Weighted Topological Entropy of the Set of Generic Points in Topological Dynamical Systems
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Abstract
This article is devoted to the investigation of the weighted topological entropy of generic points of the ergodic measures in dynamical systems. We showed that the weighted topological entropy of generic points of the ergodic measure \(\mu \) is equal to the weighted measure entropy of \(\mu ,\) which generalized the classical result of Bowen (Trans Am Math Soc 184:125–136, 1973). As an application, we also use the result to study the dimension of generic points for a class of skew product expanding maps on high dimensional tori.
Keywords
Weighted entropy Generic points Variational principleMathematics Subject Classification
37D35Notes
Acknowledgements
The first and second author were supported by NNSF of China (11671208 and 11431012). The third author was supported by NNSF of China (11601235 and 11271191), NSF of the Jiangsu Higher Education Institutions of China (16KJD110003), NSF of Jiangsu Province (BK20161014) and China Postdoctoral Science Foundation (2016M591873).
References
- 1.Bowen, R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184, 125–136 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Barral, J., Feng, D.: Weighted thermodynamic formalism on subshifts and applications. Asian J. Math. 16(2), 319–352 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Feng, D., Huang, W.: Variational principle for the weighted topological pressure. J. Math. Pures Appl. 106(3), 411–452 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Gurevich, B.M., Tempelman, A.A.: Hausdorff dimension of sets of generic points for Gibbs measures. J. Stat. Phys. 108(5–6), 1281–1301 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Kenyou, R., Peres, Y.: Measures of full dimension on affine-invariant sets. Ergod. Theory Dyn. Syst. 16, 307–323 (1996)MathSciNetGoogle Scholar
- 6.Ledrappier, F., Young, L.S.: The metric entropy of diffeomorphisms. Ann. Math. 122, 540–574 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.McMulen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1–9 (1984)MathSciNetCrossRefGoogle Scholar
- 8.Pesin, Y.: Dimension Theory in Dynamical Systems, Contemporary Views and Applications. University of Chicago Press, Chicago (1997)CrossRefzbMATHGoogle Scholar
- 9.Pesin, Y., Pitskel, B.: Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl. 18, 307–318 (1984)MathSciNetCrossRefzbMATHGoogle Scholar