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.

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.

Addendum to “Limit Sets of Typical Homeomorphisms”

2014 ◽  
Vol 57 (2) ◽  
pp. 240-244
Author(s):  
Nilson C. Bernardes
Keyword(s):  
Borel Measure ◽  
Limit Set ◽  
Positive Borel Measure ◽  
Limit Sets ◽  
Omega Limit Set

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 ƒ : X → X, it is true that for μ-almost every point x in X the restriction of ƒ (respectively of f-1) to the omega limit set ω( ƒ ; x) (respectively to the alpha limit set α( ƒ ; x)) is topologically conjugate to the universal odometer.

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

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 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.

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.

A characterization of ω-limit sets of maps of the interval with zero topological entropy

1993 ◽  
Vol 13 (1) ◽  
pp. 7-19 ◽  
Author(s):  
A. M. Bruckner ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Cantor Set ◽  
Limit Set ◽  
Limit Sets ◽  
Continuous Map

AbstractWe prove that an infiniteW⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iffW=Q∪PwhereQis a Cantor set, andPis countable, disjoint fromQ, dense inWif non-empty, and such that for any intervalJcontiguous toQ, card (J∩P) ≤ 1 if 0 or 1 is inJ, and card (J∩P) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 thatPcan contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containingQand contained inW, can be uncountable.

scholarly journals Basmajian-type identities and Hausdorff dimension of limit sets

2017 ◽  
Vol 38 (6) ◽  
pp. 2224-2244
Author(s):  
YAN MARY HE
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Complex Dynamical Systems ◽  
Limit Sets ◽  
One Dimensional ◽  
Type Series ◽  
Trivial Monodromy

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit non-trivial monodromy.

The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups

2000 ◽  
Vol 128 (1) ◽  
pp. 123-139 ◽  
Author(s):  
KATSUHIKO MATSUZAKI
Keyword(s):  
Hausdorff Dimension ◽  
Discrete Subgroup ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Full Dimension

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

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