zbMATH — the first resource for mathematics

Dynamics of isometry systems: On the endpoints of orbits. (Dynamique des systèmes d’isométries: Sur les bouts des orbites.) (French) Zbl 0880.58009
A 1-dimensional system of isometries is a finite reunion $$D$$ of compact intervals in $${\mathbb{R}}$$ endowed with a finite family $$\{\varphi _1,\dots \varphi _k\}$$ of isometries between closed subintervals from $$D$$. The systems of isometries can be studied as one-dimensional dynamical systems.
The main results from this paper are:
Theorem 0.1. The orbits of a system of isometries have a finite number of endpoints. This number is at most $$2$$ except for, possibly, a finite number of orbits.
Theorem 0.2. The reunion of the orbits of an exotic system, having a fixed endpoint, contains a dense $$G_\delta$$.
Theorem 0.3. The number of endpoints of the orbits of an exotic system is bounded and the number of the orbits having at least three endpoints is finite.
Theorem 0.4. The set of the orbits with two endpoints of an exotic system is uncountable.
Reviewer: V.Oproiu (Iaşi)

MSC:
 37E99 Low-dimensional dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
Full Text: