/pdf/p-adic-analogues-of-the-mandelbrot-set-3j9qtttkb0.pdf

Iteration of Rational Functions

p-adic Analogues of the Mandelbrot Set

background

/pdf/classification-of-periodic-fatou-components-for-rational-dx0b362ebn.pdf

Classification of periodic Fatou components for rational maps

Complex dynamics deals with the iteration of holomorphic functions. As is well- known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R^2 which are a composition of a quadratic polynomial and an affine stretch.

Quasiregular mappings of polynomial type in R^2

While the notion of roots of a quadratic polynomial is rudimentary in high school mathematics, that of its fixed points is uncommon. A real or complex number is a fixed point of a polynomial p (x) p (θ) = θ. The fact that the notion of fixed point of polynomials is not commonly covered in high school or undergraduate mathematics is surprising because the relevance of the fixed points of a quadratic can be demonstrated easily via iterative methods for the approximation of such numbers as , when the quadratic formula offers no remedy.

How Many Real Attractive Fixed Points Can A Polynomial Have

Given a polynomial p, the degree of its Chebyshev’s method \(C_p\) is determined. If p is cubic, then the degree of \(C_p\) is found to be 4, 6 or 7 and we investigate the dynamics of \(C_p\) in these cases. If a cubic polynomial p is unicritical or non-generic, then it is proved that the Julia set of \(C_p\) is connected. The family of all rational maps arising as the Chebyshev’s method applied to a cubic polynomial which is non-unicritical and generic is parametrized by the multiplier of one of its extraneous fixed points. Denoting a member of this family with an extraneous fixed point with multiplier \(\lambda \) by \(C_\lambda \), we have shown that the Julia set of \(C_\lambda \) is connected whenever \(\lambda \in [-1,1]\). 

/pdf/the-julia-sets-of-chebyshevs-method-with-small-degrees-1kwdszub.pdf

Iteration of Rational Functions

Citations

p-adic Analogues of the Mandelbrot Set

Classification of periodic Fatou components for rational maps

Quasiregular mappings of polynomial type in R^2

How Many Real Attractive Fixed Points Can A Polynomial Have

The Julia sets of Chebyshev’s method with small degrees

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