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

Author: Xiao Ping Gu
Journal: Proc. Amer. Math. Soc. 118 (1993), 1281-1289
MSC: Primary 28A78; Secondary 58F11
MathSciNet review: 1181166
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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.

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  • [B1] T. Bedford, Hausdorff dimension and box dimension in self similar sets, Proc. Conf. Topology and Measure V, Greifswald, 1988, pp. 17-26. MR 1029553 (91a:58139)
  • [B2] -, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), 89-100. MR 829390 (87g:28004)
  • [Bo] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, Berlin, 1975. MR 0442989 (56:1364)
  • [D] F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), 78-104. MR 654549 (84e:52023)
  • [F] K. Falconer, Fractal geometry, mathematical foundations and applications, Wiley, England, 1990. MR 1102677 (92j:28008)
  • [H] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [J] Y. Jiang, On non-conformal semigroups, preprint, State University of New York at Stony Brook.
  • [K] J. P. Kahane, Some random series of functions, second ed., Cambridge Univ. Press, Cambridge and New York, 1985. MR 833073 (87m:60119)
  • [MM] H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynamical Systems 3 (1983), 251-260. MR 742227 (85j:58127)
  • [MW] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811-829. MR 961615 (89i:28003)
  • [W] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, Heidelberg, and Berlin, 1982. MR 648108 (84e:28017)

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Article copyright: © Copyright 1993 American Mathematical Society

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