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Joining Polynomial and Exponential Combinatorics for Some Entire Maps

Antonio Garijo, Xavier Jarque, and Mónica Moreno Rocha

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We consider families of entire transcendental maps given by $F_{\lambda,m} (z) = \lambda z^m \exp(z) $ where $m \ge 2$. All these maps have a superattracting fixed point at $z=0$ and a free critical point at $z=-m$. In parameter planes we focus on the capture zones, i.e., we consider $\lambda$ values for which the free critical point belongs to the basin of attraction of $z=0$. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane.

Article information

Publ. Mat., Volume 54, Number 1 (2010), 113-136.

First available in Project Euclid: 8 January 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F20: Combinatorics and topology 30D20: Entire functions, general theory

Julia sets polynomial-like maps combinatorial dynamics


Garijo, Antonio; Jarque, Xavier; Moreno Rocha, Mónica. Joining Polynomial and Exponential Combinatorics for Some Entire Maps. Publ. Mat. 54 (2010), no. 1, 113--136.

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