Multifractal dimensions for projections of measures
DOI:
https://doi.org/10.5269/bspm.44913Abstract
In this paper, we calculate the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai in \cite{D} about the multifractal analysis of a measure of multifractal exact dimension.
References
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5. I. S. Baek. Multifractal by self-similar measures. J. Appl. Math. & Computing., 23 (2007), 497-503.
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7. A. Batakis and B. Testud. Multifractal analysis of inhomogeneous Bernoulli products. Journal of Statistical Physics. 142 (2011), 1105-1120. https://doi.org/10.1007/s10955-011-0147-5
8. F. Ben Nasr and J. Peyriere. Revisiting the multifractal analysis of measures. Revista Math. Ibro., 25 (2013), 315-328. https://doi.org/10.4171/RMI/721
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10. F. Ben Nasr, I. Bhouri and Y. Heurteaux. The validity of the multifractal formalism: results and examples. Adv. in Math., 165 (2002), 264-284. https://doi.org/10.1006/aima.2001.2025
11. A. Ben Mabrouk. A higher order multifractal formalism. Stat. Prob. Lett., 78 (2008), 1412-1421. https://doi.org/10.1016/j.spl.2007.12.010
12. J. Cole. Relative multifractal analysis. Chaos, Solitons & Fractals, 11: (2000), 2233-2250. https://doi.org/10.1016/S0960-0779(99)00143-5
13. M. F. Dai. Multifractal analysis of a measure of multifractal exact dimension. Nonlinear Analysis: Theory, Methods & Applications. 70 (2008), 1069-1079. https://doi.org/10.1016/j.na.2008.01.033
14. Z. Douzi and B. Selmi. Multifractal variation for projections of measures. Chaos, Solitons & Fractals. 91 (2016), 414-420. https://doi.org/10.1016/j.chaos.2016.06.026
15. Z. Douzi and B. Selmi. On the projections of mutual multifractal spectra. arXiv:1805.06866v1, (2018).
16. K. J. Falconer. Techniques in fractal geometry. Wiley. New York., (1997). https://doi.org/10.2307/2533585
17. K. J. Falconer and P. Mattila. The packing dimensions of projections and sections of measures. Math. Proc. Cambridge Philos. Soc., 119 (1996), 695-713. https://doi.org/10.1017/S0305004100074533
18. K. J. Falconer and T. C. O'Neil. Convolutions and the geometry of multifractal measures. Math. Nachr., 204 (1999), 61-82. https://doi.org/10.1002/mana.19992040105
19. A. H. Fan. Sur la dimension des mesures. Studia. Math., 111 (1994), 1-17. https://doi.org/10.4064/sm-111-1-1-17
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21. A. H. Fan, K. S. Lau and H. Rao. Relationships between different dimensios of a measures. Monatsh. Math., 135 (2002), 191-201. https://doi.org/10.1007/s006050200016
22. A. Farhat and A. Ben Mabrouk, A joint multifractal analysis of finitely many non Gibbs-Ahlfors type measures. viXra:1808.0576, (2018).
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26. Y. Heurteaux. Dimension of measures: the probabilistic approach. Publ. Mat., 51 (2007), 243-290. https://doi.org/10.5565/PUBLMAT_51207_01
27. J. Li. A note on multifractal packig dimension of measures. Anal. Theory Appl., 25 (2009), 147-153. https://doi.org/10.1007/s10496-009-0147-3
28. P. Mattila. The Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambrdige. (1995). https://doi.org/10.1017/CBO9780511623813
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30. M. Menceur, A. Ben Mabrouk and K. Betina. The multifractal formalism for measures, review and extension to mixed cases. Anal. Theory Appl., 32 (2016), 77-106. https://doi.org/10.4208/ata.2016.v32.n4.1
31. M. Menceur and A. Ben Mabrouk. A mixed multifractal formalism for finitely many non Gibbs Frostman-like measures. arXiv:1804.09034v1, (2018).
32. T. C. O'Neil. The multifractal spectra of projected measures in Euclidean spaces. Chaos, Solitons & Fractals. 11 (2000), 901-921. https://doi.org/10.1016/S0960-0779(98)00256-2
33. L. Olsen. A multifractal formalism. Advances in Mathematics. 116 (1995), 82-196. https://doi.org/10.1006/aima.1995.1066
34. L. Olsen. Multifractal dimensions of product measures. Math. Proc. Camb. Phil. Soc., 120 (1996), 709-734. https://doi.org/10.1017/S0305004100001675
35. L. Olsen. Measurability of multifractal measure functions and multifractal dimension functions. Hiroshima Math. J., 29 (1999), 435-458. https://doi.org/10.32917/hmj/1206124851
36. L. Olsen. Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures. Math. Scand., 86 (2000), 109-129. https://doi.org/10.7146/math.scand.a-14284
37. P. Y. Pesin. Dimension type characteristics for invariant sets of dynamical systems. Russian Math: Surveys. 43 (1988), 111-151. https://doi.org/10.1070/RM1988v043n04ABEH001892
38. P. Y. Pesin. Dimension theory in dynamical systems, Contemporary views and applications. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, (1997). https://doi.org/10.7208/chicago/9780226662237.001.0001
39. S. Shen. Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's b and B functions. Journal of Statistical Physics. 159 (2015), 1216-1235. https://doi.org/10.1007/s10955-015-1223-z
40. B. Selmi. A note on the effect of projections on both measures and the generalization of q-dimension capacity. Probl. Anal. Issues Anal., 5 (23) (2016), 38-51. https://doi.org/10.15393/j3.art.2016.3290
41. B. Selmi. Measure of relative multifractal exact dimensions. Adv. Appl. Math. Sci., 17 (2018), 629-643.
42. B. Selmi. Some results about the regularities of multifractal measures. Korean J. Math. 26 (2018), 271-283
43. B. Selmi. On the strong regularity with the multifractal measures in a probability space. Analysis and Mathematical Physics (to appear).
44. B. Selmi and N. Yu. Svetova. On the projections of mutual Lq,t-spectrum. Probl. Anal. Issues Anal., 6 (24) (2017), 94-108. https://doi.org/10.15393/j3.art.2017.4231
45. M. Tamashiro. Dimensions in a separable metric space. Kyushu. J. Math., 49 (1995), 143-162. https://doi.org/10.2206/kyushujm.49.143
46. M. Wu. The multifractal spectrum of some Moran measures. Sci. China. Ser. A Math., 48 (2005), 97-112. https://doi.org/10.1360/022004-10
47. M. Wu. The singularity spectrum f(α) of some Moran fractals. Monatsh Math., 144 (2005), 141-55. https://doi.org/10.1007/s00605-004-0254-3
48. M. Wu and J. Xiao. The singularity spectrum of some non-regularity moran fractals. Chaos, Solitons & Fractals. 44 (2011), 548-557. https://doi.org/10.1016/j.chaos.2011.05.002
49. J. Xiao and M. Wu. The multifractal dimension functions of homogeneous moran measure. Fractals. 16 (2008), 175-185. https://doi.org/10.1142/S0218348X08003892
2. N. Attia, B. Selmi and Ch. Souissi. Some density results of relative multifractal analysis. Chaos, Solitons & Fractals. 103 (2017), 1-11. https://doi.org/10.1016/j.chaos.2017.05.029
3. I. S. Baek. On deranged Cantor sets. Kyungpook Math. Journal. 38 (1998), 363-367.
4. I. S. Baek. Weak local dimension on deranged Cantor sets. Real Analysis Exchange. 26 (2001), 553-558. https://doi.org/10.2307/44154059
