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A correction is presented to the article "Correction To existence And Box Dimension Of General Recurrent Fractal Interpolation Functions" published in a previous issue of the periodical.

In this paper, we provide a complete characterization on when the Hausdorff and topological Hausdorff dimension of a given Bedford–McMullen carpet coincide. These two dimensions have a common value if and only if the carpet is either the Cartesian product of [ 0 , 1 ] and a self-similar set, or it is one of the two diagonals of the unit square. [ABSTRACT FROM AUTHOR]

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. ['Recurrent iterated function systems', Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$. We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting. [ABSTRACT FROM AUTHOR]

We present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on triangular domains. Then we introduce affine RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of affine RFISs under certain constraints. [ABSTRACT FROM AUTHOR]

Bilinear fractal interpolation surfaces were introduced by Ruan and Xu in 2015. In this paper, we present the formula for their box dimension under certain constraint conditions. [ABSTRACT FROM AUTHOR]

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]

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using spectral decimation, we prove that the “hot spots” conjecture holds on the level-3 Sierpinski gasket, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by J. Kigami) attains its maximum and minimum on the boundary. [Copyright &y& Elsevier]

Abstract: There are many works on the “hot spots” conjecture for domains in Euclidean space since the conjecture was posed by J. Rauch in 1974. In this paper, using spectral decimation, we prove that the conjecture holds on the Sierpinski gasket, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by J. Kigami) attains its maximum and minimum on the boundary. [Copyright &y& Elsevier]

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]

In this paper, we introduce fractal interpolation functions (FIFs) and linear FIFs on a post critically finite (p.c.f. for short) self-similar set K. We present a sufficient condition such that linear FIFs have finite energy and prove that the solution of Dirichlet problem -Δμ u = f,u|∂K = 0 is a linear FIF on K if f is a linear FIF. [ABSTRACT FROM AUTHOR]

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]

Abstract: In 1997 David and Semmes asked whether there exists a bilipschitz map between the two compact self-similar subset M and of the real line defined by the relations and . We answer this question positively. To cite this article: H. Rao et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). [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]

In this paper, we study the topological properties and the gap sequences of Bedfordâ€"McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedfordâ€"McMullen sets and prove that all normal Bedfordâ€"McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedfordâ€"McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence. [ABSTRACT FROM AUTHOR]

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far, all the known results only cover self-similar sets in with no more than three branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called substitutable. [ABSTRACT FROM AUTHOR]