The Multifractal Spectra of V-Statistics

Let (X, T) be a topological dynamical system and let Φ : X r → ℝ be a continuous function on the product space X r = X ×⋯ ×X (r ≥ 1). We are interested in the limit of V-statistics taking Φ as kernel: $$\lim\limits_{n\rightarrow \infty }{n}^{-r}\displaystyle\sum\limits_{ 1\leq i_{1},\cdots \,,i_{r}\leq n}\Phi ({T}^{i_{1} }x,\cdots \,,{T}^{i_{r} }x).$$ The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case (r = 1) where the spectrum is an analytic function when Φ is Holder continuous, the spectrum of the limit of higher-order V-statistics (r ≥ 2) may be discontinuous even for very nice kernel Φ.