Bedford, Eric Kim, Kyounghee
Published in
The Journal of Geometric Analysis

For any polynomial diffeomorphism f of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb C}^2$$\end{document} with positive entropy, the Julia set of f is neve...

Ahn, Taeyong
Published in
The Journal of Geometric Analysis

We consider a finite composition of generalized Hénon mappings f:C2→C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {f}}:{\mathbb {C}}^2\rightarrow {\mathb...

Déserti, Julie Leguil, Martin
Published in
The Journal of Geometric Analysis

The polynomial automorphisms of the affine plane have been studied a lot: if f is such an automorphism, then either f preserves a rational fibration, has an uncountable centralizer, and its first dynamical degree equals 1, or f preserves no rational curves, has a countable centralizer, and its first dynamical degree is >1. In higher dimensions ther...

Gao, Yan

The core entropy of polynomials, recently introduced by W. Thurston, is a dy-namical invariant extending topological entropy for real maps to complex polynomials, whence providing a new tool to study the parameter space of polynomials. The base is a combinatorial algorithm allowing for the computation of the core entropy given by Thurston, but with...

Matthies, Gunar Salimi, Mehdi Sharifi, Somayeh Varona, Juan Luis
Published in
Japan Journal of Industrial and Applied Mathematics

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and th...

Schleicher, Dierk
Published in
Arnold Mathematical Journal

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometr...

Bajo, Ignacio
Published in
Qualitative Theory of Dynamical Systems

We study the existence of invariants for the family of systems in an open domain D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}$$\end{document} of Rn\doc...

Sharifi, Somayeh Ferrara, Massimiliano Salimi, Mehdi Siegmund, Stefan
Published in
Open Mathematics

In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of method...

Campos, Beatriz Cordero, Alicia Torregrosa, Juan R. Vindel, Pura
Published in
Numerical Algorithms

The study of the dynamical behaviour of the operators defined by iterative methods help us to know more deeply the regions where these methods have a good performance. In this paper, we follow the dynamical study of a multipoint variant of the known Chebyshev-Halley’s family, showing the existence of attractive periodic orbits of period 2 for some ...

Levin, Genadi Przytycki, Feliks Shen, Weixiao
Published in
Inventiones mathematicae

We prove that for any polynomial map with a single critical point its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets...