scholarly journals The W. Thurston algorithm applied to real polynomial maps

2021 ◽  
Vol 25 (9) ◽  
pp. 179-199
Author(s):  
Araceli Bonifant ◽  
John Milnor ◽  
Scott Sutherland
Keyword(s):  
Effective Procedure ◽  
Real Polynomial ◽  
Polynomial Maps ◽  
Real Polynomial Maps

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

scholarly journals Real Polynomial Maps and Singular Open Books at Infinity

2016 ◽  
Vol 118 (1) ◽  
pp. 57 ◽  
Author(s):  
Raimundo N. Araújo Dos Santos ◽  
Ying Chen ◽  
Mihai Tibăr
Keyword(s):  
Open Book ◽  
Real Polynomial ◽  
Open Books ◽  
Polynomial Maps ◽  
Real Polynomial Maps

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.

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

2014 ◽  
Author(s):  
Sebastian van Strien
Keyword(s):  
Topological Entropy ◽  
State Of The Art ◽  
Rigidity Theorem ◽  
Real Polynomial ◽  
Open Problems ◽  
Polynomial Maps ◽  
History Of ◽  
The Subject ◽  
Quadratic Case ◽  
Real Polynomial Maps

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.

scholarly journals On the Euler characteristic of fibres of real polynomial maps

1998 ◽  
Vol 44 (1) ◽  
pp. 175-182 ◽  
Author(s):  
Adam Parusiński ◽  
Zbigniew Szafraniec
Keyword(s):  
Euler Characteristic ◽  
Real Polynomial ◽  
Polynomial Maps ◽  
Real Polynomial Maps

scholarly journals On open real polynomial maps

1996 ◽  
Vol 110 (3) ◽  
pp. 297-304 ◽  
Author(s):  
J.M. Gamboa ◽  
F. Ronga
Keyword(s):  
Real Polynomial ◽  
Polynomial Maps ◽  
Real Polynomial Maps

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

2007 ◽  
Vol 257 (4) ◽  
pp. 745-767 ◽  
Author(s):  
Carles Bivià-Ausina
Keyword(s):  
Real Polynomial ◽  
Polynomial Maps ◽  
Łojasiewicz Exponents ◽  
Real Polynomial Maps

CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON ℝ2

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950099 ◽  
Author(s):  
ROBERTO DE LEO
Keyword(s):  
Julia Set ◽  
Rational Functions ◽  
Real Polynomial ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Open Set ◽  
Complex Roots ◽  
Wandering Domain ◽  
Set Of Points ◽  
Real Polynomial Maps

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.

scholarly journals Toward Effective Detection of the Bifurcation Locus of Real Polynomial Maps

2016 ◽  
Vol 17 (3) ◽  
pp. 837-849 ◽  
Author(s):  
Luis Renato G. Dias ◽  
Susumu Tanabé ◽  
Mihai Tibăr
Keyword(s):  
Real Polynomial ◽  
Bifurcation Locus ◽  
Polynomial Maps ◽  
Real Polynomial Maps

scholarly journals Hyperbolicity of the invariant sets for the real polynomial maps

2012 ◽  
Vol 45 (3) ◽  
pp. 314-324 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Invariant Sets ◽  
Real Polynomial ◽  
The Real ◽  
Polynomial Maps ◽  
Real Polynomial Maps

scholarly journals Stability of Real Parametric Polynomial Discrete Dynamical Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Fermin Franco-Medrano ◽  
Francisco J. Solis
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Special Function ◽  
Real Polynomial ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Exact Boundary ◽  
The Stability ◽  
Real Polynomial Maps

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.

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