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Estimating singularity dimension

Published online by Cambridge University Press:  16 December 2014

M. POLLICOTT  and
P. VYTNOVA
M. POLLICOTT
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL. e-mails: masdbl@warwick.ac.uk; vytnova@mccme.ru
P. VYTNOVA
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL. e-mails: masdbl@warwick.ac.uk; vytnova@mccme.ru
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Abstract

In this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space. A well-known result of Falconer showed that under mild assumptions the Hausdorff dimension of typical self-affine sets is equal to its Singularity dimension. Heuter and Lalley subsequently presented a smaller open family of non-trivial examples for which there is an equality of these two dimensions. In this article we analyse the size of this family and present an efficient algorithm for estimating the dimension.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 223 - 238
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Falconer, K.The Hausdorff dimension of self-affine fractals. Math. Camb. Phil. Soc. 103 (1988), 339–350.Google Scholar
[2]Grothendieck, A.Produits tensoriels topologiques et espaces nucleaires. Mem. Amer. Math. Soc. 16 (1955), 1–140.Google Scholar
[3]Heuter, I. and Lalley, S.Falconer's formula for the Hausdorff dimension of a self-affine set in ℝ2. Ergodic Theory Dynam. System 15 (1995), 77–97.Google Scholar
[4]Jenkinson, O. and Pollicott, M.Calculating Hausdorff dimensions of Julia sets and Kleinian limit sets. Amer. J. Math. 124 (2002), 495–545.Google Scholar
[5]Luzia, N.Hausdorff dimension for an open class of repellers in ℝ2. Nonlinearity. 19 (2006), 2895–2908.Google Scholar
[6]Mayer, D.The Ruelle–Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Physics vol. 123 (Springer, Berlin, 1980).Google Scholar
[7]McMullen, C.Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math. 120 (1998), 691–721.Google Scholar
[8]Peres, Yu.Domains of analytic continuation for the top Lyapunov exponent. Ann. Inst. H. Poincaré Probab. Statis. 28 (1) (1992), 131–148.Google Scholar
[9]Pollicott, M.Maximal Lyapunov exponents for random matrix products. Invent. Math. 181 (2010), 209–226.Google Scholar
[10]Ruelle, D.Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), 231–242.Google Scholar
[11]Solomyak, B.Measure and dimension for some fractal families. Math. Proc. Camb. Phil. Soc. 124 (1998), no. 3, 531–546.Google Scholar