Abstract
We define a self-affine function whose typical example is the component function of the famous Peano curve (P 1(t),P 2(t)), 0≤t≤1. We obtain the Hausdorff and packing dimension of the graph of a self-affine function under some conditions. We also prove that the functionP 1(t−P 2(t) has a continuous occupation density with respect to time and space which will be the first example of a continuous deterministic function whose occupation density is continuous with respect to time and space.
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Kôno, N. On self-affine functions. Japan J. Appl. Math. 3, 259–269 (1986). https://doi.org/10.1007/BF03167101
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DOI: https://doi.org/10.1007/BF03167101