Lower bounds on the dimension of the Rauzy gasket. (Bornes inférieures pour la dimension de la baderne de Rauzy.) (English. French summary) Zbl 1451.37029
The authors obtain the lower bound for the Hausdorff dimension of the Rauzy gasket \(R\). It is a fractal subset of a 2-simplex appearing in different contexts, such as
in the representation of low-complexity subshifts by interval exchange transformations, in the study of frequencies of letters in the ternary episturmian words, in the dynamics of some isometries and in Novikov’s problem from conductivity theory for monocrystals.
S.P. Novikov and A. Y. Maltsev conjectured in [J. Stat. Phys. 115, No. 1–2, 31–46 (2004; Zbl 1157.82403)] that \({1<\mathrm{dim}_HR<2}.\) The right hand side of this inequality was established in [A. Avila et al., Bull. Soc. Math. Fr. 144, No. 3, 539–568 (2016; Zbl 1356.37018)]. Using numerical experiments R. DeLeo and I. A. Dynnikov [Geom. Dedicata 138, 51–67 (2009; Zbl 1165.28006)] suggested that \(\dim_HR>1.7.\)
The main result of the paper is \(\dim_HR>1.9.\) The proof relies on the thermodynamical approach due to Y. Cao et al. [Geom. Funct. Anal. 29, No. 5, 1325–1368 (2019; Zbl 1427.37018)].
S.P. Novikov and A. Y. Maltsev conjectured in [J. Stat. Phys. 115, No. 1–2, 31–46 (2004; Zbl 1157.82403)] that \({1<\mathrm{dim}_HR<2}.\) The right hand side of this inequality was established in [A. Avila et al., Bull. Soc. Math. Fr. 144, No. 3, 539–568 (2016; Zbl 1356.37018)]. Using numerical experiments R. DeLeo and I. A. Dynnikov [Geom. Dedicata 138, 51–67 (2009; Zbl 1165.28006)] suggested that \(\dim_HR>1.7.\)
The main result of the paper is \(\dim_HR>1.9.\) The proof relies on the thermodynamical approach due to Y. Cao et al. [Geom. Funct. Anal. 29, No. 5, 1325–1368 (2019; Zbl 1427.37018)].
Reviewer: Ivan Podvigin (Novosibirsk)
MSC:
| 37C45 | Dimension theory of smooth dynamical systems |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37B40 | Topological entropy |
| 28A78 | Hausdorff and packing measures |
| 28A80 | Fractals |
Keywords:
Rauzy gasket; Novikov’s problem; Hausdorff dimension; thermodynamic formalism; topological pressureReferences:
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