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

Bowen entropy for fixed-point free flows

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.

Citation: Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and 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. 

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

D. DouM. 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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

D. DouM. 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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

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