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December  2017, 37(12): 6319-6331. doi: 10.3934/dcds.2017273

Topological entropy on subsets for fixed-point free flows

Dou Dou , Meng Fan and  Hua Qiu ,

Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China

* Corresponding author

Received  January 2017 Revised  July 2017 Published  August 2017

Fund Project: The first author was supported by the National Nature Science Foundation of China (Grant No. 10901080,11271191 and 11431012) and CSC. The third author was supported by the National Nature Science Foundation of China (Grant No. 11471157).

By considering all possible reparametrizations of the flows instead of the time-$1$ maps, we introduce Bowen topological entropy and local entropy on subsets for flows. Through handling techniques for reparametrization balls, we prove a covering lemma for fixed-point free flows and then prove a variational principle.

Keywords: Entropy, Bowen entropy, variational principle, reparametrization, flow.
Mathematics Subject Classification: Primary: 37B40, 54H20.
Citation: Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273
References:
[1]

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

[2]

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

[3]

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

[4]

D. J. 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

[5]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[6]

Ya. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[7]

W. X. Sun, Measure-theoretic entropy for flows, Science in China Series A: Mathematics, 40 (1997), 725-731.  doi: 10.1007/BF02878695.  Google Scholar

[8]

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

[9]

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

[10]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

D. J. 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

[5]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[6]

Ya. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[7]

W. X. Sun, Measure-theoretic entropy for flows, Science in China Series A: Mathematics, 40 (1997), 725-731.  doi: 10.1007/BF02878695.  Google Scholar

[8]

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

[9]

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

[10]

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

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