We present a one-parameter family Fλ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of Fλ. Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and $\infty$. The family Fλ is the simplest in this class, and its parameter space shows open sets of λ-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.

Recently, Benini et al showed that, in simply connected wandering domains of entire functions, all pairs of orbits behave in the same way relative to the hyperbolic metric, thus giving us our first insight into the general internal dynamics of such domains. The author proved in a recent paper [G. R. Ferreira. Multiply connected wandering domains of meromorphic functions: internal dynamics andconnectivity. J. Lond. Math. Soc. (2) 106 (2022), 1897–1919] that the same is not true for multiply connected wandering domains, a natural question is how inhomogeneous multiply connected wandering domains can be. We give an answer to this question, in that we show that uniform dynamics inside an open subset of the domain generalizes to the whole wandering domain. As an application of this result, we construct the first example of a meromorphic function with a semi-contracting infinitely connected wandering domain.

In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function $\phi\in K(z)$ defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.

2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.