A relative multifractal analysis: box-dimensions, densities, and projections

References

• N. Attia and B. Selmi, Relative multifractal box-dimensions, Filomat 33 (2019), 2841–2859. doi: 10.2298/FIL1909841A
   Web of Science ®Google Scholar
• N. Attia and B. Selmi, On the Billingsley dimension of Birkhoff average in the countable symbolic space, Comptes rendus Mathematique 358 (2020), 255–265. doi: 10.5802/crmath.21
   Web of Science ®Google Scholar
• N. Attia, B. Selmi, and Ch. Souissi, Some density results of relative multifractal analysis, Chaos, Solitons & Fractals 103 (2017), 1–11. doi: 10.1016/j.chaos.2017.05.029
   Web of Science ®Google Scholar
• F. Bahroun and I. Bhouri, Multifractals and projections, Extr. Math. 21 (2006), 83–91.
   Google Scholar
• J. Barral, F. Ben Nasr, and J. Peyrière, Comparing multifractal formalism: the neighbouring box condition, Asian J. Math. 7 (2003), 149–166. doi: 10.4310/AJM.2003.v7.n2.a1
   Google Scholar
• F. Ben Nasr, I. Bhouri, and Y. Heurteaux, The validity of the multifractal formalism: results and examples, Adv. in Math. 165 (2002), 264–284. doi: 10.1006/aima.2001.2025
   Web of Science ®Google Scholar
• I. Bhouri, On the projections of generalized upper Lq -spectrum, Chaos, Solitons & Fractals 42 (2009), 1451–1462. doi: 10.1016/j.chaos.2009.03.056
   Web of Science ®Google Scholar
• P. Billingsley, Ergodic Theory and Information, Willey, New York, 1965.
   Google Scholar
• H. Cajar, Billingsley dimension in probability spaces, Lecture notes in mathematics, Vol. 892, Springer, New York, 1981.
   Google Scholar
• J. Cole, Relative multifractal analysis, Choas, Solitons & Fractals 11 (2000), 2233–2250. doi: 10.1016/S0960-0779(99)00143-5
   Web of Science ®Google Scholar
• C. Dai, Frostman lemma for Hausdorff measure and Packing measure in probability spaces, J. Nanjing University, Mathematical Biquarterly 12 (1995), 191–203.
   Google Scholar
• C. Dai and Y. Hou, Frostman lemmas for Hausdorff measure and packing measure in a product probability space and their physical application, Chaos, Solitons & Fractals 24 (2005), 733–744. doi: 10.1016/j.chaos.2004.05.045
   Web of Science ®Google Scholar
• C. Dai and Y. Li, A multifractal formalism in a probability space, Chaos, Solitons & Fractals 27 (2006), 57–73. doi: 10.1016/j.chaos.2004.11.088
   Web of Science ®Google Scholar
• C. Dai and Y. Li, Multifractal dimension inequalities in a probability space, Chaos, Solitons & Fractals 34 (2007), 213–223. doi: 10.1016/j.chaos.2006.04.028
   Web of Science ®Google Scholar
• M. Dai, X. Peng and W. Li, Relative Multifractal Analysis in a Probability Space, Interna Journal of Nonlinear Science 10 (2010), 313–319.
   Google Scholar
• M. Das, Local properties of self-similar measures, Illinois J. Math. 42 (1998), 313– 332. doi: 10.1215/ijm/1256045047
   Google Scholar
• M. Das, Hausdorff measures, dimensions and mutual singularity, Transactions of the American Mathematical Society 357 (2005), 4249–4268. doi: 10.1090/S0002-9947-05-04031-6
   Web of Science ®Google Scholar
• M. Das, Billingsley’s packing dimension, Proceedings of the American Mathematical Society 136 (2008), 273–278. doi: 10.1090/S0002-9939-07-09069-7
   Web of Science ®Google Scholar
• Z. Douzi and B. Selmi, Multifractal variation for projections of measures, Chaos, Solitons & Fractals 91 (2016), 414–420. doi: 10.1016/j.chaos.2016.06.026
   Web of Science ®Google Scholar
• Z. Douzi and B. Selmi, Regularities of general Hausdorff and packing functions, Chaos, Solitons & Fractals 123 (2019), 240–243. doi: 10.1016/j.chaos.2019.04.001
   Web of Science ®Google Scholar
• Z. Douzi and B. Selmi, The mutual singularity of the relative multifractal measures, Nonauton. Dyn. Syst. 8 (2021), 18-26. doi: 10.1515/msds-2020-0123
   Google Scholar
• K.J. Falconer and T. O’Neil, Convolutions and the geometry of multifractal measures, Math. Nachr. 204 (1999) 61–82. doi: 10.1002/mana.19992040105
   Web of Science ®Google Scholar
• O. Frostman, Potential dequilibre et capacit des ensembles avec quelques applications à la thorie des fonctions Meddel, Lunds Univ. Math. Sem. 3 (1935) 1–118.
