scholarly journals A mixing dynamical system on the cantor set

1995 ◽  
Vol 18 (3) ◽  
pp. 607-612
Author(s):  
Jeong H. Kim
Keyword(s):  
Dynamical System ◽  
Cantor Set ◽  
Strong Mixing ◽  
Mixing Properties ◽  
Measure Preserving Transformation ◽  
Shift Map ◽  
The One

In this paper we give mixing properties (ergodic, weak-mixng and strong-mixing) to a dynamical system on the Cantor set by showing that the one-sided(12,12)-shift map is isomorphic to a measure preserving transformation defined on the Cantor set

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

The Saxon–Hutner Theorem, its Converse Theorem and their Applications to Localization and Delocalization Problems in Binary Quasiperiodic Lattices

1997 ◽  
Vol 11 (18) ◽  
pp. 2157-2182 ◽  
Author(s):  
Kazumoto Iguchi
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Cantor Set ◽  
Localized States ◽  
Edge States ◽  
Converse Theorem ◽  
One Dimensional ◽  
The One ◽  
Fibonacci Lattice

In this paper we discuss the application of the Saxon–Hutner theorem and its converse theorem in one-dimensional binary disordered lattices to the one-dimensional binary quasiperiodic lattices. We first summarize some basic theorems in one-dimensional periodic lattices. We discuss how the bulk and edge states are treated in the transfer matrix method. Second, we review the Saxon–Hutner theorem and prove the converse theorem, using the so-called Fricke identities. Third, we present an alternative approach for a rigorous proof of the existence of a Cantor-set spectrum in the Fibonacci lattice and in the related binary quasiperiodic lattices by means of the theorems together with their trace map with the invariant I. We obtain that if I > 0, then the spectrum is always a Cantor set, which was first proved for the Fibonacci lattice by Sütö and generalized for other quasiperiodic lattices by Bellissard, Iochum, Scopolla, and Testard. Fourth, we rigorously prove the existence of extended states in the spectrum of a class of binary quasiperiodic lattices first studied by Kolář and Ali. Fifth, we discuss the so-called gap labeling theorem emphasized by Bellissard and the classic argument of Kohn and Thouless for localized states in a one-dimensional disordered lattice in terms of the language of the transfer matrix method.

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Harry Dankowicz ◽  
Frank Schilder
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
The Other ◽  
Extended Formulation ◽  
Hybrid Dynamical System ◽  
Families Of Periodic Orbits ◽  
Continuation Problem ◽  
The One ◽  
Additional Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

Mixing properties of (α,β)-expansions

2009 ◽  
Vol 29 (4) ◽  
pp. 1119-1140 ◽  
Author(s):  
KARMA DAJANI ◽  
YUSUF HARTONO ◽  
COR KRAAIKAMP
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Mixing Properties ◽  
Special Values ◽  
Image Position ◽  
Relationship Of ◽  
The Relationship

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.

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.

MINIMAL AND STRANGE ATTRACTORS

1996 ◽  
Vol 06 (06) ◽  
pp. 1177-1183 ◽  
Author(s):  
A. GORODETSKI ◽  
Yu. ILYASHENKO
Keyword(s):  
Dynamical System ◽  
General Concept ◽  
Principal Result ◽  
Strange Attractors ◽  
Mild Form ◽  
Attracting Set ◽  
Probabilistic Character ◽  
A Chain ◽  
The One ◽  
Definition Of

A general concept going back to Kolmogorov claims that if a dynamical system has a complicated attracting set then its behavior has not a deterministic, but rather probabilistic character. This concept was not formalized up to now. Even the definition of attractor has a lot of different versions. This paper presents an attempt to give some definitions and results formalizing this heuristic ideas. It contains a definition of a minimal attractor, modifying the one given in Ilyashenko [1991]. The actual minimality of the attractor is discussed. The principal result is the Triple Choice Theorem. It claims that the existence of a strange minimal attractor implies some mild form of chaos for the map itself or for a nearby one. The program of further investigation is proposed as a chain of problems at the end of the paper.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

Stability of Markovian processes I: criteria for discrete-time Chains

1992 ◽  
Vol 24 (3) ◽  
pp. 542-574 ◽  
Author(s):  
Sean P. Meyn ◽  
R. L. Tweedie
Keyword(s):  
Discrete Time ◽  
Rates Of Convergence ◽  
Strong Mixing ◽  
Test Function ◽  
The Central Limit Theorem ◽  
Markovian Processes ◽  
Continuous Components ◽  
General Test ◽  
The One ◽  
Discrete Formulation

In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets (‘petite sets').We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov–Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.

On a One-Parameter Discrete-Time Z4-Equivariant Cubic Dynamical System

2019 ◽  
Vol 29 (04) ◽  
pp. 1950052
Author(s):  
Armands Gritsans
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Invariant Curves ◽  
Discrete Time System ◽  
Time System ◽  
Multiple Periodic Orbits ◽  
The One ◽  
Planar Hamiltonian System

The lemniscate sine and cosine are solutions of a [Formula: see text]-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time [Formula: see text]-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

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