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

[1]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052

[2]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[3]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[6]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054

[7]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[8]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[9]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[10]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[11]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283

[12]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[13]

Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021017

[14]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320

[15]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[16]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021003

[17]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[18]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[19]

Zi Xu, Siwen Wang, Jinjin Huang. An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 971-979. doi: 10.3934/jimo.2020007

[20]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]