Weighted multifractal spectrum of V-statistics

MESON, ALEJANDRO M.; VERICAT, FERNANDO

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS

SPRINGER

Lugar: New York; Año: 2022 vol. 34 p. 1085 - 1105

1040-7294

We analyze and describe the weighted multifractal spectrum of V-statistics. The descriptionwill be possible when the condition of "weighted saturation" is fulfilled. This means that theweighted topological entropy of the set of generic points of a measure u equals the measure-theoretic entropy of u. Zhao et al. (J Dyn Diff Equat. 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).