Projections and slices of measures
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We consider a generalization of the $L^q$-spectrum with respect to two Borel probability measures on $\mathbb{R}^n$ having the same compact support, and also study their behavior under orthogonal projections of measures onto an $m$-dimensional subspace. In particular, we try to improve the main result of Bahroun and Bhouri \cite{1}. In addition, we are interested in studying the behavior of the generalized lower and upper $L^q$-spectrum with respect to two measures on ``sliced'' measures in an $(n-m)$-dimensional linear subspace. The results in this article establish relations with the $L^q$-spectrum with respect to two Borel probability measures and its projections and generalize some well-known results.
Abstract
We consider a generalization of the $L^q$-spectrum with respect to two Borel probability measures on $\mathbb{R}^n$ having the same compact support, and also study their behavior under orthogonal projections of measures onto an $m$-dimensional subspace. In particular, we try to improve the main result of Bahroun and Bhouri \cite{1}. In addition, we are interested in studying the behavior of the generalized lower and upper $L^q$-spectrum with respect to two measures on ``sliced'' measures in an $(n-m)$-dimensional linear subspace. The results in this article establish relations with the $L^q$-spectrum with respect to two Borel probability measures and its projections and generalize some well-known results.
참고문헌 (42)
- Najmeddine Attia, "Relative multifractal box-dimensions", Filomat, 33(9), pp.2841~2859 (2019), National Library of Serbia[DOI]
- N. Attia, "On the Billingsley dimension of Birkhoff average in the countable symbolic space", Comptes Rendus Math, 358, pp.255~265 (2020)[DOI]
- N. Attia, "Some density results of relative multifractal analysis", Chaos, Solitons & Fractals, 103, pp.1~11 (2017)[DOI]
- F. Bahroun, "Multifractals and projections", Extracta Math, 21(1), pp.83~91 (2006)
- J. Barral, "Multifractal analysis for projections of Gibbs and related measures", Ergodic Theory Dynam. Systems, 31(3,), pp.673~701 (2011)[DOI]
- F. Ben Nasr, "Analyse multifractale de mesures", C. R. Acad. Sci. Paris S´er. I Math, 319(. 8,), pp.807~810 (1994)
- Imen Bhouri, "On the projections of generalized upper Lq-spectrum", Chaos, Solitons & Fractals, 42(3), pp.1451~1462 (2009), Elsevier BV[DOI]
- Z. Douzi, "Multifractal variation for projections of measures", Chaos, Solitons & Fractals, 91, pp.414~420 (2016)[DOI]
- Z. Douzi, "On the projections of mutual multifractal spectra" (2018)
- Z. Douzi, "On the projections of the mutual multifractal R´enyi dimensions", Anal. Theory Appl, 37(4), pp.572~592 (2021), Global Science Press[DOI]
- K. J. Falconer, "Sets with large intersection properties", J. London Math. Soc, (2) 49(2), pp.267~280 (1994)[DOI]
- K. J. Falconer, "Packing dimensions of projections and dimension profiles", Math. Proc. Cambridge Philos. Soc, 121(2), pp.269~286 (1997)[DOI]
- K. J. Falconer, "The packing dimension of projections and sections of measures", Math. Proc. Cambridge Philos. Soc, 119(4), pp.695~713 (1996)[DOI]
- K. J. Falconer, "Convolutions and the geometry of multifractal measures", Math. Nachr, 204, pp.61~82 (1999)[DOI]
- Y. Heurteaux, "Estimations de la dimension inf´erieure et de la dimension sup´erieure des mesures", Ann. Inst. H. Poincar´e Probab. Statist, 34(3), pp.309~338 (1998)[DOI]
