Weighted Multifractal Spectrum of V -Statistics
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Weighted Multifractal Spectrum of V -Statistics Alejandro Mesón1,2 · Fernando Vericat1,2
Received: 18 May 2020 / Revised: 14 September 2020 / Accepted: 16 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 μ equals the measure-theoretic entropy of μ. 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). Keywords V -statistics · Weighted multifractal spectrum · Weighted saturation · Weighted specification Mathematics Subject Classification 37B40, 37C45
1 Introduction The multiple ergodic averages can be seen as a dynamical version of the Szemeredi theorem in combinatorial number theory. This kind of interplay was studied by Furstenberg [9]. He analyzed ergodic averages in a measure-preserving probability space (X , B, μ, f ) of the form
B
Fernando Vericat [email protected] Alejandro Mesón [email protected]
1
Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), CCT CONICET-La Plata, La Plata, Argentina
2
Grupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería (GAMEFI), UNLP, La Plata, Argentina
123
Journal of Dynamics and Differential Equations N −1 1 μ A ∩ f n A ∩ · · · ∩ f kn A , N−M
(1.1)
n=M
where A ∈ B and j ∈ N. Furstenberg proved that if μ (A) > 0 then lim inf N →∞
N −1 1 μ A ∩ fnA ∩ ··· ∩ f N−M
jn
A > 0.
n=M
From this can be proved, by arguments of Ergodic Theory, the Szemeredi theorem which in short says that if S is a set of integers with positive upper density then S contains arithmetic progressions of arbitrary length. The V -statistics, thus called after the article by Fan et al. [5], are multiergodic averages of the following form: let (X , f ) be a topological dynamical system with X a compact metric space and f a continuous map, let X r = X × · · · × X be the product of r -copies of X with r ≥ 1. If : X r → R is a continuous map, then we can define 1 (1.2) f i1 (x) , . . . , f ir (x) . V (n, x) = r n 1≤i 1 ,...,ir ≤n
These averages are called the V -statistics of order r with kernel . Ergodic limits of the form 1 lim r f i1 (x) , . . . , f ir (x) , n→∞ n 1≤i 1 ,...,ir ≤n
were studied among others by Furstenberg [9], Bergelson [2] and Bourgain [3]. The multifractal decomposition for the spectra of V -statistics is E (α) = x : lim V (n, x) = α . n→∞
Hereafter (X i , di , f i ) , i = 1, 2, . . . , k, with k ≥ 2,
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