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

2010 ◽  
Vol 148 (3) ◽  
pp. 531-551 ◽  
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.

The Hausdorff dimension of Julia sets of entire functions II

1996 ◽  
Vol 119 (3) ◽  
pp. 513-536 ◽  
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.

The Hausdorff dimension of Julia sets of entire functions III

1997 ◽  
Vol 122 (2) ◽  
pp. 223-244 ◽  
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.

scholarly journals Fatou’s Associates

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.

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

2000 ◽  
Vol 20 (6) ◽  
pp. 1577-1582 ◽  
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.

scholarly journals Dimensions of slowly escaping sets and annular itineraries for exponential functions

2015 ◽  
Vol 36 (7) ◽  
pp. 2273-2292 ◽  
Author(s):  
D. J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Exponential Family ◽  
Transcendental Entire Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Exponential Functions ◽  
Dynamical Properties ◽  
Necessary And Sufficient

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to$1$. We prove these results by using the idea of anannular itinerary. In the case of a general transcendental entire function we show that one of these sets, theuniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.

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

2015 ◽  
Vol 158 (2) ◽  
pp. 365-383 ◽  
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.

scholarly journals Structurally Similar Sets of Transcendental Entire Function

2018 ◽  
Vol 4 ◽  
pp. 11-16
Author(s):  
Bishnu Hari Subedi
Keyword(s):  
Entire Function ◽  
Complex Plane ◽  
Full Text ◽  
Complex Dynamics ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Classical Sense

In complex dynamics, the complex plane is partitioned into invariant subsets. In classical sense, these subsets are of course Fatou set and Julia set. Rest of the abstract available with the full text

scholarly journals Dynamics of slowly growing entire functions

2001 ◽  
Vol 63 (3) ◽  
pp. 367-377 ◽  
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.

scholarly journals Hollow quasi-Fatou components of quasiregular maps

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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