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Mellin transforms and correlation dimensions
Institution:1. Institute of Mathematics and Computer Sciences, University of São Paulo, PO Box 369, 13560-970, São Carlos, SP, Brazil;2. Indiana University Network Science Institute, Bloomington, IN, USA;3. CENTAI Institute, Corso Inghilterra, 3, 10138, Turin, Italy;4. São Carlos Institute of Physics, University of São Paulo, PO Box 369, 13560-970, São Carlos, SP, Brazil;5. ISI Foundation, Via Chisola 5, 10126, Turin, Italy;1. CHU Sainte-Justine Research Center, Department of Psychiatry, University of Montreal, Canada;2. Center for Psychedelic & Consciousness Research, John Hopkins University, United States of America;3. Department for Developmental Psychology, University of Amsterdam, Netherlands;4. Mila – Quebec AI Institute, Canada;1. Department of Neuroscience and Developmental Biology, Vienna BioCenter (VBC), University of Vienna, Djerassiplatz 1, 1030 Vienna, Austria;2. Research Institute of Molecular Pathology (IMP), Vienna BioCenter (VBC), Campus-Vienna-Biocenter 1, 1030 Vienna, Austria
Abstract:We consider the Mellin transform of the correlation integrals and show that the divergence abscissa is the correlation dimension. The analytic structure of the Mellin transform is explicitly described for some Julia and Cantor sets. The existence of oscillations in the correlation integral for the Cantor sets is proved. Extensions of the results to the order d correlation integrals are discussed.
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