# American Institute of Mathematical Sciences

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

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.
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

show all references

##### 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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