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.

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 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 } .

scholarly journals A non-Borel special alpha-limit set in the square

2021 ◽  
pp. 1-11
Author(s):  
STEPHEN JACKSON ◽  
BILL MANCE ◽  
SAMUEL ROTH
Keyword(s):  
Dynamical Systems ◽  
Limit Set ◽  
Limit Sets ◽  
Unit Square

Abstract We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

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.

scholarly journals ω-limit sets for maps of the interval

1986 ◽  
Vol 6 (3) ◽  
pp. 335-344 ◽  
Author(s):  
Louis Block ◽  
Ethan M. Coven
Keyword(s):  
Periodic Points ◽  
Compact Interval ◽  
Minimal Set ◽  
Limit Sets ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Limit Points

AbstractLet f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

Almost sure limit sets of random samples in ℝd

1988 ◽  
Vol 20 (3) ◽  
pp. 573-599 ◽  
Author(s):  
Richard A. Davis ◽  
Edward Mulrow ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Almost Sure Convergence ◽  
Random Sets ◽  
Regularity Conditions ◽  
Limit Set ◽  
Multivariate Regular Variation ◽  
Exponential Form ◽  
Limit Sets ◽  
Random Samples ◽  
Distribution Tail

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

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.

scholarly journals Periodic points for amenable group actions on uniquely arcwise connected continua

2020 ◽  
pp. 1-12
Author(s):  
ENHUI SHI ◽  
XIANGDONG YE
Keyword(s):  
Periodic Point ◽  
Amenable Group ◽  
Group Actions ◽  
Periodic Points ◽  
Arcwise Connected

Abstract We show that any action of a countable amenable group on a uniquely arcwise connected continuum has a periodic point of order $\leq 2$ .

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