Limit periodic functions, adding machines, and solenoids
type: Item
instance of: scholarly article
title: Limit periodic functions, adding machines, and solenoids
zbMATH Open document ID: 0832.34031
DOI: 10.1007/BF02219369
author: Harold Bell
author: Kenneth R. Meyer
published in: Journal of Dynamics and Differential Equations
publication date: 1995-09-25T00:00:00Z
review text: The authors prove two theorems: 1) If \(\Lambda\) is a stable invariant set for the homeomorphism \(f: \mathbb{R}^2\to \mathbb{R}^2\) such that \(f|\Lambda\) is conjugate to the adding machine then \(\Lambda\) is the limit of periodic points of \(f\). 2) Let \(\Phi_t\) be a smooth flow on a 3-manifold. If \(\Lambda\subset M\) is a stable invariant set conjugate to the solenoid minimal flow, then \(\Lambda\) is the limit of periodic orbits. The paper contains also an example of the homeomorphism of \(\mathbb{R}^3\) into itself which admits a stable adding machine invariant set, but which has no periodic points. This example shows that a result similar to 1) is false in higher dimensions.
reviewed by: J.Rogowski
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 800415
zbMATH Keywords: adding machine
zbMATH Keywords: periodic points
zbMATH Keywords: solenoid minimal flow
MaRDI profile type: Publication
cites work: A fixed point theorem for plane homeomorphisms
cites work: On the definition of almost periodicity
cites work: Liapunov stability and adding machines
cites work: Some fixed point theorems
cites work: Q5508167
cites work: Periodic Orbits and Solenoids in Generic Hamiltonian Dynamical Systems
cites work: Introduction to Hamiltonian dynamical systems and the \(N\)-body problem
cites work: Melnikov Transforms, Bernoulli Bundles, and Almost Periodic Perturbations
cites work: Q3264374
cites work: Q5775616
full work available at URL: https://doi.org/10.1007/bf02219369
OpenAlex ID: W1975814363
description: scientific article; zbMATH DE number 800415