Classification of Denjoy continua
In this interesting paper, the authors reprove \textit{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.
scientific article; zbMATH DE number 1501772