Exponential decay of Lebesgue numbers

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October 2012, 32(10): 3773-3785. doi: 10.3934/dcds.2012.32.3773

Exponential decay of Lebesgue numbers

1.	China Economics and Management Academy, Central University of Finance and Economics, No. 39 College South Road, Beijing, 100081, China

Received January 2011 Revised March 2012 Published May 2012

Abstract
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We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.

Keywords: Lipschitz constant, topological entropy, exponential decay., open cover, box dimension, Lebesgue number, Hausdorff dimension.

Mathematics Subject Classification: Primary: 37B40, 54E45, 54F4.

Citation: Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773

References:

[1]	R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar
[2]	X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies,, Sci. China Ser. A, 41 (1998), 1068. doi: 10.1007/BF02871841. Google Scholar
[3]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar
[4]	M. Misiurewicz, On Bowen's definition of topological entropy,, Discrete Contin. Dyn. Syst., 10 (2004), 827. doi: 10.3934/dcds.2004.10.827. Google Scholar
[5]	P. Sun, Exponential decay of expansive constants,, preprint, (2011). Google Scholar
[6]	P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar

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References:

[1]	R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar
[2]	X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies,, Sci. China Ser. A, 41 (1998), 1068. doi: 10.1007/BF02871841. Google Scholar
[3]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Katok and Leonardo Mendoza, 54 (1995). Google Scholar
[4]	M. Misiurewicz, On Bowen's definition of topological entropy,, Discrete Contin. Dyn. Syst., 10 (2004), 827. doi: 10.3934/dcds.2004.10.827. Google Scholar
[5]	P. Sun, Exponential decay of expansive constants,, preprint, (2011). Google Scholar
[6]	P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar

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