Local connectivity of boundaries of tame Fatou components of meromorphic functions

We prove local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class $\mathcal{B}$, and the boundaries of the basins (possibly unbounded) are allowed to contain an infinite number of post-singular values, as well as the essential singularity at infinity. A basic assumption is that the unbounded parts of the basins are contained in regions which we call repelling petals at infinity', where the map exhibits a kind of parabolic’ behaviour. In particular, our results apply to a wide class of Newton’s methods for transcendental entire maps. As an application, we prove local connectivity of the Julia set of Newton’s method for $\sin z$, providing the first non-trivial example of a locally connected Julia set of a transcendental map outside class $\mathcal{B}$, with an infinite number of unbounded Fatou components.