Limit periodic functions, adding machines, and solenoids
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.
scientific article; zbMATH DE number 800415