  For symbolic dynamics with some mild conditions, we solve the lowering
topological entropy problem for subsystems and determine the Hausdorff
dimension of the level set with given complexity, where the complexity is
represented by Hausdorff dimension of orbit closure. These results can be
applied to some dynamical systems such as $\beta$-transformations, conformal
expanding repeller, etc. We also determine the dimension of the Furstenberg
level set, which is related to a problem of Furstenberg on the orbits under two
multiplicatively independent maps.
