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 and satisfy a -pinching condition for some . Moreover, if the construction diffeomorphisms also satisfy the disjoint open set condition we then give a lower bound for the Hausdorff dimension.

**[B1]**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****[B2]**Tim Bedford,*Dimension and dynamics for fractal recurrent sets*, J. London Math. Soc. (2)**33**(1986), no. 1, 89–100. MR**829390**, 10.1112/jlms/s2-33.1.89**[Bo]**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****[D]**F. M. Dekking,*Recurrent sets*, Adv. in Math.**44**(1982), no. 1, 78–104. MR**654549**, 10.1016/0001-8708(82)90066-4**[F]**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****[H]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[J]**Y. Jiang,*On non-conformal semigroups*, preprint, State University of New York at Stony Brook.**[K]**Jean-Pierre Kahane,*Some random series of functions*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR**833073****[MM]**Heather McCluskey and Anthony Manning,*Hausdorff dimension for horseshoes*, Ergodic Theory Dynam. Systems**3**(1983), no. 2, 251–260. MR**742227**, 10.1017/S0143385700001966**[MW]**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**, 10.1090/S0002-9947-1988-0961615-4**[W]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**

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

Article copyright:
© Copyright 1993
American Mathematical Society