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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]

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]

Although the 'hot spots' conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this note, we show surprisingly that the 'hot spots' conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals. [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]

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]

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]