A note on double rotations of infinite type

Artigiani, M.; Fougeron, C.; Hubert, P.; Skripchenko, A.

doi:10.1090/mosc/311

A note on double rotations of infinite type
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by M. Artigiani, C. Fougeron, P. Hubert and A. Skripchenko

Trans. Moscow Math. Soc. 2021, 157-172

DOI: https://doi.org/10.1090/mosc/311

Published electronically: March 15, 2022

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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.

References

H. Bruin, Renormalization in a class of interval translation maps of $d$ branches, Dyn. Syst. 22 (2007), no. 1, 11–24. MR 2308208, DOI 10.1080/14689360601028084
Henk Bruin and Gregory Clack, Inducing and unique ergodicity of double rotations, Discrete Contin. Dyn. Syst. 32 (2012), no. 12, 4133–4147. MR 2966738, DOI 10.3934/dcds.2012.32.4133
Jérôme Buzzi and Pascal Hubert, Piecewise monotone maps without periodic points: rigidity, measures and complexity, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 383–405. MR 2054049, DOI 10.1017/S0143385703000488
Michael Boshernitzan and Isaac Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems 15 (1995), no. 5, 821–832. MR 1356616, DOI 10.1017/S0143385700009652
H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math. 137 (2003), 125–148. MR 2013352, DOI 10.1007/BF02785958
Julien Cassaigne and François Nicolas, Factor complexity, Combinatorics, automata and number theory, Encyclopedia Math. Appl., vol. 135, Cambridge Univ. Press, Cambridge, 2010, pp. 163–247. MR 2759107
Dynnikov I., Hubert P., Skripchenko A. Dynamical systems around the Rauzy gasket and their ergodic properties. arXiv:2011.15043v1 [math.DS]. 2020.
Fougeron C. Dynamical properties of simplicial systems and continued fraction algorithms. arXiv:2001.01367 [math.DS]. 2020.
D. Gaboriau, G. Levitt, and F. Paulin, Pseudogroups of isometries of $\textbf {R}$ and Rips’ theorem on free actions on $\textbf {R}$-trees, Israel J. Math. 87 (1994), no. 1-3, 403–428. MR 1286836, DOI 10.1007/BF02773004
N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. MR 1970385, DOI 10.1007/b13861
Hideyuki Suzuki, Shunji Ito, and Kazuyuki Aihara, Double rotations, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 515–532. MR 2152403, DOI 10.3934/dcds.2005.13.515
J. Schmeling and S. Troubetzkoy, Interval translation mappings, Dynamical systems (Luminy-Marseille, 1998) World Sci. Publ., River Edge, NJ, 2000, pp. 291–302. MR 1796167
Alexandra Skripchenko and Serge Troubetzkoy, Polygonal billiards with one sided scattering, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 1881–1896 (English, with English and French summaries). MR 3449199, DOI 10.5802/aif.2975
Troubetzkoy S. Absence of mixing for interval translation mappings and some generalizations. arXiv:2102.05904 [math.DS]. 2021.
Denis Volk, Almost every interval translation map of three intervals is finite type, Discrete Contin. Dyn. Syst. 34 (2014), no. 5, 2307–2314. MR 3124735, DOI 10.3934/dcds.2014.34.2307
William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
Jean-Christophe Yoccoz, Interval exchange maps and translation surfaces, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 1–69. MR 2648692