Dimensions and stability of invariant measures supported on fractal surfaces

The current paper deals with the estimates for dimensions of the invariant measures associated with fractal interpolation surfaces. Additionally, some bounds for the Hausdorff and the box dimension of a fractal surface are obtained with the new approach. Further, the continuous dependence of the invariant measures with respect to the Monge-Kantorovich metric on the parameters involved in the construction is explored. We also derive some conditions under which the defined measures generated by different probability vectors are mutually singular and explore their absolute continuity with respect to the Hausdorff measure. In the end, projections and restrictions of the fractal surfaces and associated measures are highlighted.