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On Khintchine exponents and Lyapunov exponents of continued fractions

Published online by Cambridge University Press:  01 February 2009

AI-HUA FAN ,
LING-MIN LIAO ,
BAO-WEI WANG  and
JUN WU
AI-HUA FAN
Affiliation:
Department of Mathematics, Wuhan University, Wuhan, 430072, PR China CNRS UMR 6140-LAMFA, Université de Picardie, 80039 Amiens, France (email: ai-hua.fan@u-picaride.fr, lingmin.liao@u-picardie.fr)
LING-MIN LIAO
Affiliation:
Department of Mathematics, Wuhan University, Wuhan, 430072, PR China CNRS UMR 6140-LAMFA, Université de Picardie, 80039 Amiens, France (email: ai-hua.fan@u-picaride.fr, lingmin.liao@u-picardie.fr)
BAO-WEI WANG
Affiliation:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, PR China (email: bwei_wang@yahoo.com.cn, wujunyu@public.wh.hb.cn)
JUN WU
Affiliation:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, PR China (email: bwei_wang@yahoo.com.cn, wujunyu@public.wh.hb.cn)
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Abstract

Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 73 - 109
Copyright
Copyright © 2008 Cambridge University Press

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