On the accumulation sets of exponential rays

2017 ◽  
Vol 39 (2) ◽  
pp. 370-391
Author(s):  
JIANXUN FU ◽  
GAOFEI ZHANG
Keyword(s):  
Jordan Arc ◽  
Indecomposable Continuum

We show that there exist non-landing exponential rays with bounded accumulation sets. By introducing folding models of certain rays, we prove that each of the corresponding accumulation sets is an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc.

Arcs with increasing chords

1972 ◽  
Vol 72 (2) ◽  
pp. 205-207 ◽  
Author(s):  
D. G. Larman ◽  
P. McMullen
Keyword(s):  
Absolute Constant ◽  
Complex Analysis ◽  
Private Communication ◽  
Arc Length ◽  
Euclidean Length ◽  
Jordan Arc ◽  
Image Position

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that.

scholarly journals Any counterexample to Makienko’s conjecture is an indecomposable continuum

2009 ◽  
Vol 29 (3) ◽  
pp. 875-883 ◽  
Author(s):  
CLINTON P. CURRY ◽  
JOHN C. MAYER ◽  
JONATHAN MEDDAUGH ◽  
JAMES T. ROGERS Jr
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Fatou Set ◽  
Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

A non-metric indecomposable continuum

1971 ◽  
Vol 38 (1) ◽  
pp. 15-20 ◽  
Author(s):  
David P. Bellamy
Keyword(s):  
Indecomposable Continuum

Singularities in the boundaries of local Siegel disks

1992 ◽  
Vol 12 (4) ◽  
pp. 803-821 ◽  
Author(s):  
James T. Rogers
Keyword(s):  
Entire Functions ◽  
Indecomposable Continuum

AbstractBounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle or it is an indecomposable continuum.

scholarly journals The Space of Composants of an Indecomposable Continuum

2002 ◽  
Vol 166 (2) ◽  
pp. 149-192 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Indecomposable Continuum

scholarly journals Intersections of Arc-Cluster Sets for Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 213-220 ◽  
Author(s):  
Charles L. Belna
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Limit Point ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Jordan Arc

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

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