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Multifractal Spectrum for Barycentric Averages

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Abstract

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any xX and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any xX (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

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Acknowledgments

The support of this work by Consejo Nacional de Investigaciones Científicas y Técnicas, Universidad Nacional de La Plata and Universidad Nacional de Rosario of Argentina is greatly appreciated. FV is a member of CONICET.

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Correspondence to Fernando Vericat.

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Mesón, A., Vericat, F. Multifractal Spectrum for Barycentric Averages. J Dyn Control Syst 22, 623–635 (2016). https://doi.org/10.1007/s10883-015-9278-3

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