a b s t r a c tA method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

We consider a construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal interpolation surface (RFIS) is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given data set on rectangular grids, we construct general recurrent iterated function systems with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

In this paper, we present some analytic properties of hidden variable bivariable fractal interpolation functions (HVBFIFs) with four function contractivity factors presented in [C. H. Yun and M. K. Li, Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors, Asian-Eur. J. Math. (2017), doi:10.1142/s1793557119500219]. Since four contractivity factors of these HVBFIFs are all functions, the construction of these HVBFIFs has more flexibility and diversity in fitting and approximation of complicated surfaces in nature and irregular experimental data with less self-similarity than one whose four contractivity factors are all constants or only one factor is function. The smoothness and stability of HVBFIFs are needed to ensure the applicability of the HVBFIFs in many practical problems such as the simulation of the objects of the nature, data fitting, etc. We first obtain the results related to their smoothness in nine different cases and then prove that the HVBFIFs are stable to the small perturbations of the interpolation points.

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