Projections of measures with small supports

Bilel Selmi

doi:10.2478/aupcsm-2021-0001

.blurhash-client-img { display: none !important; }

Projections of measures with small supports

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

Volume 20 (2021): Issue 1 (January 2021)

By: Bilel Selmi

Open Access

|Jan 2022

Abstract

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.

References

[1] Attia, Najmeddine, and Bilel Selmi. “Relative multifractal box-dimensions.” Filomat 33, no. 9 (2019): 2841-2859. Cited on 6.10.2298/FIL1909841A
Search in Google Scholar Back to article
[2] Attia, Najmeddine, Bilel Selmi, and Chouhaïd Souissi. “Some density results of relative multifractal analysis.” Chaos Solitons Fractals 103 (2017): 1-11. Cited on 6.10.1016/j.chaos.2017.05.029
Search in Google Scholar Back to article
[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. Cited on 5.10.1142/S021969131450009X
Search in Google Scholar Back to article
[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. Cited on 5 and 6.10.1017/S0143385710000143
Search in Google Scholar Back to article
[5] Cole, Julian David. “Relative multifractal analysis.” Chaos Solitons Fractals 11, no. 14 (2000): 2233-2250. Cited on 6.10.1016/S0960-0779(99)00143-5
Search in Google Scholar Back to article
[6] Dai, Meifeng, et al. “Mixed multifractal analysis of crude oil, gold and exchange rate series.” Fractals 24, no. 4 (2016): article no. 1650046. Cited on 5.10.1142/S0218348X16500468
Search in Google Scholar Back to article
[7] Douzi, Zied, and Bilel Selmi. “Multifractal variation for projections of measures.” Chaos Solitons Fractals 91 (2016): 414-420. Cited on 5 and 6.10.1016/j.chaos.2016.06.026
Search in Google Scholar Back to article
[8] Douzi, Zied, and Bilel Selmi. “On the projections of mutual multifractal spectra.” Arxiv (2018): arxiv.org/pdf/1805.06866.pdf Cited on 5 and 6.
Search in Google Scholar Back to article
[9] Douzi, Zied, and Bilel Selmi. “On the projections of the mutual multifractal Rényi dimensions.” Anal. Theory Appl.(to appear). Cited on 6.
Search in Google Scholar Back to article
[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. Cited on 5.10.1017/S0305004196001375
Search in Google Scholar Back to article
[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. Cited on 5 and 8.10.1017/S0305004100074168
Search in Google Scholar Back to article
[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. Cited on 5.10.1017/S0305004100074533
Search in Google Scholar Back to article
[13] Falconer, Kenneth John, and Toby Christopher O’Neil. “Convolutions and the geometry of multifractal measures.” Math. Nachr. 204 (1999): 61-82. Cited on 8 and 10.
Search in Google Scholar Back to article
[14] Kaufman, Robert P. “On Hausdorff dimension of projections.” Mathematika 15 (1968): 153-155. Cited on 5.10.1112/S0025579300002503
Search in Google Scholar Back to article
[15] Khelifi, Mounir, et al. “A relative multifractal analysis.” Chaos Solitons Fractals 140 (2020): article no. 110091. Cited on 6.10.1016/j.chaos.2020.110091
Search in Google Scholar Back to article
[16] Marstrand, John Martin “Some fundamental geometrical properties of plane sets of fractional dimensions.” Proc. London Math. Soc. (3) 4 (1954): 257-302. Cited on 5.10.1112/plms/s3-4.1.257
Search in Google Scholar Back to article
[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. Cited on 5.10.5186/aasfm.1975.0110
Search in Google Scholar Back to article
[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. Cited on 5, 6 and 7.10.4208/ata.2016.v32.n4.1
Search in Google Scholar Back to article
[19] Menceur, Mohamed, and Anouar Ben Mabrouk. “A joint multifractal analysis of vector valued non Gibbs measures.” Chaos Solitons Fractals 126 (2019): 203-217. Cited on 7.10.1016/j.chaos.2019.05.010
