scholarly journals On topological rank of factors of Cantor minimal systems

2021 ◽  
pp. 1-24
Author(s):  
NASSER GOLESTANI ◽  
MARYAM HOSSEINI
Keyword(s):  
Dynamical System ◽  
Finite Rank ◽  
Affirmative Answer ◽  
Cantor Set ◽  
Minimal System ◽  
Full Generality ◽  
Cantor Minimal Systems ◽  
Minimal Dynamical System ◽  
Uniformly Bounded

Abstract A Cantor minimal system is of finite topological rank if it has a Bratteli–Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite, then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite in full generality. As a consequence, we obtain the dichotomy of Downarowicz and Maass for Cantor factors of finite-rank Cantor minimal systems: they are either odometers or subshifts.

Finite-rank Bratteli–Vershik diagrams are expansive

2008 ◽  
Vol 28 (3) ◽  
pp. 739-747 ◽  
Author(s):  
TOMASZ DOWNAROWICZ ◽  
ALEJANDRO MAASS
Keyword(s):  
Finite Rank ◽  
Main Role ◽  
Fixed Level ◽  
Bounded Number ◽  
Cantor Minimal Systems ◽  
Uniformly Bounded ◽  
The Way

AbstractThe representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

Parallelisability in Banach spaces: a parallelisable dynamical system with uniformly bounded trajectories

1988 ◽  
Vol 108 (3-4) ◽  
pp. 371-378
Author(s):  
B. M. Garay
Keyword(s):  
Dynamical System ◽  
Banach Space ◽  
Banach Spaces ◽  
Supremum Norm ◽  
Uniformly Bounded ◽  
Bounded Trajectories

SynopsisIn the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

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.

Dynamics of Morse-Smale urn processes

1995 ◽  
Vol 15 (6) ◽  
pp. 1005-1030 ◽  
Author(s):  
Michel Benaïm ◽  
Morris W. Hirsch
Keyword(s):  
Dynamical System ◽  
Sample Path ◽  
Asymptotically Stable ◽  
Martingale Differences ◽  
Stable Periodic Orbit ◽  
Sample Paths ◽  
Stable Periodic ◽  
Image Position ◽  
Deterministic Dynamical System ◽  
Uniformly Bounded

AbstractWe consider stochastic processes {xn}n≥0 of the formwhere F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

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

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

scholarly journals Dynamical System on Cantor Set

1998 ◽  
Vol 21 (1) ◽  
pp. 217-231
Author(s):  
Makoto MORI
Keyword(s):  
Dynamical System ◽  
Cantor Set

scholarly journals On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

2009 ◽  
Vol 30 (3) ◽  
pp. 639-664 ◽  
Author(s):  
XAVIER BRESSAUD ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS
Keyword(s):  
Finite Rank ◽  
Additive Group ◽  
Bratteli Diagram ◽  
Necessary Condition ◽  
Incidence Matrices ◽  
Bounded Number ◽  
Stable Space ◽  
To Come ◽  
Cantor Minimal Systems ◽  
Stable Subspace

AbstractIn this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

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