A generalization of Douady's formula
Gamaliel Blé Carlos Cabrera
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6183-6188 doi: 10.3934/dcds.2017267

The Douady's formula was defined for the external argument on the boundary points of the main hyperbolic component $W_0$ of the Mandelbrot set $M$ and it is given by the map $T(θ)=1/2+θ/4$. We extend this formula to the boundary of all hyperbolic components of $M$ and we give a characterization of the parameter in $M$ with these external arguments.

keywords: Mandelbrot set quadratic polynomials tuning map summability condition absolutely continuous invariant measures
Semigroup representations in holomorphic dynamics
Carlos Cabrera Peter Makienko Peter Plaumann
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1333-1349 doi: 10.3934/dcds.2013.33.1333
We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.
keywords: Semigroup representations complex polynomials holomorphic dynamics holomorphic correspondences. rational maps

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