Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

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.