Geodesic
A note on the effect of projections on both measures and the generalization of $q$-dimension capacity
Problemy analiza, Tome 5 (2016) no. 2, pp. 38-51.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we are concerned both with the properties of the generalization of the $L^q$-spectrum relatively to two Borel probability measures and with the generalized $q$-dimension Riesz capacity. We are also interested in the study of their behaviors under orthogonal projections.
Keywords: orthogonal projection; Hausdorff measure and dimension; capacity; dimension spectra. 
@article{PA_2016_5_2_a3,
     author = {Bilel Selmi},
     title = {A note on the effect of projections on both measures and the generalization of $q$-dimension capacity},
     journal = {Problemy analiza},
     pages = {38--51},
     publisher = {mathdoc},
     volume = {5},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2016_5_2_a3/}
}
TY  - JOUR
AU  - Bilel Selmi
TI  - A note on the effect of projections on both measures and the generalization of $q$-dimension capacity
JO  - Problemy analiza
PY  - 2016
SP  - 38
EP  - 51
VL  - 5
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2016_5_2_a3/
LA  - en
ID  - PA_2016_5_2_a3
ER  - 
%0 Journal Article
%A Bilel Selmi
%T A note on the effect of projections on both measures and the generalization of $q$-dimension capacity
%J Problemy analiza
%D 2016
%P 38-51
%V 5
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2016_5_2_a3/
%G en
%F PA_2016_5_2_a3
Bilel Selmi. A note on the effect of projections on both measures and the generalization of $q$-dimension capacity. Problemy analiza, Tome 5 (2016) no. 2, pp. 38-51. http://geodesic.mathdoc.fr/item/PA_2016_5_2_a3/

[1] Barral J., Bhouri I., “Multifractal analysis for projections of Gibbs and related measures”, Ergod. Th. Dynam. Sys., 31 (2011), 673–701 | DOI | MR | Zbl

[2] Douzi Z., Selmi B., “Multifractal variation for projections of measures”, Chaos Solitons Fractals, 91 (2016), 414–420 | DOI | MR

[3] Falconer K. J., Howroyd J., “Packing Dimensions of Projections and Dimensions Profiles”, Math. Proc. Cambridge Philos. Soc., 121 (1997), 269–286 | DOI | MR | Zbl

[4] Falconer K. J., Howroyd J., “Projection theorems for box and packing dimensions”, Math. Proc. Cambridge Philos. Soc., 119 (1996), 287–295 | DOI | MR | Zbl

[5] Falconer K. J., Mattila P., “The packing dimensions of projections and sections of measures”, Math. Proc. Cambridge Philos. Soc., 119 (1996), 695–713 | DOI | MR | Zbl

[6] Kaufmann R., “On Hausdorff dimension of projections”, Mathematika, 15 (1968), 153–155 | DOI | MR

[7] Marstrand J. M., “Some fundamental geometrical properties of plane sets of fractional dimensions”, Proc. Lond. Math. Soc., 3 (1954), 257–302 | DOI | MR | Zbl

[8] Mattila P., “Hausdorff dimension. Orthogonal projections and intersections with planes”, Ann. Acad. Sci. Fenn. Ser., A I Math., 1 (1975), 227–244 | DOI | MR | Zbl

[9] O'Neil T., “The multifractal spectrum of projected measures in Euclidean spaces”, Chaos Solitons Fractals, 11 (2000), 901–921 | DOI | MR | Zbl

[10] Falconer K. J., “Fractal Geometry”, Math. Found. Appl., 1990, 27–37 | MR

[11] Hunt B. R., Kaloshin V. Y., “How projections affect the dimension spectrum of fractal measures”, Nonlinearity, 10 (1997), 1031–1046 | DOI | MR | Zbl

[12] Bahroun F., Bhouri I., “Multifractals and projections”, Extr. Math., 21 (2006), 83–91 | MR | Zbl

[13] Mattila P., The Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math., 1995 | MR

[14] Mattila P., “Orthogonal projections, Riez capacities and minkowski contents”, Ind. Univ. Math. J., 39 (1990), 185–197 | DOI | MR

[15] Bayart F., “Multifractal spectra of typical and prevalent measures”, Nonlinearity, 26 (2013), 353–367 | DOI | MR | Zbl

