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Weighted topological and measure-theoretic entropy
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China |
Feng and Huang in 2016 defined a new notion called weighted topological entropy (pressure) and obtained the corresponding variational principle for compact dynamical systems. In this paper, it was our hope to carry out a further study from the following three aspects:
(1) Inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships.
(2) The classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version.
(3) A formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.
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] |
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,
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,
The relation between topological entropy and metric entropy, Sov. Math., 11 (1970), 13-16.
|
| [6] |
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. |
| [7] |
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. |
| [8] |
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. |
| [9] |
D. J. Feng and W. Huang,
Variational principle for the weighted topological pressure, J. Math. Pures Appl., 106 (2016), 411-452.
doi: 10.1016/j.matpur.2016.02.016. |
| [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. Lond. 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: 10.2307/2036610. |
| [13] |
W. Huang, X. D. Ye and G. H. Zhang,
Lowering topological entropy over subsets, Ergodic Theory Dynam. Systems, 30 (2010), 181-209.
doi: 10.1017/S0143385709000066. |
| [14] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173.
|
| [15] |
A. N. Kolmogorov,
A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR, 119 (1958), 861-864.
|
| [16] |
P. Oprocha and G. H. Zhang,
Dimensional entropy over sets and fibres, Nonlinearity, 24 (2011), 2325-2346.
doi: 10.1088/0951-7715/24/8/009. |
| [17] |
Y. B. Pesin, Dimension Theory in Tynamical Systems: Contemporary views and applications, University of Chicago Press, Chicago, 1997.
doi: 10.7208/chicago/9780226662237.001.0001.
|
| [18] |
Ya. B. Pesin and B. S. Pitskel',
Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63.
|
| [19] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982. |
| [20] |
C. Zhao, E. C. Chen, X. Y. Zhou and Z. Yin,
Weighted topological entropy of the set of generic points in topological dynamical systems, J. Dynam. Differential Equations, 30 (2018), 937-955.
doi: 10.1007/s10884-017-9575-5. |
| [21] |
X. M. Zhou,
A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst., 36 (2016), 4063-4075.
doi: 10.3934/dcds.2016.36.4063. |
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] |
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,
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,
The relation between topological entropy and metric entropy, Sov. Math., 11 (1970), 13-16.
|
| [6] |
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. |
| [7] |
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. |
| [8] |
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. |
| [9] |
D. J. Feng and W. Huang,
Variational principle for the weighted topological pressure, J. Math. Pures Appl., 106 (2016), 411-452.
doi: 10.1016/j.matpur.2016.02.016. |
| [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. Lond. 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: 10.2307/2036610. |
| [13] |
W. Huang, X. D. Ye and G. H. Zhang,
Lowering topological entropy over subsets, Ergodic Theory Dynam. Systems, 30 (2010), 181-209.
doi: 10.1017/S0143385709000066. |
| [14] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173.
|
| [15] |
A. N. Kolmogorov,
A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR, 119 (1958), 861-864.
|
| [16] |
P. Oprocha and G. H. Zhang,
Dimensional entropy over sets and fibres, Nonlinearity, 24 (2011), 2325-2346.
doi: 10.1088/0951-7715/24/8/009. |
| [17] |
Y. B. Pesin, Dimension Theory in Tynamical Systems: Contemporary views and applications, University of Chicago Press, Chicago, 1997.
doi: 10.7208/chicago/9780226662237.001.0001.
|
| [18] |
Ya. B. Pesin and B. S. Pitskel',
Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63.
|
| [19] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982. |
| [20] |
C. Zhao, E. C. Chen, X. Y. Zhou and Z. Yin,
Weighted topological entropy of the set of generic points in topological dynamical systems, J. Dynam. Differential Equations, 30 (2018), 937-955.
doi: 10.1007/s10884-017-9575-5. |
| [21] |
X. M. Zhou,
A formula of conditional entropy and some applications, Discrete Contin. Dyn. Syst., 36 (2016), 4063-4075.
doi: 10.3934/dcds.2016.36.4063. |
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