scholarly journals Dynamical System on Cantor Set

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

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.

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 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.

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