It is suggested that the scaling laws satisfied by fluvial erosion topography and river networks reflect a basic self-similarity or multifractality property of the topographic surface within river basins. By analyzing the symmetries of fluvial topography, we conclude that this self-similarity or multifractality condition should be expressed in a particular way in terms of the topographic increments within subbasins. We then analyze whether self-similar or multifractal topographies can be stationary or transient solutions of dynamic evolution models of the type ∂h/∂t=U-f{&bgr;,&tgr;}, where U is the uplift rate, f is the fluvial erosion rate, &bgr; is a vector of erodibility parameters, and &tgr; is hydraulic shear stress. The hydraulic stress on a channel bed is assumed to satisfy &tgr;∝AmSn, where A is contributing area, S is slope, and m and n are parameters. We allow U to vary randomly in time and &bgr; to vary randomly in space and determine conditions on these random functions as well as the parameters m and n under which the topography may remain in a self-similar or multifractal state. Simulation shows that self-similar states are attractive also for non-self-similar boundary and initial conditions. ¿ 2000 American Geophysical Union