We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class B), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1.
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(z 0 ) of an attracting fixed point z 0 . We study the structure of the Julia set of f , which is equal to the boundary of B(z 0 ), and the behaviour of the Riemann mapping ϕ onto B(z 0 ) using the technique of geometric coding trees of preimages of points from B(z 0 ). 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.
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