Abstract
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:
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 Hölder continuous, the spectrum of the limit of higher-order V-statistics (r ≥ 2) may be discontinuous even for very nice kernel Φ.
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Fan, Ah., Schmeling, J., Wu, M. (2013). The Multifractal Spectra of V-Statistics. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_7
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