The Hausdorff dimension of self-similar sets under a pinching condition

Gu, Xiao Ping

doi:10.1090/S0002-9939-1993-1181166-1

The Hausdorff dimension of self-similar sets under a pinching condition
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by Xiao Ping Gu

Proc. Amer. Math. Soc. 118 (1993), 1281-1289

DOI: https://doi.org/10.1090/S0002-9939-1993-1181166-1

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Abstract:

We study self-similar sets in the case where the construction diffeomorphisms are not necessarily conformal. Using topological pressure we give an upper estimate of the Hausdorff dimension, when the construction diffeomorphisms are ${C^{1 + \kappa }}$ and satisfy a $\kappa$-pinching condition for some $\kappa \leqslant 1$. Moreover, if the construction diffeomorphisms also satisfy the disjoint open set condition we then give a lower bound for the Hausdorff dimension.

References

Tim Bedford, Hausdorff dimension and box dimension in self-similar sets, Proceedings of the Conference: Topology and Measure, V (Binz, 1987) Wissensch. Beitr., Ernst-Moritz-Arndt Univ., Greifswald, 1988, pp. 17–26. MR 1029553
Tim Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), no. 1, 89–100. MR 829390, DOI 10.1112/jlms/s2-33.1.89
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), no. 1, 78–104. MR 654549, DOI 10.1016/0001-8708(82)90066-4
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055

On non-conformal semigroups

Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
Heather McCluskey and Anthony Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 251–260. MR 742227, DOI 10.1017/S0143385700001966
R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108