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We present a one-parameter family $F_\lambda$ of transcendental entirefunctions with zeros, whose Newton's method yields wandering domains,coexisting with the basins of the roots of $F_\lambda$. Wandering domains forNewton maps of zero-free functions have been built before by, e.g., Buff andR\"uckert based on the lifting method. This procedure is suited to our Newtonmaps as members of the class of projectable functions (or maps of thecylinder), i.e. transcendental meromorphic functions $f(z)$ in the complexplane that are semiconjugate, via the exponential, to some map $g(w)$, whichmay have at most a countable number of essential singularities.  In this paper we make a systematic study of the general relation (dynamicaland otherwise) between $f$ and $g$, and inspect the extension of thelogarithmic lifting method of periodic Fatou components to our context,especially for those $g$ of finite-type. We apply these results to characterizethe entire functions with zeros whose Newton's method projects to some map $g$which is defined at both $0$ and $\infty$. The family $F_\lambda$ is thesimplest in this class, and its parameter space shows open sets of$\lambda$-values in which the Newton map exhibits wandering or Baker domains,in both cases regions of initial conditions where Newton's root-finding methodfails.

Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps