On the Irregular Part of V -Statistics Multifractal Spectra for Systems with Non-Uniform Specification

Abstract

Let (X,f) be a dynamical system with X a compact metric space. Let X^{r} be the product of r-copies of X, r≥1, and Φ:X^{r} →R. The multifractal decomposition for V-statistics for Φ,f is defined as E_{Φ,f}(α)={x:lim_{n→∞}(1/(n^{r}))∑_{0≤i₁,...,i_{r}≤n-1}Φ(f^{i₁}(x),...,f^{i_{r(x))=α}. The set of points x∈X, for which the limit does not exist is called the irregular part, or historic set, of the spectrum. In this article we analyze the irregular part of the V-statistics for systems satisfying a weak form of the known Bowen specification property, called the non-uniform specification property. This concept was introduced by P. Varandas and allows to work in a nonuniformly hyperbolic context.