scholarly journals On the differentiability of hairs for Zorich maps

2017 ◽  
Vol 39 (7) ◽  
pp. 1824-1842 ◽  
Author(s):  
PATRICK COMDÜHR
Keyword(s):  
Pairwise Disjoint ◽  
Exponential Family ◽  
Julia Set ◽  
Exponential Map ◽  
Simple Curves

Devaney and Krych showed that, for the exponential family $\unicode[STIX]{x1D706}e^{z}$, where $0\,<\,\unicode[STIX]{x1D706}\,<\,1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$. Viana proved that these curves are smooth. In this article, we consider quasiregular counterparts of the exponential map, the so-called Zorich maps, and generalize Viana’s result to these maps.

scholarly journals Julia sets of Zorich maps

2021 ◽  
pp. 1-37
Author(s):  
ATHANASIOS TSANTARIS
Keyword(s):  
Complex Plane ◽  
Pairwise Disjoint ◽  
Exponential Family ◽  
Julia Set ◽  
Julia Sets ◽  
Three Dimensional ◽  
Periodic Points ◽  
Exponential Map ◽  
Simple Curves ◽  
Entire Complex

Abstract 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.

scholarly journals Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

2011 ◽  
Vol 33 (1) ◽  
pp. 284-302 ◽  
Author(s):  
JÖRN PETER
Keyword(s):  
Finite Order ◽  
Hausdorff Measure ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Gauge Function ◽  
Exponential Map ◽  
Hausdorff Measures ◽  
Transcendental Entire Functions ◽  
Bounded Type Functions

AbstractWe show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become ‘smaller’ as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

scholarly journals Knobbly but nice

2020 ◽  
pp. 1-7
Author(s):  
NEIL DOBBS
Keyword(s):  
Complex Plane ◽  
Jordan Curve ◽  
Julia Set ◽  
Piecewise Smooth ◽  
Exponential Map ◽  
Geometric Properties ◽  
Smooth Jordan Curve

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

2008 ◽  
Vol 28 (3) ◽  
pp. 915-946 ◽  
Author(s):  
VOLKER MAYER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Finite Order ◽  
Exponential Family ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Riemannian Metrics ◽  
Elliptic Functions ◽  
Real Analyticity ◽  
Conformal Measures

AbstractWorking with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

Ramanujan and the Julia Set of the Iterated Exponential Map

1988 ◽  
Vol 43 (7) ◽  
pp. 681-683
Author(s):  
Akhlesh Lakhtakia ◽  
Mercedes N. Lakhtakia
Keyword(s):  
Julia Set ◽  
Exponential Map

Abstract The Julia set of Ramanujan's iterated exponential map, Fr+1(z) = exp {Fr(z)} -1, is investigated for positive integral r. Connection with Devaney's work on the map zr+1 = λ exp[zr] is also made.

HAIRS FOR THE COMPLEX EXPONENTIAL FAMILY

1999 ◽  
Vol 09 (08) ◽  
pp. 1517-1534 ◽  
Author(s):  
CLARA BODELÓN ◽  
ROBERT L. DEVANEY ◽  
MICHAEL HAYES ◽  
GARETH ROBERTS ◽  
LISA R. GOLDBERG ◽  
...  
Keyword(s):  
Complex Plane ◽  
Exponential Family ◽  
Julia Set ◽  
Dynamical Plane ◽  
Entire Complex ◽  
Complex Exponential

In this paper we consider both the dynamical and parameter planes for the complex exponential family Eλ(z)=λez where the parameter λ is complex. We show that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under Eλ tend to infinity and hence are in the Julia set. We also show that there are similar hairs in the λ-plane. In this case, the hairs contain λ-values for which the orbit of 0 tends to infinity under the corresponding exponential. In this case it is known that the Julia set of Eλ is the entire complex plane.

An Introduction to Statistical Models for Networks

2020 ◽  
pp. 218-233
Author(s):  
Valentina Kuskova ◽  
Stanley Wasserman
Keyword(s):  
Social System ◽  
Exponential Family ◽  
Behavioral Science ◽  
Social Science Research ◽  
Science Research ◽  
Uniform Distributions ◽  
The Social ◽  
Special Cases ◽  
Social And Behavioral Science ◽  
Network Methods

Network theoretical and analytic approaches have reached a new level of sophistication in this decade, accompanied by a rapid growth of interest in adopting these approaches in social science research generally. Of course, much social and behavioral science focuses on individuals, but there are often situations where the social environment—the social system—affects individual responses. In these circumstances, to treat individuals as isolated social atoms, a necessary assumption for the application of standard statistical analysis is simply incorrect. Network methods should be part of the theoretical and analytic arsenal available to sociologists. Our focus here will be on the exponential family of random graph distributions, p*, because of its inclusiveness. It includes conditional uniform distributions as special cases.

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

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