Projections of measures with small supports

Selmi, Bilel

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, Tome 20 (2021), pp. 5-15.

Résumé

In this paper, we use a characterization of the mutual multifractal Hausdorff dimension in terms of auxiliary measures to investigate the projections of measures with small supports.

Détail

Mots-clés : Multifractal analysis, Orthogonal projection, s-Ahlfors regular

Bibliographie

[1] Attia, Najmeddine, and Bilel Selmi. "Relative multifractal box-dimensions." Filomat 33, no. 9 (2019): 2841-2859.

[2] Attia, Najmeddine, Bilel Selmi, and Chouhaïd Souissi. "Some density results of relative multifractal analysis." Chaos Solitons Fractals 103 (2017): 1-11.

[3] Aouidi, Jamil, and Anouar Ben Mabrouk. "A wavelet multifractal formalism for simultaneous singularities of functions." Int. J. Wavelets Multiresolut. Inf. Process. 12, no. 1 (2014): article no. 1450009.

[4] Barral, Julien, and Imen Bhouri. "Multifractal analysis for projections of Gibbs and related measures." Ergodic Theory Dynam. Systems 31, no. 3 (2011): 673-701.

[5] Cole, Julian David. "Relative multifractal analysis." Chaos Solitons Fractals 11, no. 14 (2000): 2233-2250.

[6] Dai, Meifeng, et al. "Mixed multifractal analysis of crude oil, gold and exchange rate series." Fractals 24, no. 4 (2016): article no. 1650046.

[7] Douzi, Zied, and Bilel Selmi. "Multifractal variation for projections of measures." Chaos Solitons Fractals 91 (2016): 414-420.

[8] Douzi, Zied, and Bilel Selmi. "On the projections of mutual multifractal spectra." Arxiv (2018): arxiv.org/pdf/1805.06866.pdf

[9] Douzi, Zied, and Bilel Selmi. "On the projections of the mutual multifractal Rényi dimensions." Anal. Theory Appl.(to appear).

[10] Falconer, Kenneth John, and John D. Howroyd. "Packing dimensions of projections and dimension profiles." Math. Proc. Cambridge Philos. Soc. 121, no. 2 (1997): 269-286.

[11] Falconer, Kenneth John, and John D. Howroyd. "Projection theorems for box and packing dimensions." Math. Proc. Cambridge Philos. Soc. 119, no. 2 (1996): 287-295.

[12] Falconer, Kenneth John, and Pertti Mattila. "The packing dimension of projections and sections of measures." Math. Proc. Cambridge Philos. Soc. 119, no. 4 (1996): 695-713.

[13] Falconer, Kenneth John, and Toby Christopher O’Neil. "Convolutions and the geometry of multifractal measures." Math. Nachr. 204 (1999): 61-82.

[14] Kaufman, Robert P. "On Hausdorff dimension of projections." Mathematika 15 (1968): 153-155.

[15] Khelifi, Mounir, et al. "A relative multifractal analysis." Chaos Solitons Fractals 140 (2020): article no. 110091.

[16] Marstrand, John Martin "Some fundamental geometrical properties of plane sets of fractional dimensions." Proc. London Math. Soc. (3) 4 (1954): 257-302.

[17] Mattila, Pertti. "Hausdorff dimension, orthogonal projections and intersections with planes." Ann. Acad. Sci. Fenn. Ser. A I Math. 1, no. 2 (1975): 227-244.

[18] Menceur, Mohamed, and Anouar Ben Mabrouk, and Kamel Betina. "The multifractal formalism for measures, review and extension to mixed cases." Anal. Theory Appl. 32, no. 4 (2016): 303-332.

[19] Menceur, Mohamed, and Anouar Ben Mabrouk. "A joint multifractal analysis of vector valued non Gibbs measures." Chaos Solitons Fractals 126 (2019): 203-217.

[20] O’Neil, Toby Christopher. "The multifractal spectra of projected measures in Euclidean spaces." Chaos Solitons Fractals 11, no. 6 (2000): 901-921.

[21] O’Neil, Toby Christopher. "The multifractal spectrum of quasi-self-similar measures." J. Math. Anal. Appl. 211, no. 1 (1997): 233-257.

[22] Olsen, Lars Ole Ronnow. "A multifractal formalism." Adv. Math. 116, no. 1 (1995): 82-196.

[23] Olsen, Lars Ole Ronnow. "Mixed generalized dimensions of self-similar measures." J. Math. Anal. Appl. 306, no. 2 (2005): 516-539.

[24] Rogers, Claude Ambrose. Hausdorff Measures. Cambridge: Cambridge University Press, 1970.

[25] Selmi, Bilel. "A note on the effect of projections on both measures and the generalization of q-dimension capacity." Probl. Anal. Issues Anal. 5(23), no. 2 (2016): 38-51.

[26] Selmi, Bilel. "Measure of relative multifractal exact dimensions." Advances and Applications in Mathematical Sciences 17, no. 10 (2018): 629-643.

[27] Selmi, Bilel. "Multifractal dimensions for projections of measures." Bol. Soc. Paran. Mat. (to appear).

[28] Selmi, Bilel. "On the projections of the multifractal packing dimension for q > 1." Ann. Mat. Pura Appl. (4) 199, no. 4 (2020): 1519-1532.

[29] Selmi, Bilel. "On the strong regularity with the multifractal measures in a probability space." Anal. Math. Phys. 9, no. 3 (2019): 1525-1534.

[30] Selmi, Bilel. "Projection estimates for mutual multifractal dimensions." J. Pure Appl. Math. Adv. 22, no. 1 (2020): 71-89.

[31] Selmi, Bilel. "Appendix to the paper "On the Billingsley dimension of Birkhoff average in the countable symbolic space"." C. R. Math. Acad. Sci. Paris 358, no. 8 (2020): 939.

[32] Selmi, Bilel. "Some new characterizations of Olsen’s multifractal functions." Results Math. 75, no. 4 (2020): paper no. 147.

[33] Selmi, Bilel. "The relative multifractal analysis, review and examples." Acta Sci. Math. (Szeged) 86, no. 3-4 (2020): 635-666.

[34] Selmi, Bilel. "The relative multifractal densities: a review and application." J. Interdiscip. Math. (to appear).

[35] Selmi, Bilel. "On the effect of projections on the Billingsley dimensions." Asian-Eur. J. Math. 13, no. 7 (2020): 2050128.

[36] Selmi, Bilel, and Nina Yuryevna Svetova. "On the projections of mutual Lq,tspectrum." Probl. Anal. Issues Anal. 6(24), no. 2 (2017): 94-108.

[37] Selmi, Bilel, and Nina Yuryevna Svetova. "Projections and Slices of measures." Commun. Korean Math. Soc. (to appear).

[38] Svetova, Nina Yuryevna. "Mutual multifractal spectra. II. Legendre and Hentschel-Procaccia spectra, and spectra defined for partitions." Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 11 (2004): 47-56.

[39] Svetova, Nina Yuryevna. "Mutual multifractal spectra. I. Exact spectra." Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 11 (2004): 41-46.