Global dynamics of the real secant method

We investigate the root finding algorithm given by the secant method applied to a real polynomial p as a discrete dynamical system defined on R2. We study the shape and distribution of the basins of attraction associated to the roots of p, and we also show the existence of other stable dynamics that might affect the efficiency of the algorithm. Finally we extend the secant map to the punctured torus T2∞ and which allow us to better understand the dynamics of the secant method near ∞ and facilitate the use of the secant map as a method to find all roots of a polynomial

The secant map applied to a real polynomial with multiple roots

We investigate the plane dynamical system given by the secant map applied to a polynomial p having at least one multiple root of multiplicity d > 1. We prove that the local dynamics around the fixed points associated to the roots of p depend on the parity of d

Topological properties of the immediate basins of attraction for the secant method

We study the discrete dynamical system defined on a subset of R given by the iterates of the secant method applied to a real polynomial p. Each simple real root α of p has associated its basin of attraction A(α) formed by the set of points converging towards the fixed point (α, α) of S. We denote by A(α) its immediate basin of attraction, that is, the connected component of A(α) which contains (α, α). We focus on some topological properties of A(α), when α is an internal real root of p. More precisely, we show the existence of a 4-cycle in ∂A(α) and we give conditions on p to guarantee the simple connectivity of A(α)

Wandering domains for composition of entire functions

C. Bishop in [Bis, Theorem 17.1] constructs an example of an entire function $f$ in class $\mathcal {B}$ with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, $f$ has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps $f$ and $g$ in class $\mathcal {B}$ such that the Fatou set of $f \circ g$ has a wandering domain, while all Fatou components of $f$ or $g$ are preperiodic. This complements a result of A. Singh in [Sin03, Theorem 4] and results of W. Bergweiler and A.Hinkkanen in [BH99] related to this problem

Joining polynomial and exponential combinatorics for some entire maps

We consider families of entire transcendental maps given by Fλ,m (z) = λzm exp (z) where m ≥ 2. All these maps have a superattracting fixed point at z = 0 and a free critical point at z = −m. In parameter planes we focus on the capture zones, i.e., we consider λ values for which the free critical point belongs to the basin of attraction of z = 0. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane

An effective algorithm to compute Mandelbrot sets in parameter planes

Agraïments: The second author is partially supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168 and the MDM-2014-445 Maria de Maeztu.In 2000 McMullen proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algo- rithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps

An effective algorithm to compute Mandelbrot sets in parameter planes.

McMullen in 2000 proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps

On Newton's method applied to real polynomials

Agraïments: The first author would also like to thank the Ministerio de Ciencia y Innovacion of Spain for the financial support when visiting Boston University.It is known that if we apply Newton's method to the complex function F(z) = P(z)e Q(z), with deg(Q) > 2, then the immediate basin of attraction of the roots of P has finite area. In this paper we show that under certain conditions on P, if deg(Q) = 1, then there is at least one immediate basin of attraction having infinite area

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's method of a non- constant polynomial is connected ([18]). This is, in fact, a consequence of a much more general result that establishes the relationship between simple connectivity of Fatou components of rational maps and fixed points which are repelling or parabolic with multiplier 1. In this paper we study Fatou components of transcendental mero- morphic functions, namely, we show the existence of such fixed points provided that immediate attractive basins or preperiodic components be multiply connected

Accesses to infinity from Fatou components

Agraïments: Supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168.We study the boundary behaviour of a meromorphic map f: \C C on its invariant simply connected Fatou component U. To this aim, we develop the theory of accesses to boundary points of U and their relation to the dynamics of f. In particular, we establish a correspondence between invariant accesses from U to infinity or weakly repelling points of f and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps