ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS*

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doi:10.1007/s10473-023-0610-9

Abstract: In this paper, we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions. More precisely, we show that, given a real number $1\leq\beta\leq2$ , any real-valued continuous function in C([0,1]) can be decomposed into a product of two real-valued continuous functions, each having a graph of Hausdorff dimension $\beta$ . In addition, a product decomposition result for the packing dimension is obtained. This work answers affirmatively two questions raised by Verma and Priyadarshi [14].

Key words: Hausdorff dimension, packing dimension, graph of function, product of functions

CLC Number:

28A80

Jia LIU, Saisai SHI, Yuan ZHANG. ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS^*[J].Acta mathematica scientia,Series B, 2023, 43(6): 2483-2492.

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[1] Balka R. Dimensions of graphs of prevalent continuous maps. J Fractal Geom, 2016, 3: 407-428
[2] Balka R, Buczolich Z, Elekes M. The topological Hausdorff dimension and level sets of generic continuous functions on fractals. Chaos, Solitons & Fractals, 2011, 45(12): 1579-1589
[3] Balka R, Buczolich Z, Elekes M. A new fractal dimension: the topological Hausdorff dimension. Adv Math, 2015, 274: 881-927
[4] Bayart F, Heurteaux Y.On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets//Barral J, Seuret S. Further Developments in Fractals and Related Fields: Mathematical Foundations and Connections (Trends in Mathematics). New York: Birkhauser/Spinger, 2013: 25-34
[5] Dougherty R. Examples of non-shy sets. Fund Math, 1994, 144: 73-88
[6] Falconer K J.Fractal Geometry: Mathematical Foundations and Applications. 3rd ed. Chichester: John Wiley & Sons, 2014
[7] Falconer K J, Fraser J M. The horizon problem for prevalent surfaces. Math Proc Cambridge Philos Soc, 2011, 152(2): 355-372
[8] Gruslys V, Jonušas J, Mijović V, et al. Dimensions of prevalent continuous functions. Monatsh Math, 2012, 166(2): 153-180
[9] Hyde J, Laschos V, Olsen L, et al. On the box dimensions of graphs of typical continuous functions. J Math Anal Appl, 2012, 391(2): 567-581
[10] Liu J, Liu D Z. On the decomposition of continuous functions and dimensions. Fractals, 2021, 28(1): 2050007-1-6
[11] Liu J, Tan B, Wu J. Graphs of continuous functions and packing dimension. J Math Anal Appl, 2016, 435(2): 1099-1106
[12] Liu J, Wu J. A remark on decomposition of continuous functions. J Math Anal Appl, 2013, 401(1): 404-406
[13] Mauldin R D, Williams S C. On the Hausdorff dimension of some graphs. Trans Amer Math Soc, 1986, 298(2): 793-803
[14] Verma M, Priyadarshi A. Graph of continuous functions and fractal dimension. arxiv:2202.11502v1
[15] Wingren P. Dimensions of graphs of functions and lacunary decompositions of spline approximations. Real Anal Exchange, 2000, 26(1): 17-26
[16] Xiao Z Y.The classification of states for skew product Markov chains
(in Chinese). Acta Math Sci, 2003, 23A(3): 306-313
[17] Zhang X M, Hu D H. The dimensions of the range of random walks in time-random environments. Acta Math Sci, 2006, 26B(4): 615-628

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