Exploring Higher Order Local Entropy Functions via Baire Category Theory

• Tingting Wang ^[1] ; Bilel Selmi ^[2] ; Zhiming Li ^[1]
  1. [1] Northwest University

     Northwest University

     China

     
  2. [2] University of Monastir

     University of Monastir

     Túnez

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this study, we rigorously prove that for a typical measure, the local entropy function of a continuous flow does not exist. Specifically, the pattern demonstrated by a typical measure is incredibly complex and irregular, and even after employing widely accepted and significantly effective smoothing techniques, including higher-order Riesz-Hardy logarithmic averages and Cesàro averages, the local entropy function of the flow still does not exist. More precisely, the lower average local entropy function is zero, while the upper average local entropy function is infinity.

• Referencias bibliográficas
  • 1. Aaronson, J., Denker, M., Fisher, A.: Second order ergodic theorems for ergodic transformations of infinite measure spaces. Proc. Amer....
  • 2. Adam-Day, B., Ashcroft, C., Olsen, L., Pinzani, N., Rizzoli, A., Rowe, J.: On the average box dimensions of graphs of typical continuous...
  • 3. Allen, D., Edwards, H., Harper, S., Olsen, L.: Average distances on self-similar sets and higher order average distances of self-similar...
  • 4. Bedford, T., Fisher, A.: Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. 64,...
  • 5. Bi´s, A., Mihailescu, E.: Inverse pressure for finitely generated semigroups. Nonlinear Anal. 222, 112942 (2022)
  • 6. Bowen, R.: Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184, 125–136 (1973)
  • 7. Brin, M., Katok, A.: On local entropy. Lecture Notes in Mathematics, vol. 1007. BerlinNewYork, Springer-Verlag (1983)
  • 8. Dou, D., Fan, M., Qiu, H.: Topological entropy on subsets for fixed-point free flows. Discrete Contin. Dyn. Syst. 37, 6319–6331 (2017)
  • 9. Fisher, A.: Integer Cantor sets and an order-two ergodic theorem. Ergod. Th. Dynam. Sys. 13, 45–64 (1993)
  • 10. Feng, D., Huang, W.: Variational principles for topological entropies of subsets. J. Funct. Anal. 263, 2228–2254 (2012)
  • 11. Graf, S.: On Bandt’s tangential distribution for self-similar measures. Monatsh. Math. 120, 223–246 (1995)
  • 12. Ji, Y., Chen, E., Wang, Y., Zhao, C.: Bowen entropy for fixed-point free flows. Discrete Contin. Dyn. Syst. 39, 6231–6239 (2019)
  • 13. Katok, A.: Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dynam. 1, 545–596 (2007)
  • 14. Li, Z., Selmi, B., Zyoudi, H.: A comprehensive approach to multifractal analysis. Expositiones Mathematicae 43(5), 125690 (2025). (ISSN...
  • 15. Mañé, R.: Ergodic theory and differentiable dynamics. Springer-Verlag, Berlin (1987)
  • 16. Mihailescu, E.: Ergodic properties for some non-expanding non-reversible systems. Nonlinear Anal. 73(12), 3779–3787 (2010)
  • 17. Mörters, P.: Average densities and linear rectifiability of measures. Mathematika 44, 313–324 (1997)
  • 18. Mörters, P.: The average density of the path of planar Brownian motion. Stochastic Process. Appl. 74, 133–149 (1998)
  • 19. Olsen, L.: Higher order local dimensions and Baire category. Studia Math. 204, 1–20 (2011)
  • 20. Olsen, L.: On average Hewitt-Stromberg measures of typical compact metric spaces. Math. Z. 293, 1201–1225 (2019)
  • 21. Olsen, L.: On the average Lq -dimensions of typical measures belonging to the Gromov-HausdorffProhoroff space. J. Math. Anal. Appl. 469,...
  • 22. Olsen, L.: Average box dimensions of typical compact sets. Ann. Acad. Sci. Fenn. Math. 44, 141–165 (2019)
  • 23. Olsen, L., Richardson, A.: Average distances between points in graph-directed self-similar fractals. Math Nachr 292, 170–194 (2019)
  • 24. Olsen, L., West, M.: Average frequencies of digits in infinite IFS’s and applications to continued fractions and Liiroth expansions. Monatshefte...
  • 25. Olsen, L.: On average Hewitt-Stromberg measures of typical compact metric spaces. Math. Z. 293, 1201–1225 (2019)
  • 26. Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)
  • 27. Romeo, R.F.: Entropy of expansive flows. Ergod. Th. Dynam. Sys. 7, 611–625 (1987)
  • 28. Romeo, R.F.: Topological entropy of fixed-point free flows. Trans. Amer. Math. Soc. 319, 601–618 (1990)
  • 29. Sun, W.: Measure-theoretic entropy for flows. Sci. China Ser. A 40, 725–731 (1997)
  • 30. Sun, W., Vargas, E.: Entropy of flows, revisited. Bol. Soc. Brasil. Mat. (N.S.) 30, 315–333 (1999)
  • 31. Selmi, B.: Average Hewitt-Stromberg and box dimensions of typical compact metric spaces. Quaest. Math. 46, 411–444 (2023)
  • 32. Selmi, B.: Average general fractal dimensions of typical compact metric spaces. Fuzzy Sets and Systems 499, 109192 (2025)
  • 33. Sun, W.: Entropy of orthonormal n-frame flows. Nonlinearity 14, 829 (2001)
  • 34. Shen, J., Zhao, Y.: Entropy of a flow on non-compact sets. Open Syst. Inf. Dyn. 19 02, 1250015 (2012)
  • 35. Wang, Y., Chen, E., Lin, Z., Wu, T.: Bowen entropy of sets of generic points for fixed-point free flows. J. Differential Equations 269,...
  • 36. Xu, L., Zhou, X.: Variational principles for entropies of nonautonomous dynamical systems. J. Dynam. Differential Equations 30, 1053–1062...
  • 37. Zähle, M.: The average density of self-conformal measures. J. London Math. Soc. 63, 721–734 (2001)