| Abstract: |
| Barnsley introduced the theory of non-smooth interpolation for finite data in his seminal work ``Fractal Functions and Interpolation, Constr. Approx 2 (1986) 303-329 by exploiting an iterated function system concept. After that, numerous fractal functions are constructed corresponding to real/vector-valued functions. Recently, fractal interpolation functions corresponding to a set-valued function on a compact interval of the real line have been constructed. Set-valued functions played a significant role in applied areas such as mathematical modeling, game theory, control theory, and many more. Dimension estimation of any set or the graph of a function remains a vibrant area of research in the literature. In this talk, we first construct the set-valued fractal function corresponding to any continuous set-valued function defined on the compact subset of the real line using the metric sum of sets. Subsequently, some results are obtained for the bounds estimation of fractal dimensions, such as the Hausdorff and box dimension for the graphs of constructed fractal functions. Further, some bounds on the dimension of the distance sets of graphs of these functions are also discussed. In the end, we shed some light on the celebrated Falconer`s distance-set conjecture regarding the graphs of set-valued functions. |
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