scholarly journals Lebesgue measure of Feigenbaum Julia sets

2022 ◽  
Vol 195 (1) ◽  
Author(s):  
Artur Avila ◽  
Mikhail Lyubich
Keyword(s):  
Lebesgue Measure ◽  
Julia Sets

scholarly journals A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

scholarly journals The solar Julia sets of basic quadratic Cremer polynomials

2009 ◽  
Vol 30 (1) ◽  
pp. 51-65 ◽  
Author(s):  
A. BLOKH ◽  
X. BUFF ◽  
A. CHÉRITAT ◽  
L. OVERSTEEGEN
Keyword(s):  
Lebesgue Measure ◽  
Topological Structure ◽  
Julia Set ◽  
Julia Sets ◽  
Full Lebesgue Measure

AbstractIn general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

scholarly journals Measure of the Julia set of the Feigenbaum map with infinite criticality

2009 ◽  
Vol 30 (3) ◽  
pp. 855-875 ◽  
Author(s):  
GENADI LEVIN ◽  
GRZEGORZ ŚWIA̧TEK
Keyword(s):  
Stochastic Process ◽  
Fixed Points ◽  
Lebesgue Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Martingale Theory ◽  
Hyperbolic Dimension

AbstractWe consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

scholarly journals On transcendental meromorphic functions which are geometrically finite

2002 ◽  
Vol 72 (1) ◽  
pp. 93-108 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Meromorphic Function ◽  
Lebesgue Measure ◽  
Meromorphic Functions ◽  
Julia Sets ◽  
Geometrically Finite ◽  
Definition Of

AbstractIn this paper we give the definition of a meromorphic function which is geometrically finite and investigate some properties of geometrically finite meromorphic functions and the Lebesgue measure of their Julia sets.

FIXED POINTS-JULIA SETS

2020 ◽  
Vol 9 (9) ◽  
pp. 6759-6763
Author(s):  
G. Subathra ◽  
G. Jayalalitha
Keyword(s):  
Fixed Points ◽  
Julia Sets

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

LEBESGUE MEASURE AND GAMBLING

1993 ◽  
Vol 19 (1) ◽  
pp. 40
Author(s):  
Kanovei ◽  
Linton
Keyword(s):  
Lebesgue Measure
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