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Remarks on nonlinear elastic waves in the radial symmetry in 2-D
A formula of conditional entropy and some applications
| 1. | Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, School of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China |
References:
| [1] |
R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.
doi: 10.1090/S0002-9947-1965-0175106-9. |
| [2] |
T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116. |
| [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] |
M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro,,(1981),in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
| [5] |
E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22. |
| [6] |
T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), 217-247.
doi: 10.4064/fm172-3-2. |
| [7] |
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, 2011.
doi: 10.1007/978-0-85729-021-2. |
| [8] |
C. Fang, W. Huang, Y. Yi and P. Zhang, Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628.
doi: 10.1017/S0143385710000982. |
| [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] |
E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/surv/101. |
| [11] |
T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.
doi: 10.1112/blms/3.2.176. |
| [12] |
L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.
doi: 10.1090/S0002-9939-1969-0247030-3. |
| [13] |
W. Huang, Stable sets and $\epsilon$-stable sets in positive-entropy systems, Comm. Math. Phys., 279 (2008), 535-557.
doi: 10.1007/s00220-008-0430-8. |
| [14] |
W. Huang, J. Li and X.D. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394.
doi: 10.1016/j.jfa.2014.01.005. |
| [15] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576. |
| [16] |
P. D. Liu, A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 593-603.
doi: 10.1017/S0143385704000586. |
| [17] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
| [18] |
Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986.
doi: 10.1007/978-1-4684-9175-3. |
| [19] |
A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, (Russian) Dokl. Akad. Nauk. SSSR (N.S.), 119 (1958), 861-864. |
show all references
References:
| [1] |
R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.
doi: 10.1090/S0002-9947-1965-0175106-9. |
| [2] |
T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116. |
| [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] |
M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro,,(1981),in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
| [5] |
E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22. |
| [6] |
T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), 217-247.
doi: 10.4064/fm172-3-2. |
| [7] |
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, 2011.
doi: 10.1007/978-0-85729-021-2. |
| [8] |
C. Fang, W. Huang, Y. Yi and P. Zhang, Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628.
doi: 10.1017/S0143385710000982. |
| [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] |
E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/surv/101. |
| [11] |
T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.
doi: 10.1112/blms/3.2.176. |
| [12] |
L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.
doi: 10.1090/S0002-9939-1969-0247030-3. |
| [13] |
W. Huang, Stable sets and $\epsilon$-stable sets in positive-entropy systems, Comm. Math. Phys., 279 (2008), 535-557.
doi: 10.1007/s00220-008-0430-8. |
| [14] |
W. Huang, J. Li and X.D. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394.
doi: 10.1016/j.jfa.2014.01.005. |
| [15] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc.(2), 16 (1977), 568-576. |
| [16] |
P. D. Liu, A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 593-603.
doi: 10.1017/S0143385704000586. |
| [17] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
| [18] |
Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986.
doi: 10.1007/978-1-4684-9175-3. |
| [19] |
A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, (Russian) Dokl. Akad. Nauk. SSSR (N.S.), 119 (1958), 861-864. |
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