   Google Scholar
• U. Frish, and G. Parisi, Fully Developed Turbulence and Intermittency, In: Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi, eds., International school of physics “Enrico Fermi”, course 88, pp. 71–88, North-Holland, New York, 1985.
   Google Scholar
• R. Kaufmann, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155. doi: 10.1112/S0025579300002503
   Web of Science ®Google Scholar
• M. Khelifi, H. Lotfi, A. Samti, and B. Selmi, A relative multifractal analysis, Choas, Solitons & Fractals 140 (2020), 110091. doi: 10.1016/j.chaos.2020.110091
   Web of Science ®Google Scholar
• P. Mattila, Geometry of sets and Measures in Euclidian Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995. doi:10.1017/CBO9780511623813
   Google Scholar
• T.C. O’Neil, The multifractal spectrum of quasi self-similar measures, Journal of Mathematical Analysis and Applications 211 (1997), 233–257. doi: 10.1006/jmaa.1997.5458
   Web of Science ®Google Scholar
• T.C. O’Neil, The multifractal spectra of projected measures in Euclidean spaces, Chaos, Solitons & Fractals 11 (2000), 901–921. doi: 10.1016/S0960-0779(98)00256-2
   Web of Science ®Google Scholar
• L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82–196. doi: 10.1006/aima.1995.1066
   Web of Science ®Google Scholar
• Y. Pesin, Dimension theory in dynamical systems, Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001
   Google Scholar
• J. Peyrière, Multifractal measures, Vol. 372, pp.175–186, Kluwer Academic Press, Dordrecht, 1992.
   Google Scholar
• J. Peyrière, A vectorial multifractal formalism, Proc. Sympos. Pure Math. 72 (2004), 217–230. doi: 10.1090/pspum/072.2/2112124
   Google Scholar
• B. Selmi, Measure of relative multifractal exact dimensions, Adv. Appl. Math. Sci. 17 (2018), 629–643.
   Google Scholar
• B. Selmi, On the strong regularity with the multifractal measures in a probability space, Anal. Math. Phys. 9 (2019), 1525–1534. doi: 10.1007/s13324-018-0261-5
   Web of Science ®Google Scholar
• B. Selmi, On the effect of projections on the Billingsley dimensions, Asian-European Journal of Mathematics 13 (2020), 2050128, 1–17. doi: 10.1142/S1793557120501284
   Web of Science ®Google Scholar
• B. Selmi, On the projections of the multifractal packing dimension for q > 1, Ann. Mat. Pura Appl. 199 (2020), 1519–1532. doi: 10.1007/s10231-019-00929-7
   Web of Science ®Google Scholar
• B. Selmi, The relative multifractal analysis, review and examples, Acta Scientiarum Mathematicarum 86 (2020), 635–666. doi: 10.14232/actasm-020-801-8
   Web of Science ®Google Scholar
• B. Selmi, Appendix to the paper “On the Billingsley dimension of Birkhoff average in the countable symbolic space”, Comptes rendus Mathematique 358 (2020), 939. doi: 10.5802/crmath.116
   Web of Science ®Google Scholar
• B. Selmi, Multifractal Geometry of Slices of Measure, Z. Anal. Anwend. 40 (2021), 237–253. doi: 10.4171/ZAA/1682
   Web of Science ®Google Scholar
• B. Selmi, Multifractal dimensions for projections of measures, Bol. Soc. Paran. Mat., to appear.
   Google Scholar
• B. Selmi, The relative multifractal densities: a review and application, Journal of Interdisciplinary Mathematics, to appear.
   Web of Science ®Google Scholar
• C. Tricot, General Hausdorff functions and the notion of one-sided measure and dimension, Ark. Math. 48 (2010), 149–176. doi: 10.1007/s11512-008-0087-8
   Web of Science ®Google Scholar
• J. Xiao and M. Wu, The multifractal dimension functions of homogeneous Moran measure, Fractals 16 (2008), 175–185. doi: 10.1142/S0218348X08003892
   Web of Science ®Google Scholar