The surface structure of an equilibrium restricted curvature (RC) model on a Sierpinski gasket substrate is studied. The surface width W increases as tβ at early time t and becomes saturated at Lα for , where L is the system size. The growth exponent β≈0.323, the roughness exponent α≈1.54 and the dynamic exponent z≈4.78 are obtained numerically. They satisfy the scaling relations 2α + df = z and z = 2zrw very well, where zrw is the random walk exponent of the Sierpinski gasket. We introduce a fractional Langevin equation to describe the model.

A conserved noise restricted solid-on-solid model on both a Sierpinski gasket substrate and a checkerboard fractal substrate is studied. The interface width W grows as t(β) at early time t and becomes saturated at L(α) for t >> L(z), where L is the system size. We obtain β ≈ 0.0788, α ≈ 0.377 for a Sierpinski gasket, and β ≈ 0.100, α ≈ 0.516 for a checkerboard fractal. The dynamic exponent z ≈ 4.79 for a Sierpinski gasket and z ≈ 5.16 for a checkerboard fractal are obtained by the relation z = α/β. They satisfy the scaling relations 4 α + 2 d(f) = z and z = 2 z(rw), where z(rw) is the random-walk exponent of the fractal substrate. A fractional Langevin equation is introduced to describe the model.

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