Abstract
We define the Rauzy gasket as a subset of the standard two-dimensional simplex associated with letter frequencies of ternary episturmian words. We prove that the Rauzy gasket is homeomorphic to the usual Sierpiński gasket (by a two-dimensional generalization of the Minkowski ? function) and to the Apollonian gasket (by a map which is smooth on the boundary of the simplex). We prove that it is also homothetic to the invariant set of the fully subtractive algorithm, hence of measure 0.
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References
Arnoux, P., Rauzy, G.: Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119, 199–215 (1991)
Boshernitzan, M.: A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math. 46, 196–213 (1993)
Cassaigne, J., Ferenczi, S., Zamboni, L.Q.: Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50, 1265–1276 (2000)
Droubay, X., Justin, J., Pirillo, G.: Episturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci. 255(1-2), 539–553 (2001)
Fogg, N.P.: Substitutions in Arithmetics, Dynamics and Combinatorics, 1st edn. Springer, New York (2002)
Glen, A., Justin, J.: Episturmian words: a survey. Theoret. Inf. Appl. 49(3), 403–442 (2009)
Justin, J., Pirillo, G.: Episturmian words and episturmian morphisms. Theoret. Comput. Sci. 276(1-2), 281–313 (2002). doi:http://dx.doi.org/10.1016/S0304-3975(01)00207-9
Kraaikamp, C., Meester, R.: Ergodic properties of a dynamical system arising from percolation theory. Ergod. Theor. Dyn. Syst. 15, 653–661 (1995)
Mauldin, R.D., Urbański, M.: Dimension and measures for a curvilinear Sierpiński gasket or Apollonian packing. Adv. Math. 136(1), 26–38 (1998)
McMullen, C.T.: Hausdorff dimension and conformal dynamics III: computation of dimension. Am. J. Math. 120(4), 691–72 (1998)
Meester, R.W.J., Nowicki, T.: Infinite clusters and critical values in two dimensional circle percolation. Isr. J. Math. 68(1), 63–81 (1989)
Panti, G.: Multidimensional continued fractions and a Minkowski function. Monatsh. Math. 154, 247–264 (2008)
Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110, 147–178 (1982)
Rauzy, G.: Suites à termes dans un alphabet fini. Séminaire de Théorie des Nombres de Bordeaux Année 1982–1983(exposé 25) (1983)
Sullivan, D.: Entropy, Hausdorff measure old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153, 259–277 (1984)
Wozny, N., Zamboni, L.Q.: Frequencies of factors in Arnoux-Rauzy sequences. Acta Arith. 96, 261–278 (2001)
Acknowledgments
The second author acknowledges financial support by the Czech Science Foundation grant GAČR 201/09/0584, by the grants MSM6840770039 and LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic, and by the grant of the Grant Agency of the Czech Technical University in Prague grant No. SGS11/162/OHK4/3T/14.
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Arnoux, P., Starosta, Š. (2013). The Rauzy Gasket. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_1
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DOI: https://doi.org/10.1007/978-0-8176-8400-6_1
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