Edit: According to comment of Andre Herniques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?