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 x∈X 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 x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:
and we will obtain a variational formula for this multifractal spectrum.
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References
Austin T. A C A T(0)−valued pointwise ergodic theorem. J Topol Anal 2011;3:145–152.
Bowen R. Topological entropy for non-compact sets. Trans Amer Math Soc 1973; 184:125–136.
Cartan E. Leçons sur la géométrie des espaces de Riemann. Paris: Guathier-Villars; 1951.
Navas A. A L 1 ergodic theorem with values in a nonpositively curved space with a canonical barycenter map. Ergo Th Dynam Sys 2012;33(2):609–623.
Pfister C.E., Sullivan W.G. On topological entropy of saturated sets. Ergod Th Dynam Sys 2007;27:1–29.
Sturm K.T. Probability measures on metric spaces, of non-positive curvature. Heat kernels and analysis on manifolds, graphs and metric spaces, (Paris 2002), vol. 338 of Contemp. Math.; 2003. p. 357–390.
Takens F., Verbitski E. On the variational principle for the topological entropy of certain non-compact sets. Ergod Th Dynam Sys 2003;23:317–348.
Walters P. An introduction to ergodic theory. Berlin: Springer; 1982.
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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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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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DOI: https://doi.org/10.1007/s10883-015-9278-3