Abstract
We analyze and describe the weighted multifractal spectrum of V-statistics. The description will be possible when the condition of “weighted saturation” is fulfilled. This means that the weighted topological entropy of the set of generic points of measure \(\mu \) equals the measure-theoretic entropy of \(\mu \). Zhao et al. (J Dyn Differ Equ 30:937–955, 2018) proved that for any ergodic measure weighted saturation is verified, generalizing a result of Bowen. Here we prove that under a property of “weighted specification” the saturation holds for any measure. From this we obtain the description of the spectrum of V-statistics. This generalizes the variational result that Fan, Schmeling and Wu obtained for the non-weighted case (arXiv:1206.3214v1, 2012).
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References
Barral, J., Feng, D.: Weighted thermodynamic formalism on subshifts and applications. Asian J. Math. 6, 319–352 (2012)
Bergelson, V.: Weakly mixing PET. Ergod. Theory Dyn. Syst. 7, 337–349 (2012)
Bourgain, J.: Double recurrence and almost sure convergence. J. Reine Angew. Math. 404, 140–161 (1990)
Bowen, R.: Topological entropy for non-compact sets. Trans. Am. Math. Soc. 184, 125–136 (1973)
Fan, A.H., Schmeling, J., Wu, J.: The multifractal spectra of \(V-\)statistics. In: Barral, J., Seuret, S. (eds.) Further Developments in Fractals and Related Fields. Trends Math, pp. 135–141. Birkhäuser/Springer, New York (2013)
Fan, A.H., Feng, D.J., Wu, J.: Recurrence, dimension and entropy. J. Lond. Math. Soc. 64, 229–244 (2001)
Fan, A.H., Liao, I.M., Peyrière, J.: Generic points in systems of specification and Banach valued Birkhoff averages. Discrete Contin. Dyn. Syst. 21, 1103–1128 (2008)
Feng, D.J., Huang, W.: Variational principle for the weighted topological pressure. J. Math. Pures Appl. 106, 411–452 (2016)
Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szmerédi on arithmetic progressions. J. d’Analyse Math. 31, 204–256 (1977)
Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 51, 137–173 (1980)
Li, B., Wu, J.: Beta-expansion and continued fraction expansion. J. Math. Anal. Appl. 339, 1322–1331 (2008)
Meson, A.M., Vericat, F.: On the topological entropy of the irregular part of V-statistics multifractal spectra. J. Dyn. Syst. Geom. Theories 11, 1–12 (2013)
Pomeau, Y., Manneville, P.P.: Intermittent transition to turbulence in dissipative dynamica systems. Commun. Math. Phys. 74, 189–197 (1980)
Takens, F., Verbitskiy, E.: On the variational principle for the topological entropy of certain non-compact sets. Ergod. Theory Dyn. Syst. 23, 317–348 (2003)
Varandas, V.: Non-uniform specification and large deviations for weak Gibbs measures. J. Stat. Phys. 146, 330–358 (2012)
Young, L.S.: Large deviations in dynamical systems. Trans. Am. Math. Soc. 318, 525–543 (1990)
Zhao, C., Chen, E., Zhou, X., Yin, X.: Weighted topological entropy of the set of generic points in topological dynamical systems. J. Dyn. Differ. Equ. 30, 937–955 (2018)
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Mesón, A., Vericat, F. Weighted Multifractal Spectrum of V-Statistics. J Dyn Diff Equat 34, 1085–1105 (2022). https://doi.org/10.1007/s10884-020-09915-7
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DOI: https://doi.org/10.1007/s10884-020-09915-7