- X. Hu, "Fractal properties of products and projections of measures in Rd", Math. Proc. Cambridge Philos. Soc, 115(3), pp.527~544 (1994)[DOI]
- B. R. Hunt, "How projections affect the dimension spectrum of fractal measures", Nonlinearity, 10(5), pp.1031~1046 (1997)[DOI]
- E. J¨arvenp¨a¨a, "Linear mappings and generalized upper spectrum for dimensions", Nonlinearity, 12(3), pp.475~493 (1999)[DOI]
- E. J¨arvenp¨a¨a, "One-dimensional families of projections", Nonlinearity, 21(3), pp.453~463 (2008)[DOI]
- E. J¨arvenp¨a¨a, "Local dimensions of sliced measures and stability of packing dimensions of sections of sets", Adv. Math, 183(1,), pp.127~154 (2004)[DOI]
- E. J¨arvenp¨a¨a, "Hausdorff dimension and non-degenerate families of projections", J. Geom. Anal, 24(4,), pp.2020~2034 (2014)[DOI]
- M. J¨arvenp¨a¨a, "Hausdorff and packing dimensions and sections of measures", Mathematika, 45(1,), pp.55~77 (1998)[DOI]
- R. Kaufman, "On Hausdorff dimension of projections", Mathematika, 15, pp.153~155 (1968)[DOI]
- M. Khelifi, "A relative multifractal analysis", Choas, Solitons & Fractals, 140, pp.110091~ (2020)[DOI]
- J. M. Marstrand, "Some fundamental geometrical properties of plane sets of fractional dimensions", Proc. London Math. Soc, (3) 4, pp.257~302 (1954)[DOI]
- P. Mattila, "Hausdorff dimension, orthogonal projections and intersections with planes,", Ann. Acad. Sci. Fenn. Ser. A I Math, 1(2), pp.227~244 (1975)
- P. Mattila, "Geometry of Sets and Measures in Euclidean Space" (1995), Cambridge University Press
- Sze-Man Ngai, "A dimension result arising from the $L^q$-spectrum of a measure", Proceedings of the American Mathematical Society, 125(10), pp.2943~2951 (1997), American Mathematical Society (AMS)[DOI]
- L. Olsen, "Fractal geometry and stochastics, II", pp.3~37 (1998), Birkh¨auser
- L. Olsen, "Random Geometrically Graph Directed Self-Similar Multifractals" (1994), Pitman
- T. C. O’Neil, "The multifractal spectra of projected measures in Euclidean spaces", Chaos, Solitons & Fractals, 11(6), pp.901~921 (2000)[DOI]
- T. Orponen, "On the packing measure of slices of self-similar sets", J. Fractal Geom, 2(4), pp.389~401 (2015)[DOI]
- D. Preiss, "Geometry of measures in Rn: distribution, rectifiability, and densities", Ann. of Math, (2) 125(3), pp.537~643 (1987)[DOI]
- A. Rapaport, "On the Hausdorff and packing measures of slices of dynamically defined sets", J. Fractal Geom, 3(1), pp.33~74 (2016)[DOI]
- 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), pp.38~51 (2016)[DOI]
- B. Selmi, "Multifractal dimensions for projections of measures", Bol. Soc. Paran. Mat[DOI]
- Bilel Selmi, "On the projections of the multifractal packing dimension for $$q>1$$", Annali di Matematica Pura ed Applicata (1923 -), 199(4), pp.1519~1532 (2020), Springer Science and Business Media LLC[DOI]
- B. Selmi, "On the effect of projections on the Billingsley dimensions", Asian-Eur. J. Math, 13, pp.2050128~ (2020)[DOI]
- B. Selmi, "Projection estimates for mutual multifractal dimensions", J. Pure Appl. Math. Adv. Appl, 22, pp.71~89 (2020)[DOI]
- B. Selmi, "Svetova, On the projections of mutual Lq,t-spectrum", Probl. Anal. Issues Anal, 6(24)(2), pp.94~108 (2017)[DOI]
- "Local dimensions of intersection measures: similarities, linear maps and continuously differentiable functions"[URL]
- M. Z¨ahle, "The average fractal dimension and projections of measures and sets in Rn", Fractals, 3(4,), pp.747~754 (1995)[DOI]
