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Asymptotic Hausdorff dimensions of Cantor sets associated with an asymptotically non-hyperbolic family

Published online by Cambridge University Press:  12 September 2005

AIHUA FAN
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, People's Republic of China LAMFA, UMR 6140, CNRS University of Picardie 33, rue Saint Leu 80039, Amiens, France (e-mail: ai-hua.fan@u-picardie.fr, wujunyu@public.wh.hb.cn)
YUNPING JIANG
Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA Department of Mathematics, CUNY Graduate School, New York, NY 10016, USA Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People's Republic of China (e-mail: yunqc@forbin.qc.edu)
JUN WU
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, People's Republic of China LAMFA, UMR 6140, CNRS University of Picardie 33, rue Saint Leu 80039, Amiens, France (e-mail: ai-hua.fan@u-picardie.fr, wujunyu@public.wh.hb.cn)

Abstract

The geometry of Cantor systems associated with an asymptotically non-hyperbolic family $(f_\epsilon)_{0\le\epsilon\le \epsilon_0}$ was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc.351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of $f_\epsilon$ behaves like $1- K\epsilon^{1/\gamma}$ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Neumann transformations. PhD Thesis, CUNY Graduate Center, May 1999).

Type
Research Article
Copyright
2005 Cambridge University Press

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