A NOTE ON THE UNIMODAL FEIGENBAUM'S MAPS

2009 ◽  
Vol 23 (14) ◽  
pp. 3101-3111
Author(s):  
GUIFENG HUANG ◽  
LIDONG WANG ◽  
GONGFU LIAO
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Periodic Points ◽  
Limit Set ◽  
Almost Periodic ◽  
Limit Sets ◽  
Shift Map ◽  
Renormalization Operator

We mainly investigate the likely limit sets and the kneading sequences of unimodal Feigenbaum's maps (Feigenbaum's map can be regarded as the fixed point of the renormalization operator [Formula: see text], where λ is to be determined). First, we estimate the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum's map and then for every decimal s ∈ (0, 1), we construct a unimodal Feigenbaum's map which has a likely limit set with Hausdorff dimension s. Second, we prove that the kneading sequences of unimodal Feigenbaum's maps are uniformly almost periodic points of the shift map but not periodic ones.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS

2011 ◽  
Vol 21 (11) ◽  
pp. 3205-3215 ◽  
Author(s):  
ISSAM NAGHMOUCHI
Keyword(s):  
Periodic Point ◽  
Periodic Points ◽  
Nonempty Interior ◽  
Limit Set ◽  
Limit Sets ◽  
Dendrite Map ◽  
Graph Map ◽  
Monotone Graph

We show that, for monotone graph map f, all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li–Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.

scholarly journals Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry

2019 ◽  
Author(s):  
Olivier Glorieux ◽  
Daniel Monclair
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Hyperbolic Geometry ◽  
Limit Set ◽  
Rigidity Result ◽  
Famous Theorem ◽  
Limit Sets ◽  
Convex Cocompact

AbstractThe aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\textrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud, and Kassel, called ${\mathbb{H}}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ${\mathbb{H}}^{2,1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$, which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$D hyperbolic geometry.

Limit Sets of Typical Homeomorphisms

2012 ◽  
Vol 55 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Nilson C. Bernardes
Keyword(s):  
Hausdorff Dimension ◽  
Borel Measure ◽  
Cantor Set ◽  
Limit Set ◽  
Positive Borel Measure ◽  
Dense Orbit ◽  
Limit Sets ◽  
Dimension Zero

AbstractGiven an integer n ≥ 3, a metrizable compact topological n-manifold X with boundary, and a finite positive Borel measure μ on X, we prove that for the typical homeomorphism f : X → X, it is true that for μ-almost every point x in X the limit set ω( f, x) is a Cantor set of Hausdorff dimension zero, each point of ω(f, x) has a dense orbit in ω(f, x), f is non-sensitive at each point of ω(f, x), and the function a → ω(f, a) is continuous at x.

ABSTRACT ω-LIMIT SETS

2018 ◽  
Vol 83 (2) ◽  
pp. 477-495 ◽  
Author(s):  
WILL BRIAN
Keyword(s):  
Dynamical System ◽  
Hausdorff Space ◽  
Complete Characterization ◽  
Topological Dynamics ◽  
Limit Set ◽  
Limit Sets ◽  
Shift Map ◽  
Image Position ◽  
Continuous Surjection

AbstractThe shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

scholarly journals Conformality for a robust class of non-conformal attractors

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Beatrice Pozzetti ◽  
Andrés Sambarino ◽  
Anna Wienhard
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Convergence Property ◽  
Limit Set ◽  
Limit Sets ◽  
Differentiability Property ◽  
Anosov Representations

AbstractIn this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.

scholarly journals On Limit Sets of Monotone Maps on Dendroids

2020 ◽  
Vol 5 (2) ◽  
pp. 311-316
Author(s):  
E.N. Makhrova
Keyword(s):  
Periodic Orbit ◽  
Periodic Points ◽  
Cantor Set ◽  
List Type ◽  
Limit Set ◽  
Limit Sets ◽  
Monotone Map ◽  
Monotone Maps

AbstractLet X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1)\omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ;(2)ω(x, f ) is either a periodic orbit or a minimal Cantor set.In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3)\Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f.The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

scholarly journals On ω-Limit Sets of Zadeh’s Extension of Nonautonomous Discrete Systems on an Interval

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1116
Author(s):  
Guangwang Su ◽  
Taixiang Sun
Keyword(s):  
Fixed Points ◽  
Fuzzy Numbers ◽  
Discrete Systems ◽  
Periodic Points ◽  
Limit Set ◽  
Limit Sets

Let I = [ 0 , 1 ] and f n be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by F ( I ) the set of fuzzy numbers on I, and denote by ( F ( I ) , f ^ ) and ( F ( I ) , f ^ n ) the Zadeh ′ s extensions of ( I , f ) and ( I , f n ) , respectively. In this paper, we study the ω -limit sets of ( F ( I ) , f ^ n ) and show that, if all periodic points of f are fixed points, then ω ( A , f ^ n ) ⊂ F ( f ^ ) for any A ∈ F ( I ) , where ω ( A , f ^ n ) is the ω -limit set of A under ( F ( I ) , f ^ n ) and F ( f ^ ) = { A ∈ F ( I ) : f ^ ( A ) = A } .

A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains

2007 ◽  
Vol 143 (1) ◽  
pp. 157-164 ◽  
Author(s):  
BRIAN LINS
Keyword(s):  
Fixed Point ◽  
Convex Subset ◽  
Limit Set ◽  
Hilbert Metric ◽  
Limit Sets ◽  
Nonexpansive Map ◽  
Nonexpansive Maps ◽  
Polyhedral Domain ◽  
Omega Limit Set

AbstractFor a polyhedral domain $\Sigma \subset \mathbb{R}^n$, and a Hilbert metric nonexpansive map T:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any point x ∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone $\mathbb{R}^n_+$ which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.

On the sofic limit sets of cellular automata

1995 ◽  
Vol 15 (4) ◽  
pp. 663-684 ◽  
Author(s):  
Alejandro Maass
Keyword(s):  
Fixed Point ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Finite Time ◽  
Limit Set ◽  
Limit Sets ◽  
One Dimensional ◽  
Sofic System

AbstractIt is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

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