Consensus of Julia Sets

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods.

Radial distributions of Julia sets of difference operators of entire solutions of complex differential equations

<abstract><p>In this paper, we mainly investigate the radial distribution of Julia sets of difference operators of entire solutions of complex differential equation $ F(z)f^{n}(z)+P(z, f) = 0 $, where $ F(z) $ is a transcendental entire function and $ P(z, f) $ is a differential polynomial in $ f $ and its derivatives. We obtain that the set of common limiting directions of Julia sets of non-trivial entire solutions, their shifts have a definite range of measure. Moreover, an estimate of lower bound of measure of the set of limiting directions of Jackson difference operators of non-trivial entire solutions is given.</p></abstract>

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

Let $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ 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.

Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

Julia Sets and Their Control in a Three-Dimensional Discrete Fractional-Order Financial Model

It is of great significance to study the three-dimensional financial system model based on the discrete fractional-order theory. In this paper, the Julia set of the three-dimensional discrete fractional-order financial model is identified to show its fractal characteristics. The sizes of the Julia sets need to be changed in some situations, so it is necessary to achieve control of the Julia sets. In combination with the characteristics of the model, two different controllers based on the fixed point are designed, and the control of the three-dimensional Julia sets is realized by adding the controllers into the model in different ways. Finally, the simulation graphs show that the controllers can effectively control the fractal behaviors.

Asymptotics of the Hausdorff dimensions of the Julia sets of McMullen maps with error bounds*

For McMullen maps f λ (z) = z p + λ/z p , where λ ∈ C \ { 0 } , it is known that if p ⩾ 3 and λ is small enough, then the Julia set J(f λ ) of f λ is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f λ ) has the following asymptotic behavior dim H J ( f λ ) = 1 + log 2 log p + O ( | λ | 2 − 4 / p ) , as λ → 0 . An explicit error estimation of the remainder is also obtained. We also observe a ‘dimension paradox’ for the Julia set of Cantor set of circles.

Transcendental Julia sets with fractional packing dimension

We construct transcendental entire functions whose Julia sets have packing dimension in ( 1 , 2 ) (1,2) . These are the first examples where the computed packing dimension is not 1 1 or 2 2 . Our analysis will allow us further show that the set of packing dimensions attained is dense in the interval ( 1 , 2 ) (1,2) , and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension.

Julia sets of Zorich maps

The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.