Coding trees and boundaries of attracting basins for some entire maps

Abstract

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.

Trees and hairs for some hyperbolic entire maps of finite order

“…The next fact follows easily from (1), the univalency of f on logarithmic tracts and the fact that L r s does not contain vertical segments of length 2π (see [4] for details). It shows that F is uniformly expanding (note that, unlike the rational case, hyperbolicity does not imply expanding for a general entire map).…”

“…In fact, the clusters of ϕ are either singletons {∞} or the sets of points in the Julia set J(f ) sharing the same symbolic itinerary (together with ∞). Both cases happen on dense sets in the unit circle (see [4]). …”

“…6. This paper is the second part of the series of three papers concerning the properties of the Julia set of hyperbolic entire maps with one Fatou component. In the first part [4] we show (without the finite order assumption) that for given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow not faster than the iterates of an exponential function, then the itinerary is allowable, i.e. there exists a point in the Julia set with this itinerary.…”

“…, R}. We use (slightly modified) logarithmic coordinates introduced by Eremenko and Lyubich (see [13,4]). Let …”

Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

“…In contrast to the case of hyperbolic rational functions, the dynamics of a general hyperbolic transcendental function on the Julia set can not be represented by a symbol dynamics. However, some particular entire functions act on a dynamically important subset of the Julia set, the so called set of "landing-" or "end-"points, in a similar way as countable Markov shifts that we consider in here (see [2,3]). …”