5. I. S. Baek. Multifractal by self-similar measures. J. Appl. Math. & Computing., 23 (2007), 497-503.
6. I. S. Baek. On multifractal of Cantor dust. Acta Mathematica Sinica, English Series. 25 (2009), 1175-1182. https://doi.org/10.1007/s10114-009-7111-1
7. A. Batakis and B. Testud. Multifractal analysis of inhomogeneous Bernoulli products. Journal of Statistical Physics. 142 (2011), 1105-1120. https://doi.org/10.1007/s10955-011-0147-5
8. F. Ben Nasr and J. Peyriere. Revisiting the multifractal analysis of measures. Revista Math. Ibro., 25 (2013), 315-328. https://doi.org/10.4171/RMI/721
9. F. Ben Nasr and I. Bhouri. Spectre multifractal de mesures borliennes sur Rd, C.R. Acad. Sci. Paris Ser. I Math., 325 (1997), 253-256. https://doi.org/10.1016/S0764-4442(97)83950-X
10. F. Ben Nasr, I. Bhouri and Y. Heurteaux. The validity of the multifractal formalism: results and examples. Adv. in Math., 165 (2002), 264-284. https://doi.org/10.1006/aima.2001.2025
11. A. Ben Mabrouk. A higher order multifractal formalism. Stat. Prob. Lett., 78 (2008), 1412-1421. https://doi.org/10.1016/j.spl.2007.12.010
12. J. Cole. Relative multifractal analysis. Chaos, Solitons & Fractals, 11: (2000), 2233-2250. https://doi.org/10.1016/S0960-0779(99)00143-5
13. M. F. Dai. Multifractal analysis of a measure of multifractal exact dimension. Nonlinear Analysis: Theory, Methods & Applications. 70 (2008), 1069-1079. https://doi.org/10.1016/j.na.2008.01.033
14. Z. Douzi and B. Selmi. Multifractal variation for projections of measures. Chaos, Solitons & Fractals. 91 (2016), 414-420. https://doi.org/10.1016/j.chaos.2016.06.026
15. Z. Douzi and B. Selmi. On the projections of mutual multifractal spectra. arXiv:1805.06866v1, (2018).
16. K. J. Falconer. Techniques in fractal geometry. Wiley. New York., (1997). https://doi.org/10.2307/2533585
17. K. J. Falconer and P. Mattila. The packing dimensions of projections and sections of measures. Math. Proc. Cambridge Philos. Soc., 119 (1996), 695-713. https://doi.org/10.1017/S0305004100074533
18. K. J. Falconer and T. C. O'Neil. Convolutions and the geometry of multifractal measures. Math. Nachr., 204 (1999), 61-82. https://doi.org/10.1002/mana.19992040105
19. A. H. Fan. Sur la dimension des mesures. Studia. Math., 111 (1994), 1-17. https://doi.org/10.4064/sm-111-1-1-17
20. A. H. Fan. On ergodicity and unidimensionality. Kyushu. J. Math., 48 (1994), 249-255. https://doi.org/10.2206/kyushujm.48.249
21. A. H. Fan, K. S. Lau and H. Rao. Relationships between different dimensios of a measures. Monatsh. Math., 135 (2002), 191-201. https://doi.org/10.1007/s006050200016
22. A. Farhat and A. Ben Mabrouk, A joint multifractal analysis of finitely many non Gibbs-Ahlfors type measures. viXra:1808.0576, (2018).
23. B. Testud. Etude d'une classe de mesures autosimilaires : calculs de dimensions et analyse multifractale. Th'ese. (2004).
24. Y. Heurteaux. Sur la comparaison des mesures avec les mesures de Hausdorff. C. R. Acad. Sci. Paris Ser. I Math., 321 (1995), 61-65.
