Zakeri (2019) where it is assumed that Kp is disconnected and Kf is a periodic component of Kp .

On Lebesgue measure and Hausdorff dimension of Julia sets of real periodic points of renormalization

Dudko Artem

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 2

151--168

We give a new sufficient condition for the Julia set of a real analytic function which is a periodic point of renormalization to have Hausdorff dimension less than 2. This condition can be verified numerically. We present results of computer experiments suggesting that this condition is satisfied for real periodic points of renormalization with low periods. Our results support the conjecture that all real Feigenbaum maps have Julia sets of Hausdorff dimension less than 2.