Trees and hairs for some hyperbolic entire maps of finite order

Abstract

Let f be an entire transcendental map of finite order, such that all the singularities of f −1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from C. This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps.

Dynamic rays of bounded-type entire functions

“…After our proof of Theorem 1.2 was first announced, Barański [Bar07] independently obtained a proof of this result for hyperbolic finite-order functions f ∈ B whose Fatou set consists of a single basin of attraction. (In fact, Barański showed that for these functions every component of the Julia set is a curve to ∞; compare Theorem 5.10.)…”

“…Let us also say that F is of disjoint type if T ⊂ H. It is easy to see that a function f ∈ B has a disjoint-type logarithmic transform if and only if S(f ) is contained in the immediate basin of an attracting fixed point of f . (This is the setting considered by Barański [Bar07].) Note that we might not be able to normalize such a function in the above-mentioned manner without losing the disjoint-type property.…”

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

“…Proof of Theorem 5.8 (2). Since A, H ≥ 2, we have that 1 1 + h x : h x α < 1/4 ⊂ C x , for all x ∈ X.…”

“…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]). …”

“…x g = ηL N θ N (x) χ l,θ N (x) + (1 − η l,θ N (x) )ηχ l,θ 2N (x) + (1 − η l,θ N (x) )(1 − η l,θ 2N (x) )g (2) for some g (2) θ 2N (x) ∈ Λ θ 2N (x),0 . Inductively, it follows that for every k ≥ 1 there is a function g (k)…”