[16] Hunt B. R., Kaloshin V. Y., “The effect pical and prevalent measures”, Nonlinearity, 26 (2013), 353–367 | DOI | MR

[17] Olsen L., “A multifractal formalism”, Adv. Math., 116 (1995), 82–196 | DOI | MR | Zbl

[18] Olsen L., “Mixed generalized dimensions of self-similar measures”, J. Math. Anal. Appl., 306 (2005), 516–539 | DOI | MR | Zbl

[19] Olsen L., “Typical $L^q$-Dimensions of Measures”, Monatsh. Math., 146 (2005), 143–157 | DOI | MR | Zbl

[20] Olsen L., “Typical upper $L^q$-dimensions of measures for $q\in[0, 1]$”, Bull. Sci. math., a132 (2008), 551–561 | DOI | MR | Zbl

[21] Shi Y., Dai M., “Typical Lower $L^q$-dimensions of Measures for $q \le 1$”, Int. J. Nonlinear Sci., 7 (2009), 231–236 | MR | Zbl

[22] Cole J., “Relative multifractal analysis”, Chaos Solitons Fractals, 11 (2000), 2233–2250 | DOI | MR | Zbl

[23] Cole J., Olsen L., “Multifractal variation measures and multifractal density theorems”, Real. Anal. Exchange, 28 (2003), 501–514 | MR | Zbl

[24] Svetova N. Yu., “The relative Renyi dimention”, Probl. Anal. Issues Anal., 1 (2012), 15–23 | DOI | MR | Zbl

[25] Dai M., Li Y., “A multifractal formalism in a probability space”, Chaos Solitons Fractals, 27 (2006), 57–73 | DOI | MR | Zbl

[26] Dai M., “The equivalence of measures on Moran set in general metric space”, Chaos Solitons Fractals, 29 (2006), 55–64 | DOI | MR | Zbl

[27] Dai M., Li Y., “Multifractal dimension inequalities in a probability space”, Chaos Solitons Fractals, 34 (2007), 213–223 | DOI | MR | Zbl

[28] Dai M., Peng X., Li W., “Relative Multifractal Analysis in a Probability Space”, Int. J. Nonlinear Sci., 10 (2010), 313–319 | MR | Zbl

[29] Dai M., Jiang Y., “The equivalence of multifractal measures on cookie-cutter-like sets”, Chaos Solitons Fractals, 41 (2009), 1408–1415 | DOI | MR | Zbl

[30] Naschie E., “The Fractal dimension of spacetime-remarks on theoretical derivation and experimental verification”, Chaos Solitons Fractals, 9 (1998), 1211–1217 | DOI | MR | Zbl

[31] Naschie E., “Remarks on superstrings, fractal gravity, Nagasawas diffusion and Cantorian spacetime”, Chaos Solitons Fractals, 8 (1997), 187–386 | DOI | MR

[32] Naschie E., “The Bethe lattice and the dimension of micro-spacetime”, Chaos Solitons Fractals, 8 (1997), 1873–1886 | DOI | MR | Zbl

[33] Naschie E., “Silver mean Hausdorff dimension and Cantor sets”, Chaos Solitons Fractals, 4 (1994), 170–186

[34] Naschie E., “Average symmetry stability and ergodicity of multi-dimensional Cantor sets, II”, Nuo Cim., 109 (1994), 149–157 | DOI

[35] Ord G., Nottale L., Naschie E., “The impact of nonlinear dynamics and fractals on quantum physics and relativity”, Chaos Solitons Fractals, 9 (1998), 100–110 | MR

[36] Frostman O., “Potential dequilibre et capacit des ensembles avec quelques applications à la thorie des fonctions Meddel”, Lunds Univ. Math. Sem., 3 (1935), 1–118

[37] Gol'dshtein V., Troyanov M., “Capacities on metric spaces”, Integral Equations Oper. Theory, 44 (2002), 212–242 | DOI | MR | Zbl

[38] Nuutinen J., Silvestre P., The Riesz capacity in metric spaces, 2015, arXiv: 1501.05746v2

[39] Lehrback J., Shanmugalingam N., Quasiadditivity of variational capacity, 2012, arXiv: 1211.6933v1 | MR