Hi,
   Please consider this object:  Start with a realization of Brownian motion in 2D, which I'll denote by rho(t) where -infinity < t <  +infinity.  Next, lets smooth rho. There are various ways of doing this. To save space, I'll leave it to your imagination.   But the point is that at short length scales the smoothed rho looks like a smooth continuous curve, but as you zoom out, it looks more and more like the original Brownian path. By the way, in the process of smoothing rho,  I think the original time parameter is effectively lost.     

So I claim two things. 

1) You can find a circle, C, such that the smoothed rho never enters inside C.

2) Let us invert the smoothed rho about the circle C, so that it now lies entirely within C. I claim that everywhere inside C, the smoothed and inverted rho will have dimension 1, except at the center of C  where it has a point fractal dimension of 2.

I'm actually a physicist, and don't have a great mathematical background, but I believe the above claims are valid. Would you agree with them?  Is any of this is interesting, or has it been covered many times before?

Thanks, and I look forward  to any comments, Chris