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

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

MathSciNet review:
1181166

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28A78,
58F11

Retrieve articles in all journals with MSC: 28A78, 58F11

Additional Information

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

Article copyright:
© Copyright 1993
American Mathematical Society