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Coding trees and boundaries of attracting basins for some entire maps

Let f be an entire transcendental map, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B(z0) of an attracting fixed point z0. We study the structure of the Julia set of f, which is equal to the boundary of B(z0), and the behaviour of the Riemann mapping φ onto B(z0) using the technique of geometric coding trees of preimages of points from B(z0). We show that for a given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow no faster than the iterates of an exponential function, then there exists a point in the Julia set with this itinerary. Moreover, we determine cluster sets for φ and show that φ has an unrestricted limit equal to ∞ at points of a dense uncountable set in the unit circle.