COMBINATORICS OF THE KNEADING MAP

1995 ◽  
Vol 05 (05) ◽  
pp. 1339-1349 ◽  
Author(s):  
H. BRUIN
Keyword(s):  
Topological Structure ◽  
Limit Set ◽  
Geometric Properties ◽  
Unimodal Maps ◽  
Omega Limit Set ◽  
Kneading Theory

The kneading map and the Hofbauer tower are tools, developed by F. Hofbauer and G. Keller, to study unimodal maps and the kneading theory. In this paper we survey the geometric properties of these tools. Results concerning the topological structure of the critical omega-limit set are obtained.

Adding machines and wild attractors

1997 ◽  
Vol 17 (6) ◽  
pp. 1267-1287 ◽  
Author(s):  
HENK BRUIN ◽  
GERHARD KELLER ◽  
MATTHIAS ST. PIERRE
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Cantor Set ◽  
Limit Set ◽  
Unimodal Maps ◽  
Irrational Rotations ◽  
Circle Rotation ◽  
Omega Limit Set ◽  
Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

Non-genericity of initial data with punctual $$\omega $$-limit set

2019 ◽  
Vol 114 (2) ◽  
pp. 185-193
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi
Keyword(s):  
Initial Data ◽  
Limit Set ◽  
Omega Limit Set

DEVIL'S STAIRCASE OF GAP MAPS

2003 ◽  
Vol 13 (01) ◽  
pp. 115-122 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN
Keyword(s):  
Topological Entropy ◽  
Devil's Staircase ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Entropy Functions ◽  
Kneading Theory

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

scholarly journals On the fixed points in an $\omega$-limit set

1992 ◽  
Vol 117 (4) ◽  
pp. 349-364
Author(s):  
J. G. Ceder
Keyword(s):  
Fixed Points ◽  
Limit Set ◽  
Omega Limit Set

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.

On the Quadratic Endomorphisms of the Plane

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
F. Bofill ◽  
J.L. Garrido ◽  
F. Vilamajó ◽  
N. Romero ◽  
A. Rovella
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Degenerate Case ◽  
Parameter Family ◽  
Limit Set ◽  
Quadratic Mappings ◽  
Omega Limit Set

AbstractIn this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.

Metric properties of non-renormalizableS-unimodal maps. Part I: Induced expansion and invariant measures

1994 ◽  
Vol 14 (4) ◽  
pp. 721-755 ◽  
Author(s):  
Michael Jakobson ◽  
Grzegorz Światek
Keyword(s):  
Critical Point ◽  
Schwarzian Derivative ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Limit Set ◽  
Countable Union ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Starting Condition ◽  
Metric Properties

AbstractFor an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

Poincaré–Bendixson theory for certain reaction–diffusion boundary-value problems

1994 ◽  
Vol 124 (1) ◽  
pp. 33-69
Author(s):  
Russell A. Smith
Keyword(s):  
Simple Structure ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Periodic Trajectory ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Limit Set ◽  
Diffusion Boundary ◽  
Two Dimensional System ◽  
Omega Limit Set

The paper studies orbits in a function space described by solutions of a system of reaction–diffusion equations in a bounded domain with a boundary condition of homogeneous Robin type. The omega-limit set of a bounded semi-orbit is shown to have a simple structure, provided that certain hypotheses are satisfied. For a two-dimensional system of Fitz-Hugh Nagumo type, these hypotheses yield explicit sufficient conditions for the existence of at least one periodic trajectory.

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