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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?

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    A version of this question was kicked here from physics.SE. I've marked as duplicate and merged.2011-07-07

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If a map $f$ is Lipschitz ( that is, $|f(x)-f(y)| \le M |x-y|$ for some constant $M$ ), then is does not increase Hausdorff dimension: $\dim f(K) \le \dim K$. An example of a Lipschitz function is one that is differentiable with bounded derivative. Dimension does not increase, but it may decrease.

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    I have found a reference. Lemma 6.1 in ``The Geometry of Fractal Sets'' by K. Falconer is the lemma I have been looking for. Thanks again to GEdgar and Robert Israel.2011-07-08