Classification of Denjoy continua. (English) Zbl 0983.37013

Barge, Marcy; Williams, R. F.

Topology Appl. 106, No. 1, 77-89 (2000).

In this interesting paper, the authors reprove R. Fokkink’s theorem [The structure of trajectories, Thesis, TU Delft (1991)]: Let \(\alpha\) and \(\beta\) be two irrationals with associated Denjoy continua \(\mathbb{D}_\alpha\) and \(\mathbb{D}_\beta\). Then \(\mathbb{D}_\alpha\) and \(\mathbb{D}_\beta\) are homeomorphic iff \(\alpha\) and \(\beta\) are equivalent. Two irrationals are equivalent if they are in the same orbit of the group action of the group \(SL(2,\mathbb{Z})\) on the irrationals, when the group \(SL(2,\mathbb{Z})\) acts on irrationals by \(\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix} \right)\alpha = {a+ b\alpha \over c+d\alpha}\). The authors’ approach emphasizes the connection between the topology of a Denjoy continuum and the geometry of the continued fraction expansion of its slope. A novel approach to continued fractions is developed via geometric bifurcation theory.

Reviewer: Alois Klíč (Praha)

MSC:

37B45	Continua theory in dynamics
54F15	Continua and generalizations
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54H20	Topological dynamics (MSC2010)
37E10	Dynamical systems involving maps of the circle

Keywords:

Farey array; suspension; Denjoy continua; continued fractions; geometric bifurcation