Abstract:
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.
Key words and phrases:
interval translation mappings, unique ergodicity, Rauzy induction.
Citation:
M. Artigiani, Ch. Fougeron, P. Hubert, A. Skripchenko, “A note on double rotations of infinite type”, Tr. Mosk. Mat. Obs., 82, no. 1, MCCME, M., 2021, 185–203; Trans. Moscow Math. Soc., 82 (2021), 157–172
\Bibitem{ArtFouHub21}
\by M.~Artigiani, Ch.~Fougeron, P.~Hubert, A.~Skripchenko
\paper A note on double rotations of infinite type
\serial Tr. Mosk. Mat. Obs.
\yr 2021
\vol 82
\issue 1
\pages 185--203
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo654}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2021
\vol 82
\pages 157--172
\crossref{https://doi.org/10.1090/mosc/311}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85127460428}
Linking options:
https://www.mathnet.ru/eng/mmo654
https://www.mathnet.ru/eng/mmo/v82/i1/p185
This publication is cited in the following 2 articles:
A. S. Skripchenko, “Renormalization in one-dimensional dynamics”, Russian Math. Surveys, 78:6 (2023), 983–1021
Sergey Kryzhevich, Viktor Avrutin, Nikita Begun, Dmitrii Rachinskii, Khosro Tajbakhsh, “Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps”, Axioms, 10:2 (2021), 80