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Infinite iterated function systems with overlaps

Published online by Cambridge University Press:  10 November 2014

SZE-MAN NGAI  and
JI-XI TONG
SZE-MAN NGAI
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA email [email protected]
JI-XI TONG
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China email [email protected]
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Abstract

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 890 - 907
Copyright
© Cambridge University Press, 2014 

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References

Deng, Q.-R. and Ngai, S.-M.. Conformal iterated function systems with overlaps. Dyn. Syst. 26 (2011), 103–123.Google Scholar
Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, 85). Cambridge University Press, Cambridge, 1986.Google Scholar
Falconer, K. J.. Fractal geometry. Mathematical Foundations and Applications, 2nd edn. John Wiley & Sons, Inc., Hoboken, NJ, 2003.Google Scholar
Fan, A. H. and Lau, K.-S.. Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231 (1999), 319–344.Google Scholar
Fernau, H.. Infinite iterated function systems. Math. Nachr. 170 (1994), 79–91.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
Käenmäki, A. and Reeve, W. J.. Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets. J. Fractal Geom. 1 (2014), 83–152.Google Scholar
Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math. 141 (1999), 45–96.Google Scholar
Lau, K.-S. and Ngai, S.-M.. A generalized finite type condition for iterated function systems. Adv. Math. 208 (2007), 647–671.CrossRefGoogle Scholar
Lau, K.-S., Ngai, S.-M. and Wang, X.-Y.. Separation conditions for conformal iterated function systems. Monatsh. Math. 156 (2009), 325–355.CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. (3) 73 (1996), 105–154.CrossRefGoogle Scholar
Moran, M.. Hausdorff measure of infinitely generated self-similar sets. Monatsh. Math. 122 (1996), 387–399.Google Scholar
Ngai, S.-M. and Wang, Y.. Hausdorff dimension of self-similar sets with overlaps. J. London Math. Soc. (2) 63 (2001), 655–672.Google Scholar
Szarek, T. and Wedrychowicz, S.. The OSC does not imply the SOSC for infinite iterated function systems. Proc. Amer. Math. Soc. 133 (2005), 437–440.Google Scholar
Zerner, M. P. W.. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc. 124 (1996), 3529–3539.CrossRefGoogle Scholar