scholarly journals Convex hulls of polynomial Julia sets

2020 ◽  
Vol 149 (1) ◽  
pp. 245-250
Author(s):  
Małgorzata Stawiska
Keyword(s):  
Julia Sets ◽  
Convex Hulls

scholarly journals Cubic critical portraits and polynomials with wandering gaps

2012 ◽  
Vol 33 (3) ◽  
pp. 713-738 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
CLINTON CURRY ◽  
LEX OVERSTEEGEN
Keyword(s):  
Branch Point ◽  
Pairwise Disjoint ◽  
Julia Sets ◽  
Convex Hulls ◽  
Dense Orbit ◽  
New Approach ◽  
Quotient Spaces ◽  
Cubic Polynomials ◽  
Induced Maps ◽  
Critical Portraits

AbstractThurston introduced $\sigma _d$-invariant laminations (where $\sigma _d(z)$ coincides with $z^d:\mathbb S ^1\to \mathbb S ^1$, $d\ge 2$) and defined wandering $k$-gons as sets ${\mathbf {T}}\subset \mathbb S ^1$ such that $\sigma _d^n({\mathbf {T}})$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma _d^n({\mathbf {T}})$ in the plane are pairwise disjoint. He proved that $\sigma _2$ has no wandering $k$-gons. Call a lamination with wandering $k$-gons a WT-lamination. In a recent paper, it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding quotient spaces $J$, are realizable as cubic polynomials on their (locally connected) Julia sets. Here we use a new approach to construct cubic WT-laminations with the above properties so that any wandering branch point of $J$ has a dense orbit in each subarc of $J$ (we call such orbits condense), and show that critical portraits corresponding to such laminations are dense in the space ${\mathcal A}_3$of all cubic critical portraits.

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.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

Convex hulls of objects bounded by algebraic curves

Algorithmica ◽  
1991 ◽  
Vol 6 (1-6) ◽  
pp. 533-553 ◽  
Author(s):  
Chanderjit Bajaj ◽  
Myung -Soo Kim
Keyword(s):  
Algebraic Curves ◽  
Convex Hulls

scholarly journals Computable representations for convex hulls of low-dimensional quadratic forms

2010 ◽  
Vol 124 (1-2) ◽  
pp. 33-43 ◽  
Author(s):  
Kurt M. Anstreicher ◽  
Samuel Burer
Keyword(s):  
Quadratic Forms ◽  
Convex Hulls ◽  
Low Dimensional

scholarly journals Locally connected models for Julia sets

2011 ◽  
Vol 226 (2) ◽  
pp. 1621-1661 ◽  
Author(s):  
Alexander M. Blokh ◽  
Clinton P. Curry ◽  
Lex G. Oversteegen
Keyword(s):  
Julia Sets
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