Double rotations and their ergodic properties

Abstract: Double rotations are the simplest subclass of interval translation mappings. A double rotation is of finite type if its attractor is an interval and of infinite type if it is a Cantor set. It is easy to see that the restriction of a double rotation of finite type to its attractor is simply a rotation. It is known due to Suzuki - Ito - Aihara and Bruin - Clark that double rotations of infinite type are defined by a subset of zero measure in the parameter set. We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic. In my talk I plan to outline this proof that is based on the recent result by Ch. Fougeron for simplicial systems. I also hope to discuss briefly some challenging open questions and further research plans related to double rotations.

The talk is based on a joint work with Mauro Artigiani, Charles Fougeron and Pascal Hubert.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

---

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

---