Real cross section of the connectedness locus of the family of polynomials ( z 2 n + 1 + a ) 2 n + 1 + b Hisashi ISHIDA Tsubasa KAMEI

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

The Combinatorial Rigidity Conjecture Is False for Cubic Polynomials

We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995. Introduction and result Let Pd = {z + ad−2z + · · · + a0} ↔ Cd−1 be the space of monic centered polynomials of degree d > 1. Our object is to show that there exists a cubic polynomial...

On the Dynamics of the Family axd(x − 1) + x

In this paper we consider the dynamics of the real polynomials of degree d + 1 with a fixed point of multiplicity d ≥ 2. Such polynomials are conjugate to fa,d(x) = axd(x−1)+x, a ∈ R{0}, d ∈ N. Our aim is to study the dynamics fa,d in some special cases.

Dynamics and Rational Maps a New Section for Chapter 7 of the Arithmetic of Dynamical Systems

N be a rational map, where f 0 ,. .. , f N are homogeneous polynomials of degree d with no common factors. Then φ is defined at all points not in the indeterminacy locus Z(φ) = { P ∈ P N : f 0 (P) = · · · = f N (P) = 0 }. The map φ is said to be dominant if the image φ (P N Z(φ)) is Zariski dense, i.e., does not lie in a proper algebraic subset of P N. Alternatively, the map φ is dominant if th...

Multicorns are not path connected

The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z 7! z2 + c. We prove that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. We extend this discussion more generally for antiholomorphic unicritical polynomials of degrees d 2 and their connectedness loci, known as multicorns.