The Julia set of transcendental shift-like maps on C^N is non-empty, and there exists an example with an escaping wandering domain unlike the polynomial case.
Wendt, The entropy of entire transcendental functions, arXiv:2011.02163 [math.DS]
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Two remarks on transcendental shift-like maps on $\mathbb{C}^N$
The Julia set of transcendental shift-like maps on C^N is non-empty, and there exists an example with an escaping wandering domain unlike the polynomial case.