The iteration of cubic polynomials Part I: The global topology of parameter space

Abstract In [21], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.

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Purely Iterative Algorithms for Newton’s Maps and General Convergence

Mathematics ◽

10.3390/math8071158 ◽

2020 ◽

Vol 8 (7) ◽

pp. 1158

Author(s):

Sergio Amat ◽

Rodrigo Castro ◽

Gerardo Honorato ◽

Á. A. Magreñán

Keyword(s):

Fixed Points ◽

Large Class ◽

Parameter Space ◽

Critical Points ◽

Broad Class ◽

Iterative Algorithms ◽

Dynamical Behaviour ◽

Root Finding ◽

Cubic Polynomials ◽

General Convergence

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space.

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ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403008259 ◽

2003 ◽

Vol 13 (10) ◽

pp. 2825-2844 ◽

Cited By ~ 15

Author(s):

SHIZUO NAKANE ◽

DIERK SCHLEICHER

Keyword(s):

Parameter Space ◽

Bifurcation Diagrams ◽

The Family ◽

Real Analytic ◽

Cubic Polynomials

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

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The parameter space of cubic laminations with a fixed critical leaf

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.22 ◽

2016 ◽

Vol 37 (8) ◽

pp. 2453-2486

Author(s):

ALEXANDER BLOKH ◽

LEX OVERSTEEGEN ◽

ROSS PTACEK ◽

VLADLEN TIMORIN

Keyword(s):

Parameter Space ◽

Mandelbrot Set ◽

Quadratic Invariant ◽

The Family ◽

Combinatorial Model ◽

Cubic Polynomials ◽

Quadratic Case

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

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