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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 |
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.
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. 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. |
| [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. 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. |
| [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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