Geodesic
Remarks on Garsia entropy and multidimensional Erdös measures
Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 141-149 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We conjecture that the Garsia entropy coincides with the entropy of the invariant multidimensional Erdös measure. This conjecture is true for all Garsia numbers. We also specify the algebraic unit being non-Pisot number, for which this conjecture is true. We prove a theorem, which generalizes the Garsia theorem on the absolute continuity of the infinite Bernoulli convolution for the Garsia numbers. The proof uses relations between the multidimensional Erdös problem and the one-dimensional Erdös problem. We discuss a connection between the entropy of the invariant Erdös measure and the conditional Ledrappier–Young entropies. We also formulate three conjectures and obtain some consequences from them. In particular, we conjecture that the Hausdorff dimension of the Erdös measure is equal to the Ledrappier–Young dimension of conditional measure for the multidimensional invariant Erdös measure along the unstable foliation corresponding to the top Lyapunov exponent of multiplicity 1. For 2-numbers, we obtain formulae for the Hausdorff dimension of Erdös measures on the unstable plane.
Keywords: Garsia entropy, Hausdorff dimension of the measure, Erdös measure, Hochman formula, Lyapunov exponent. 
@article{UFA_2022_14_4_a10,
     author = {V. I. Oseledets and V. L. Kulikov and E. F. Olekhova},
     title = {Remarks on {Garsia} entropy and multidimensional {Erd\"os} measures},
     journal = {Ufa mathematical journal},
     pages = {141--149},
     year = {2022},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2022_14_4_a10/}
}
TY  - JOUR
AU  - V. I. Oseledets
AU  - V. L. Kulikov
AU  - E. F. Olekhova
TI  - Remarks on Garsia entropy and multidimensional Erdös measures
JO  - Ufa mathematical journal
PY  - 2022
SP  - 141
EP  - 149
VL  - 14
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2022_14_4_a10/
LA  - en
ID  - UFA_2022_14_4_a10
ER  - 
%0 Journal Article
%A V. I. Oseledets
%A V. L. Kulikov
%A E. F. Olekhova
%T Remarks on Garsia entropy and multidimensional Erdös measures
%J Ufa mathematical journal
%D 2022
%P 141-149
%V 14
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2022_14_4_a10/
%G en
%F UFA_2022_14_4_a10
V. I. Oseledets; V. L. Kulikov; E. F. Olekhova. Remarks on Garsia entropy and multidimensional Erdös measures. Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 141-149. http://geodesic.mathdoc.fr/item/UFA_2022_14_4_a10/

[1] P. Erdös, “On a family of symmetric Bernoulli convolutions”, Amer. J. Math., 61 (1939), 974–976 | DOI | MR | Zbl

[2] Z.I. Bezhaeva, V.I. Oseledets, “Erdös measures, sofic measures, and Markov chains”, J. Math. Sci., 140:3 (2007), 357–368 | DOI | MR

[3] D.-J. Feng, Z. Feng, “Estimates on the dimension of self-similar measures with overlaps”, J. London Math. Soc., 105:4 (2022), 2104–2135 | DOI | MR

[4] J.E. Hutchinson, “Fractals and self similarity”, Indiana Univ. Math. J., 30:5 (1981), 713–747 | DOI | MR | Zbl

[5] A. Garsia, “Entropy and singularity of infinite Bernoulli convolutions”, Pacific J. Math., 13:4 (1963), 1159–1167 | DOI | MR

[6] Sh. Akiyama, D.-J. Feng, T. Kempton, T. Persson, “On the Hausdorff dimension of Bernoulli convolutions”, Int. Math. Res. Not., 2020:19 (2018), 6569–6595 | DOI | MR

[7] M. Hochman, “On self-similar sets with overlaps and for entropy”, Ann. Math., 140:2 (2014), 773–822 | DOI | MR

[8] Z.I. Bezhaeva, V.L. Kulikov, E.F. Olekhova, V.I. Oseledets, “Erdös measures on the Euclidean space and on the group of A-adic integers”, Proc. Steklov Inst. Math., 297:1 (2017), 28–34 | DOI | MR | Zbl

[9] F. Ledrappier, L.-S. Young, “The metric entropy of diffeomorphisms”, Bull. Amer. Math. Soc. New Ser., 11:2 (1984), 343–346 | DOI | MR | Zbl

[10] Z.I. Bezhaeva, V.I. Oseledets, “The entropy of the Erdös measure for the pseudogolden ratio”, Theory Probab. Appl., 57:1 (2013), 135–144 | DOI | MR | Zbl