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Fractal sets typically have very complex geometric structures, and a fundamental problem in fractal geometry is to characterize how “similar” different fractal sets are. The Lipschitz equivalence of fractal sets is often used to classify fractal sets that are geometrically similar. Interesting links between Lipschitz equivalence and algebraic properties of contraction ratios for self-similar sets have been uncovered and widely analyzed. However, with the exception of very few papers, the study of Lipschitz equivalence has largely focused on totally disconnected self-similar sets. For connected self-similar sets this problem becomes rather challenging, even for well known fractal models such as fractal squares. In this paper, we introduce geometric and topological methods to study the Lipschitz equivalence of connected fractal squares. In particular we completely characterize the Lipschitz equivalence of fractal squares of order 3 in which one or two squares are removed. We also discuss the Lipschitz equivalence of fractal squares of more general orders. Our paper is the first study of Lipschitz equivalence for nontrivial connected self-similar sets, and it raises also some interesting questions for the more general setting. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min–max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs. [Copyright &y& Elsevier]

Fractal interpolation functions provide a new method to model experimental data. Dalla and Drakopoulos got some conditions that a vertical scaling factor must obey to model effectively an arbitrary function (J. Approx. Theory 101 (1999) 289). In this paper, we present certain counterexamples to show that the converse does not hold. [Copyright &y& Elsevier]