25. Y. Heurteaux. Estimations de la dimension inf'erieure et de la dimension sup'erieure des mesures. Ann. Inst. H. Poincar'e Probab. Statist., 34 (1998), 309-338. https://doi.org/10.1016/S0246-0203(98)80014-9
26. Y. Heurteaux. Dimension of measures: the probabilistic approach. Publ. Mat., 51 (2007), 243-290. https://doi.org/10.5565/PUBLMAT_51207_01
27. J. Li. A note on multifractal packig dimension of measures. Anal. Theory Appl., 25 (2009), 147-153. https://doi.org/10.1007/s10496-009-0147-3
28. P. Mattila. The Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambrdige. (1995). https://doi.org/10.1017/CBO9780511623813
29. P. Mattila. Hausdorff dimension, orthogonal projections and intersections with planes. Annales Academiae Scientiarum Fennicae. Series A Mathematica. 1 (1975), 227-244. https://doi.org/10.5186/aasfm.1975.0110
30. M. Menceur, A. Ben Mabrouk and K. Betina. The multifractal formalism for measures, review and extension to mixed cases. Anal. Theory Appl., 32 (2016), 77-106. https://doi.org/10.4208/ata.2016.v32.n4.1
31. M. Menceur and A. Ben Mabrouk. A mixed multifractal formalism for finitely many non Gibbs Frostman-like measures. arXiv:1804.09034v1, (2018).
32. T. C. O'Neil. The multifractal spectra of projected measures in Euclidean spaces. Chaos, Solitons & Fractals. 11 (2000), 901-921. https://doi.org/10.1016/S0960-0779(98)00256-2
33. L. Olsen. A multifractal formalism. Advances in Mathematics. 116 (1995), 82-196. https://doi.org/10.1006/aima.1995.1066
34. L. Olsen. Multifractal dimensions of product measures. Math. Proc. Camb. Phil. Soc., 120 (1996), 709-734. https://doi.org/10.1017/S0305004100001675
35. L. Olsen. Measurability of multifractal measure functions and multifractal dimension functions. Hiroshima Math. J., 29 (1999), 435-458. https://doi.org/10.32917/hmj/1206124851
36. L. Olsen. Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures. Math. Scand., 86 (2000), 109-129. https://doi.org/10.7146/math.scand.a-14284
37. P. Y. Pesin. Dimension type characteristics for invariant sets of dynamical systems. Russian Math: Surveys. 43 (1988), 111-151. https://doi.org/10.1070/RM1988v043n04ABEH001892
38. P. Y. Pesin. Dimension theory in dynamical systems, Contemporary views and applications. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, (1997). https://doi.org/10.7208/chicago/9780226662237.001.0001
39. S. Shen. Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's b and B functions. Journal of Statistical Physics. 159 (2015), 1216-1235. https://doi.org/10.1007/s10955-015-1223-z
40. B. Selmi. A note on the effect of projections on both measures and the generalization of q-dimension capacity. Probl. Anal. Issues Anal., 5 (23) (2016), 38-51. https://doi.org/10.15393/j3.art.2016.3290
41. B. Selmi. Measure of relative multifractal exact dimensions. Adv. Appl. Math. Sci., 17 (2018), 629-643.
42. B. Selmi. Some results about the regularities of multifractal measures. Korean J. Math. 26 (2018), 271-283
43. B. Selmi. On the strong regularity with the multifractal measures in a probability space. Analysis and Mathematical Physics (to appear).
44. B. Selmi and N. Yu. Svetova. On the projections of mutual Lq,t-spectrum. Probl. Anal. Issues Anal., 6 (24) (2017), 94-108. https://doi.org/10.15393/j3.art.2017.4231
45. M. Tamashiro. Dimensions in a separable metric space. Kyushu. J. Math., 49 (1995), 143-162. https://doi.org/10.2206/kyushujm.49.143
46. M. Wu. The multifractal spectrum of some Moran measures. Sci. China. Ser. A Math., 48 (2005), 97-112. https://doi.org/10.1360/022004-10
47. M. Wu. The singularity spectrum f(α) of some Moran fractals. Monatsh Math., 144 (2005), 141-55. https://doi.org/10.1007/s00605-004-0254-3
48. M. Wu and J. Xiao. The singularity spectrum of some non-regularity moran fractals. Chaos, Solitons & Fractals. 44 (2011), 548-557. https://doi.org/10.1016/j.chaos.2011.05.002
49. J. Xiao and M. Wu. The multifractal dimension functions of homogeneous moran measure. Fractals. 16 (2008), 175-185. https://doi.org/10.1142/S0218348X08003892
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