Some Notes on Topological and Measure-Theoretic Entropy


In this note, we introduce four measure-theoretic quantities for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies and establish two new variational principles for Bowen topological entropy and packing topological entropy. Besides, we prove that the packing topological entropy of the set of generic points for any invariant ergodic Borel probability measure is equal to the measure-theoretic entropy.

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  1. 1.

    Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114(2), 309–319 (1965)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bowen, R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184, 125–136 (1973)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Brin, M., Katok, A.: On Local Entropy. Lecture Notes in Mathematics, vol. 1007. Springer-Verlag, Berlin (1983)

    Google Scholar 

  5. 5.

    Dou, D., Zheng, D.M., Zhou, X.M.: Packing topological entropy for amenable group actions. arXiv:2010.14719

  6. 6.

    Feng, D.J., Huang, W.: Variational principles for topological entropies of subsets. J. Funct. Anal. 263, 2228–2254 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51, 137–173 (1980)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  9. 9.

    Pesin, Y.B.: Dimension Theory in Dynamical Systems. Contemporary Views and Applications. University of Chicago Press, Chicago (1997)

    Google Scholar 

  10. 10.

    Pfister, C.-E., Sullivan, W.G.: On the topological entropy of saturated sets. Ergodic Theory Dyn. Syst. 27, 929–956 (2007)

    MathSciNet  Article  Google Scholar 

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Correspondence to Tao Wang.

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This work was partly supported by National Nature Science Foundation of China (12001192).

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Wang, T. Some Notes on Topological and Measure-Theoretic Entropy. Qual. Theory Dyn. Syst. 20, 13 (2021).

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  • Topological entropies
  • Measure-theoretic entropies
  • Variational principles
  • Generic points

Mathematics Subject Classification

  • 37A35
  • 37B40
  • 37C45