-
Previous Article
Cohomological equation and cocycle rigidity of discrete parabolic actions
- DCDS Home
- This Issue
-
Next Article
The twisted cohomological equation over the geodesic flow
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. |
[1] |
Andrzej Biś. Entropies of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 639-648. doi: 10.3934/dcds.2004.11.639 |
[2] |
Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic and Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373 |
[3] |
Mickaël Crampon. Entropies of strictly convex projective manifolds. Journal of Modern Dynamics, 2009, 3 (4) : 511-547. doi: 10.3934/jmd.2009.3.511 |
[4] |
Zhiming Li, Yujun Zhu. Entropies of commuting transformations on Hilbert spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5795-5814. doi: 10.3934/dcds.2020246 |
[5] |
Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219 |
[6] |
Eduard Feireisl. Relative entropies in thermodynamics of complete fluid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3059-3080. doi: 10.3934/dcds.2012.32.3059 |
[7] |
Ansgar Jüngel, Ingrid Violet. First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 861-877. doi: 10.3934/dcdsb.2007.8.861 |
[8] |
José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415 |
[9] |
Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052 |
[10] |
Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1 |
[11] |
Ana-Maria Acu, Laura Hodis, Ioan Rasa. Multivariate weighted kantorovich operators. Mathematical Foundations of Computing, 2020, 3 (2) : 117-124. doi: 10.3934/mfc.2020009 |
[12] |
Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems and Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111 |
[13] |
Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723 |
[14] |
Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649 |
[15] |
Tim McGraw, Baba Vemuri, Evren Özarslan, Yunmei Chen, Thomas Mareci. Variational denoising of diffusion weighted MRI. Inverse Problems and Imaging, 2009, 3 (4) : 625-648. doi: 10.3934/ipi.2009.3.625 |
[16] |
Ronan Costaouec, Haoyun Feng, Jesús Izaguirre, Eric Darve. Analysis of the accelerated weighted ensemble methodology. Conference Publications, 2013, 2013 (special) : 171-181. doi: 10.3934/proc.2013.2013.171 |
[17] |
Igor E. Pritsker and Richard S. Varga. Weighted polynomial approximation in the complex plane. Electronic Research Announcements, 1997, 3: 38-44. |
[18] |
Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 |
[19] |
Qing Sun. Irrigable measures for weighted irrigation plans. Networks and Heterogeneous Media, 2021, 16 (3) : 493-511. doi: 10.3934/nhm.2021014 |
[20] |
Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]