Julia sets of random exponential maps. (English) Zbl 1490.37063
Summary: 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 \(p < {1}/{2}\), the Julia set is the whole plane.
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37H12 | Random iteration |
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