The W. Thurston algorithm applied to real polynomial maps

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON ℝ2

We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics of Newton maps [Formula: see text] associated to real polynomial maps [Formula: see text] with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map [Formula: see text] has no wandering domain, almost every point under [Formula: see text] is asymptotic to a fixed point and there is some non-empty open set of points whose [Formula: see text]-limit equals the set of non-regular points of the Julia set of [Formula: see text]. The first two points were proved by B. Barna in the real one-dimensional case.

Real Polynomial Maps and Singular Open Books at Infinity

We provide significant conditions under which we prove the existence of stable open book structures at infinity, i.e. on spheres $S^{m-1}_R$ of large enough radius $R$. We obtain new classes of real polynomial maps $\mathsf{R}^m \to \mathsf{R}^p$ which induce such structures.

Stability of Real Parametric Polynomial Discrete Dynamical Systems

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept ofcanonical polynomial mapswhich are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termedProduct Position Functionfor a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termedstability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

Milnor's conjecture on monotonicity of topological entropy: Results and questions

This chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.