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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]

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]

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]