American Institute of Mathematical Sciences

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

Exponential decay of Lebesgue numbers

Peng Sun 1,

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.
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

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

[1]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
[2]

Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099
[3]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[4]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[5]

Anna-Lena Trautmann. Isometry and automorphisms of constant dimension codes. Advances in Mathematics of Communications, 2013, 7 (2) : 147-160. doi: 10.3934/amc.2013.7.147
[6]

Natalia Silberstein, Tuvi Etzion. Large constant dimension codes and lexicodes. Advances in Mathematics of Communications, 2011, 5 (2) : 177-189. doi: 10.3934/amc.2011.5.177
[7]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[8]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[9]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[10]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[11]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[12]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[13]

Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287
[14]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055
[15]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[16]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[17]

Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058
[18]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[19]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[20]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences