Let's assume X(s) is a fractal surface with Hausdorff dimension D. Now we take a nonlinear transformation f which transforms X(s) to f(X(s)). In this case, what will be the Hausdorff dimension of the transformed surface f(X(s))?
More clarification (asked by Theo Buehler) > Let's start with a simple example of one dimensional random walk. The path of the random walker becomes a fractal with the Hausdorff dimension 1.5. Let's call the path $X(t)$ at time $t$. Then we can think about the path $Y(t)=X(t)^3−2X(t)$. What will be the Hausdorff dimension of $Y(t)$?
Added (by anon) > Let's add the condition for $f$. $f$ is continuous, differentiable and bounded. In this case, will the Hausdorff dimension remain the same?