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

Topological entropy on subsets for fixed-point free flows

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.

Citation: Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete and 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

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