Structurally Similar Sets of Transcendental Entire Function

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

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Fatou’s Associates

Arnold Mathematical Journal ◽

10.1007/s40598-020-00148-6 ◽

2020 ◽

Vol 6 (3-4) ◽

pp. 459-493

Author(s):

Vasiliki Evdoridou ◽

Lasse Rempe ◽

David J. Sixsmith

Keyword(s):

Entire Function ◽

Blaschke Product ◽

Entire Functions ◽

Julia Set ◽

Singular Values ◽

Strong Relationship ◽

Inner Function ◽

Transcendental Entire Function ◽

Fatou Component ◽

Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

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Entire functions of slow growth whose Julia set coincides with the plane

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700000869 ◽

2000 ◽

Vol 20 (6) ◽

pp. 1577-1582 ◽

Cited By ~ 3

Author(s):

WALTER BERGWEILER ◽

ALEXANDRE EREMENKO

Keyword(s):

Entire Function ◽

Slow Growth ◽

Entire Functions ◽

Julia Set ◽

Transcendental Entire Function

We construct a transcendental entire function $f$ with $J(f)=\mathbb{C}$ such that $f$ has arbitrarily slow growth; that is, $\log |f(z)|\leq\phi(|z|)\log |z|$ for $|z|>r_0$, where $\phi$ is an arbitrary prescribed function tending to infinity.

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On the Hausdorff dimension of the Julia set of a regularly growing entire function

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109990491 ◽

2010 ◽

Vol 148 (3) ◽

pp. 531-551 ◽

Cited By ~ 11

Author(s):

WALTER BERGWEILER ◽

BOGUSŁAWA KARPIŃSKA

Keyword(s):

Entire Function ◽

Hausdorff Dimension ◽

Julia Set ◽

Transcendental Entire Function ◽

Dimension 2

AbstractWe show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

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The Hausdorff dimension of Julia sets of entire functions II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074387 ◽

1996 ◽

Vol 119 (3) ◽

pp. 513-536 ◽

Cited By ~ 12

Author(s):

Gwyneth M. Stallard

Keyword(s):

Entire Function ◽

Hausdorff Dimension ◽

Entire Functions ◽

Julia Set ◽

Julia Sets ◽

Transcendental Entire Function ◽

Bounded Set

AbstractLetfbe a transcendental entire function such that the finite singularities of f−1lie in a bounded set. We show that the Hausdorff dimension of the Julia set of such a function is strictly greater than one.

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Uniformly bounded components of normality

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000060 ◽

2007 ◽

Vol 143 (1) ◽

pp. 85-101

Author(s):

XIAOLING WANG ◽

WANG ZHOU

Keyword(s):

Entire Function ◽

Transcendental Entire Function ◽

Fatou Set ◽

Image Position ◽

Uniformly Bounded

AbstractSuppose that f(z) is a transcendental entire function and that the Fatou set F(f)≠∅. Set and where the supremum supU is taken over all components of F(f). If B1(f)<∞ or B2(f)<∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded.

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Maximally and non-maximally fast escaping points of transcendental entire functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000018 ◽

2015 ◽

Vol 158 (2) ◽

pp. 365-383 ◽

Cited By ~ 7

Author(s):

D. J. SIXSMITH

Keyword(s):

Entire Function ◽

Entire Functions ◽

Julia Set ◽

Transcendental Entire Function ◽

Transcendental Entire Functions ◽

Dynamical Properties

AbstractWe partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

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Dynamics of slowly growing entire functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270001947x ◽

2001 ◽

Vol 63 (3) ◽

pp. 367-377 ◽

Cited By ~ 2

Author(s):

I. N. Baker

Keyword(s):

Entire Function ◽

Normal Family ◽

Entire Functions ◽

Julia Set ◽

Maximum Modulus ◽

Stable Set ◽

Transcendental Entire Function ◽

Transcendental Entire Functions ◽

Transcendental Functions ◽

Simply Connected

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

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Hollow quasi-Fatou components of quasiregular maps

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000840 ◽

2016 ◽

Vol 162 (3) ◽

pp. 561-574

Author(s):

DANIEL A. NICKS ◽

DAVID J. SIXSMITH

Keyword(s):

Entire Function ◽

Julia Set ◽

The Other ◽

Transcendental Entire Function ◽

Connected Component ◽

Fatou Component ◽

Multiply Connected ◽

Complementary Component ◽

Fatou Components ◽

Isolated Point

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

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The Hausdorff dimension of Julia sets of entire functions III

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004197001722 ◽

1997 ◽

Vol 122 (2) ◽

pp. 223-244 ◽

Cited By ~ 5

Author(s):

GWYNETH M. STALLARD

Keyword(s):

Entire Function ◽

Hausdorff Dimension ◽

Entire Functions ◽

Julia Set ◽

Julia Sets ◽

Transcendental Entire Function ◽

Transcendental Entire Functions

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

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