COMBINATORIAL STRUCTURE OF THE PARAMETER PLANE OF THE FAMILY λ tan z2

2021 ◽  
Vol 33 (1) ◽  
pp. 1-38
Author(s):  
Santanu Nandi
Keyword(s):  
Combinatorial Structure ◽  
Parameter Plane ◽  
The Family

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

2015 ◽  
Vol 59 (3) ◽  
pp. 671-690
Author(s):  
Piotr Gałązka ◽  
Janina Kotus
Keyword(s):  
Hausdorff Dimension ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Elliptic Functions ◽  
Critical Value ◽  
Parameter Plane ◽  
Dynamical Plane ◽  
The Family ◽  
Maximal Multiplicity ◽  
Additional Assumptions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

scholarly journals Topological structure and entropy of mixing graph maps

2013 ◽  
Vol 34 (5) ◽  
pp. 1587-1614 ◽  
Author(s):  
GRZEGORZ HARAŃCZYK ◽  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Structure ◽  
Structure Theorem ◽  
Main Tool ◽  
Combinatorial Structure ◽  
Topological Graph ◽  
Topological Mixing ◽  
Entropy Of Mixing ◽  
The Family ◽  
Topologically Mixing ◽  
Refined Version

AbstractLet ${ \mathcal{P} }_{G} $ be the family of all topologically mixing, but not exact self-maps of a topological graph $G$. It is proved that the infimum of topological entropies of maps from ${ \mathcal{P} }_{G} $ is bounded from below by $\log 3/ \Lambda (G)$, where $\Lambda (G)$ is a constant depending on the combinatorial structure of $G$. The exact value of the infimum on ${ \mathcal{P} }_{G} $ is calculated for some families of graphs. The main tool is a refined version of the structure theorem for mixing graph maps. It also yields new proofs of some known results, including Blokh’s theorem (topological mixing implies the specification property for maps on graphs).

scholarly journals Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$

10.53733/135 ◽  
2021 ◽  
Vol 52 ◽  
pp. 469-510
Author(s):  
Tao Chen ◽  
Linda Keen
Keyword(s):  
Meromorphic Functions ◽  
Julia Set ◽  
Singular Values ◽  
Parameter Plane ◽  
New Techniques ◽  
Asymptotic Values ◽  
The Family ◽  
Dynamical Properties ◽  
Set Of Points ◽  
Dense Set

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite.   Here we  look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$.  These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values.   Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here.   Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set.   We also give a description of the parameter plane of the family $f_{\lambda}$.  Again there are similarities to and differences from  the parameter plane of the family $P_a$ and again  there are new techniques.   In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these  points.

LIMITING JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2008 ◽  
Vol 18 (10) ◽  
pp. 3175-3181 ◽  
Author(s):  
MARK MORABITO ◽  
ROBERT L. DEVANEY
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Julia Set ◽  
Singularly Perturbed ◽  
Parameter Plane ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Open Set ◽  
The Family

In this paper, we consider the family of rational maps given by [Formula: see text] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere.

scholarly journals Turán and Ramsey Properties of Subcube Intersection Graphs

2012 ◽  
Vol 22 (1) ◽  
pp. 55-70 ◽  
Author(s):  
J. ROBERT JOHNSON ◽  
KLAS MARKSTRÖM
Keyword(s):  
Pairwise Disjoint ◽  
Ramsey Number ◽  
Combinatorial Structure ◽  
Edge Density ◽  
Type Problem ◽  
Free Graph ◽  
Intersection Graphs ◽  
Empty Intersection ◽  
The Family ◽  
Ramsey Type

The discrete cube {0, 1}d is a fundamental combinatorial structure. A subcube of {0, 1}d is a subset of 2k of its points formed by fixing k coordinates and allowing the remaining d − k to vary freely. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r + 1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no k which have non-empty intersection and no l which are pairwise disjoint?These questions are naturally expressed using intersection graphs. The intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Let $\I(n,d)$ be the set of all n vertex graphs which can be represented as the intersection graphs of subcubes in {0, 1}d. With this notation our first question above asks for the largest number of edges in a Kr+1-free graph in $\I(n,d)$. As such it is a Turán-type problem. We answer this question asymptotically for some ranges of r and d. More precisely we show that if $(k+1)2^{\lfloor\frac{d}{k+1}\rfloor}<n\leq k2^{\lfloor\frac{d}{k}\rfloor}$ for some integer k ≥ 2 then the maximum edge density is $\bigl(1-\frac{1}{k}-o(1)\bigr)$ provided that n is not too close to the lower limit of the range.The second question can be thought of as a Ramsey-type problem. The maximum such n can be defined in the same way as the usual Ramsey number but only considering graphs which are in $\I(n,d)$. We give bounds for this maximum n mainly concentrating on the case that l is fixed, and make some comparisons with the usual Ramsey number.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

Sign in / Sign up

Export Citation Format

Share Document