A generalization of O'Neil's theorems for projections of measures and dimensions
DOI:
https://doi.org/10.22199/issn.0717-6279-5156Keywords:
projection, φ-multifractal Hausdorff and packing measures, φ-multifractal Hausdorff and packing dimensions, Bouligand-Minkowski φ-dimensionsAbstract
In this paper, more general versions of O’Neil’s projection theorems and other related theorems. In particular, we study the relationship between the φ-multifractal dimensions and its orthogonal projections in Euclidean space.
References
N. Attia, B. Selmi. “Relative multifractal box-dimensions”, Filomat, vol. 33, pp. 2841-2859, 2019.
I. Bhouri. “On the projections of generalized upper Lq-spectrum”, Chaos, Solitons & Fractals, vol. 42, pp. 1451-1462, 2009.
J. Barral, I. Bhouri, “Multifractal analysis for projections of Gibbs and related measures”, Ergodic Theory and Dynamic systems., vol. 31, pp. 673-701, 2011.
P. Billingsley, Ergodic theory and information. Wiley, New York. 1965.
L. Barreira, P. Doutor, “Almost additive multifractal analysis”, J. Math. Pures Appl., vol. 92, pp. 1-17, 2009.
G. Brown, G. Michon, J. Peyrire,”On the multifractal analysis of measures”, J. Stat. Phys., vol. 66, pp. 775-790, 1992.
L. Barreira, B. Saussol, “Variational principles and mixed multifractal spectra”, Trans. Amer. Math. Soc., vol. 353, pp. 3919-3944, 2001.
L. Barreira, B. Saussol, J. Schmeling, “Higher-dimensional multifractal analysis”, J. Math. Pures Appl., vol. 81, pp. 67-91, 2002.
J. Cole, “Relative multifractal analysis”, Chaos, Solitons & Fractals, vol. 11, pp. 2233-2250, 2000.
M. Dai, “Mixed self-conformal multifractal measures”, Analysis in Theory and Applications, vol. 25, pp. 154-165, 2009.
M. Dai, W. Li, “The mixed Lq-spectra of self-conformal measures satisfying the weak separation condition”, J. Math. Anal. Appl., vol. 382, pp. 140-147, 2011.
M. Dai, Y. Shi, “Typical behavior of mixed Lq-dimensions”, Nonlinear Analysis: Theory, Methods & Applications, vol. 72, pp. 2318-2325, 2010.
Z. Douzi, B. Selmi, “Multifractal variation for projections of measures”, Chaos, Solitons & Fractals, vol. 91, pp. 414-420, 2016.
Z. Douzi, B. Selmi, “A relative multifractal analysis: Box-dimensions, densities, and projections”, Quaestiones Mathematicae, vol. 45, no. 8, pp. 1243-1296, 2022.
Z. Douzi, B. Selmi, “On the projections of the mutual multifractal Rényi dimensions”, Anal. Theory Appl., vol. 36, no. 2, pp. 1-20, 2020.
M. Dai, C. Wang, H. Sun, “Mixed generalized dimensions of random self-similar measures”, Int. J. Nonlinear. Sci., vol. 13, pp. 123-128, 2012.
K. J. Falconer. The geometry of fractal sets, Cambridge univ. Press New. York-London, 1985.
K. J. Falconer, J. D. Howroyd, “Packing Dimensions of Projections and Dimensions Profiles”, Math. Proc. Cambridge Philos. Soc., vol. 121, pp. 269-286, 1997.
K. J. Falconer, X. Jim, “Exact dimensionality and projections of random self-similar measures and sets”, J. Lond. Math. Soc., vol. 90, pp. 388-412, 2014.
K. J. Falconer, T. C. O’Neil, “Convolutions and the geometry of multifractal measures”, Mathematische Nachrichten, vol. 204, pp. 61-82, 1999.
K. J. Falconer, P. Mattila, “Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes”, Journal of Fractal Geometry, vol. 3, pp. 319-329, 2016.
B. R. Hunt, V. Y. Kaloshin, “How projections affect the dimension spectrum of fractal measures”, Nonlinearity, vol. 10, pp. 1031-1046, 1997.
M. Hochman, P. Shmerkin, “Local entropy averages and projections of fractal measures”, 2009. [On line]. Available: https://arxiv.org/abs/0910.1956
R. Kaufman, “On Hausdorff dimension of projections”, Mathematika, vol. 15, pp. 153-155, 1968.
M. Kessebohmer, “Large deviation for weak Gibbs measures and multifractal spectra”, Nonlinearity, vol. 14, pp. 395-409, 2001.
B. Mandelbrot, “Negative Fractal Dimensions and Multifractals”, Physica A., vol. 163, pp. 306-315, 1990.
J. M. Marstrand, “Some fundamental geometrical properties of plane sets of fractional dimensions”, Proceedings of the London Mathematical Society, vol. 4, pp 257-302, 1954.
P. Mattila, The geometry of sets and measures in Euclidean spaces, Cambridge University Press. Cambrdige, 1995.
M. Menceur, A. Ben Mabrouk, “A mixed multifractal formalism for finitely many non Gibbs Frostman-like measures”, 2018. [On line]. Available: https://arxiv.org/abs/1804.09034
M. Menceur, A. Ben Mabrouk, K. Betina, “The multifractal formalism for measures, review and extension to mixed cases”, Anal. Theory Appl., vol. 32, pp. 77-106, 2016.
L. Olsen, “A multifractal formalism”, Advances in Mathematics, vol. 116, pp. 82-196, 1995.
L. Olsen, “Mixed generalized dimensions of self-similar measures”, J. Math. Anal. Appl., vol. 306, pp. 516-539, 2005.
T. C. O’Neil, “The multifractal spectrum of quasi self-similar measures”, Journal of Mathematical Analysis and its Applications, vol. 211, pp. 233-257, 1997.
T. C. O’Neil, “The multifractal spectra of projected measures in Euclidean spaces”, Chaos, Solitons & Fractals, vol. 11, pp. 901-921, 2000.
Y. Peres, W. Schalg, “Smoothness of projections, bernoulli convolutions, and the dimensions of exceptions”, Duke Math. J. , vol. 102, no. 2, pp. 193-251, 2000.
R. H. Riedi, “Conditional and relative multifractal spectra”, Fractals, vol. 5, pp. 153-168, 1997.
B. Selmi, “On the projections of the multifractal packing dimension for q > 1”, Annali di Matematica Pura ed Applicata, vol. 199, pp. 1519-1532, 2020.
B. Selmi, “On the effect of projections on the Billingsley dimensions”, Asian-Eur. J. Math., vol. 13, 2020.
N. Yu. Svetova. “The relative Renyi dimention”, Probl. Anal. Issues Anal, vol. 1, pp. 15-23, 2012.
B. Selmi and N. Yu. Svetova, “On the projections of mutual Lq,t-spectrum”, Probl. Anal. Issues Anal., vol 6, no. 24, pp. 94-108, 2017.
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