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November  2019, 39(11): 6231-6239. doi: 10.3934/dcds.2019271

Bowen entropy for fixed-point free flows

Yong Ji 1, , Ercai Chen 1, , Yunping Wang 1, and  Cao Zhao 2,,

1. 

School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210046, China
2. 

School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China

* Corresponding author

Received  September 2018 Revised  May 2019 Published  August 2019

In this paper, we devote to the study of the Bowen's entropy for fixed-point free flows and show that the Bowen entropy of the whole compact space is equal to the topological entropy. To obtain this result, we establish the Brin-Katok's local entropy formula for fixed-point free flows in ergodic case.

Keywords: Fixed-point free flows, reparametrization balls, Brin-Katok's formula, topological entropy.
Mathematics Subject Classification: Primary: 37B40, 37C45.
Citation: Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
References:
[1]

L. M. Abramov, On the entropy of a flow, Dok. Akad. Nauk SSSR, 128 (1959), 873-875.   Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.  doi: 10.2307/2373590.  Google Scholar

[4]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[5]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[6]

L. Breiman, The individual theorem of information theory, Ann. Math. Statist., 28 (1957), 809-811.  doi: 10.1214/aoms/1177706899.  Google Scholar

[7]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., 1007 (1983), 30-38.  doi: 10.1007/BFb0061408.  Google Scholar

[8]

D. Dou, M. Fan and H. Qiu, Topological entropy on subsets for fixed-point free flows, Disc. Contin. Dyn. Syst., 37 (2017), 6319-6331.  doi: 10.3934/dcds.2017273.  Google Scholar

[9]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.  Google Scholar

[10]

B. McMillan, The basic theorems of information theory, Ann. Math. Statist., 24 (1953), 196-219.  doi: 10.1214/aoms/1177729028.  Google Scholar

[11]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 623-656.  doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[12]

J. Shen and Y. Zhao, Entropy of a flow on non-compact sets, Open Syst. Inf. Dyn., 19 (2012), 1250015, 10 pp. doi: 10.1142/S1230161212500151.  Google Scholar

[13]

W. Sun, Measure-theoretic entropy for flows, Sci. China Ser. A, 40 (1997), 725-731.  doi: 10.1007/BF02878695.  Google Scholar

[14]

W. Sun and E. Vargas, Entropy of flows, revisited, Bol. Soc. Brasil. Mat. (N.S), 30 (1999), 315-333.  doi: 10.1007/BF01239009.  Google Scholar

[15]

R. F. Thomas, Entropy of expansive flows, Ergod. Th. Dynam. Sys., 7 (1987), 611-625.  doi: 10.1017/S0143385700004235.  Google Scholar

[16]

R. F. Thomas, Topological entropy of fixed-point free flows, Trans. Amer. Math. Soc., 319 (1990), 601-618.  doi: 10.1090/S0002-9947-1990-1010414-5.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[18]

D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel J. Math., 212 (2016), 895-911.  doi: 10.1007/s11856-016-1312-y.  Google Scholar

show all references

References:
[1]

L. M. Abramov, On the entropy of a flow, Dok. Akad. Nauk SSSR, 128 (1959), 873-875.   Google Scholar

[2]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[3]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.  doi: 10.2307/2373590.  Google Scholar

[4]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[5]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[6]

L. Breiman, The individual theorem of information theory, Ann. Math. Statist., 28 (1957), 809-811.  doi: 10.1214/aoms/1177706899.  Google Scholar

[7]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., 1007 (1983), 30-38.  doi: 10.1007/BFb0061408.  Google Scholar

[8]

D. Dou, M. Fan and H. Qiu, Topological entropy on subsets for fixed-point free flows, Disc. Contin. Dyn. Syst., 37 (2017), 6319-6331.  doi: 10.3934/dcds.2017273.  Google Scholar

[9]

D. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.  Google Scholar

[10]

B. McMillan, The basic theorems of information theory, Ann. Math. Statist., 24 (1953), 196-219.  doi: 10.1214/aoms/1177729028.  Google Scholar

[11]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 623-656.  doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[12]

J. Shen and Y. Zhao, Entropy of a flow on non-compact sets, Open Syst. Inf. Dyn., 19 (2012), 1250015, 10 pp. doi: 10.1142/S1230161212500151.  Google Scholar

[13]

W. Sun, Measure-theoretic entropy for flows, Sci. China Ser. A, 40 (1997), 725-731.  doi: 10.1007/BF02878695.  Google Scholar

[14]

W. Sun and E. Vargas, Entropy of flows, revisited, Bol. Soc. Brasil. Mat. (N.S), 30 (1999), 315-333.  doi: 10.1007/BF01239009.  Google Scholar

[15]

R. F. Thomas, Entropy of expansive flows, Ergod. Th. Dynam. Sys., 7 (1987), 611-625.  doi: 10.1017/S0143385700004235.  Google Scholar

[16]

R. F. Thomas, Topological entropy of fixed-point free flows, Trans. Amer. Math. Soc., 319 (1990), 601-618.  doi: 10.1090/S0002-9947-1990-1010414-5.  Google Scholar

[17]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[18]

D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel J. Math., 212 (2016), 895-911.  doi: 10.1007/s11856-016-1312-y.  Google Scholar

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