Search in Google Scholar Back to article
[20] O’Neil, Toby Christopher. “The multifractal spectra of projected measures in Euclidean spaces.” Chaos Solitons Fractals 11, no. 6 (2000): 901-921. Cited on 5, 6, 7, 8, 9 and 11.
Search in Google Scholar Back to article
[21] O’Neil, Toby Christopher. “The multifractal spectrum of quasi-self-similar measures.” J. Math. Anal. Appl. 211, no. 1 (1997): 233-257. Cited on 9.10.1006/jmaa.1997.5458
Search in Google Scholar Back to article
[22] Olsen, Lars Ole Ronnow. “A multifractal formalism.” Adv. Math. 116, no. 1 (1995): 82-196. Cited on 6.10.1006/aima.1995.1066
Search in Google Scholar Back to article
[23] Olsen, Lars Ole Ronnow. “Mixed generalized dimensions of self-similar measures.” J. Math. Anal. Appl. 306, no. 2 (2005): 516-539. Cited on 6.10.1016/j.jmaa.2004.12.022
Search in Google Scholar Back to article
[24] Rogers, Claude Ambrose. Hausdorff Measures. Cambridge: Cambridge University Press, 1970. Cited on 7.
Search in Google Scholar Back to article
[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. Cited on 5.10.15393/j3.art.2016.3290
Search in Google Scholar Back to article
[26] Selmi, Bilel. “Measure of relative multifractal exact dimensions.” Advances and Applications in Mathematical Sciences 17, no. 10 (2018): 629-643. Cited on 10.
Search in Google Scholar Back to article
[27] Selmi, Bilel. “Multifractal dimensions for projections of measures.” Bol. Soc. Paran. Mat. (to appear). Cited on 5 and 10.
Search in Google Scholar Back to article
[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. Cited on 5.10.1007/s10231-019-00929-7
Search in Google Scholar Back to article
[29] Selmi, Bilel. “On the strong regularity with the multifractal measures in a probability space.” Anal. Math. Phys. 9, no. 3 (2019): 1525-1534. Cited on 6.10.1007/s13324-018-0261-5
Search in Google Scholar Back to article
[30] Selmi, Bilel. “Projection estimates for mutual multifractal dimensions.” J. Pure Appl. Math. Adv. 22, no. 1 (2020): 71-89. Cited on 6.10.18642/jpamaa_7100122121
Search in Google Scholar Back to article
[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. Cited on 6.10.5802/crmath.116
Search in Google Scholar Back to article
[32] Selmi, Bilel. “Some new characterizations of Olsen’s multifractal functions.” Results Math. 75, no. 4 (2020): paper no. 147. Cited on 10.
Search in Google Scholar Back to article
[33] Selmi, Bilel. “The relative multifractal analysis, review and examples.” Acta Sci. Math. (Szeged) 86, no. 3-4 (2020): 635-666. Cited on 6.10.14232/actasm-020-801-8
Search in Google Scholar Back to article
[34] Selmi, Bilel. “The relative multifractal densities: a review and application.” J. Interdiscip. Math. (to appear). Cited on 6.
Search in Google Scholar Back to article
[35] Selmi, Bilel. “On the effect of projections on the Billingsley dimensions.” Asian-Eur. J. Math. 13, no. 7 (2020): 2050128. 010.1142/S1793557120501284
Search in Google Scholar Back to article
[36] Selmi, Bilel, and Nina Yuryevna Svetova. “On the projections of mutual L^q,t-spectrum.” Probl. Anal. Issues Anal. 6(24), no. 2 (2017): 94-108. Cited on 5. Cited on 6.10.15393/j3.art.2017.4231
Search in Google Scholar Back to article
[37] Selmi, Bilel, and Nina Yuryevna Svetova. “Projections and Slices of measures.” Commun. Korean Math. Soc. (to appear). Cited on 5.
Search in Google Scholar Back to article
[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. Cited on 5 and 7.
Search in Google Scholar Back to article
[39] Svetova, Nina Yuryevna. “Mutual multifractal spectra. I. Exact spectra.” Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 11 (2004): 41-46. Cited on 5 and 7.
Search in Google Scholar Back to article