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Asymptotic behavior of distribution of frequencies of digits

Published online by Cambridge University Press:  01 July 2008

LUIS BARREIRA  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal. e-mail: barreira@math.ist.utl.pt and cvalls@math.ist.utl.pt
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal. e-mail: barreira@math.ist.utl.pt and cvalls@math.ist.utl.pt
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Abstract

We present an approach to compute the Hausdorff dimension of a class of sets of real numbers that are defined in terms of nonlinear relations between frequencies of digits in some integer base m. We consider the model case of quadratic perturbations, for which the computations are already rather involved. We show that the Hausdorff dimension is analytic in the parameter determining the perturbation. Our approach also allows to estimate the asymptotic behavior of the Taylor coefficients of the dimension in terms of m.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 177 - 195
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Barreira, L.. Hyperbolicity and recurrence in dynamical systems: a survey of recent results. Resenhas IME-USP 5 (2002), 171–230.Google Scholar
[2]Barreira, L., Saussol, B. and Schmeling, J.. Distribution of frequencies of digits via multifractal analysis. J. Number Theory 97 (2002), 413–442.CrossRefGoogle Scholar
[3]Barreira, L., Saussol, B. and Schmeling, J.. Higher-dimensional multifractal analysis. J. Math. Pure Appl. 81 (2002), 67–91.CrossRefGoogle Scholar
[4]Besicovitch, A.. On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110 (1934), 321–330.CrossRefGoogle Scholar
[5]Billingsley, P.. Ergodic Theory and Information (Wiley and Sons, 1965).Google Scholar
[6]Borel, E.. Sur les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 26 (1909), 247–271.CrossRefGoogle Scholar
[7]Eggleston, H.. The fractional dimension of a set defined by decimal properties. Quart. J. Math. Oxford Ser. 20 (1949), 31–36.CrossRefGoogle Scholar
[8]Fan, A., Feng, D. and Wu, J.. Recurrence, dimension and entropy. J. London Math. Soc. 64 (2001), 229–244.CrossRefGoogle Scholar
[9]Pesin, Ya.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. (Chicago University Press, 1997).CrossRefGoogle Scholar
[10]Pfister, C.-E. and Sullivan, W.. On the topological entropy of saturated sets. Ergodic Theory Dynam. Systems 27 (2007), 929–956.CrossRefGoogle Scholar
[11]Takens, F. and Verbitskiy, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergodic Theory Dynam. Systems 23 (2003), 317–348.CrossRefGoogle Scholar