Lebesgue measure of escaping sets of entire functions
نویسندگان
چکیده
منابع مشابه
Lebesgue measure of escaping sets of entire functions of completely regular growth
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.
متن کاملHausdorff Dimensions of Escaping Sets of Transcendental Entire Functions
Let f and g be transcendental entire functions, each with a bounded set of singular values, and suppose that g ◦ φ = ψ ◦ f , where φ, ψ : C → C are affine. We show that the escaping sets of f and g have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equa...
متن کاملEntire Functions with Julia Sets of Positive Measure
Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log logM(r, f) ≥ 1 2 N log r − O(1), where M(r, f) denotes the maximum modulus of f . We show that if f doe...
متن کاملOn a Question of Eremenko concerning Escaping Sets of Entire Functions (draft)
Let f be an entire function of finite order whose set of singular values is bounded or, more generally, a finite composition of such functions. We show that every escaping point of f can be connected to ∞ by a curve in I(f). This provides a positive answer to a question of Eremenko for a large class of entire functions.
متن کاملEscaping Points of Entire Functions of Small Growth
Abstract. Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Ereme...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Ergodic Theory and Dynamical Systems
سال: 2018
ISSN: 0143-3857,1469-4417
DOI: 10.1017/etds.2018.31