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Julia sets of random exponential maps

Volume 255 / 2021

Krzysztof Lech Fundamenta Mathematicae 255 (2021), 159-180 MSC: Primary 37F10. DOI: 10.4064/fm959-10-2020 Published online: 19 April 2021

Abstract

For a bounded sequence $\omega = ( \lambda _n )_{n = 1}^{\infty }$ of positive real numbers we consider the exponential functions $f_{\lambda _n} (z) = \lambda _n e^z$ and the compositions $F_{\omega }^n := f_{\lambda _n} \circ f_{\lambda _{n-1}} \circ \cdots \circ f_{\lambda _1}$. The definitions of Julia and Fatou sets are naturally generalized to this setting. We study how the Julia set depends on the sequence $\omega $. Among other results, we prove that for the sequence $\lambda _n = {1}/{e} + {1}/{n^p}$ with $p \lt {1}/{2}$, the Julia set is the whole plane.

Authors

  • Krzysztof LechFaculty of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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