Fractal dimension of the boundary of a fractal

Question

Sorry if this is a stupid question, but I'm a physicist, not a mathematician, and fractals are pretty new to me.

Is there a simple relationship between the fractal dimension of a set and the fractal dimension of that set's boundary?

For non-fractals, the relationship is of course that the boundary dimension is one-less than the bulk dimension (e.g. the boundary of a 3d sphere is a 2d surface).

Any help (including the ways in which my thinking is totally wrong!) would be greatly appreciated.

Answer 1

It may depend on how you define "fractal", but typically a fractal is closed and nowhere dense, so the boundary is the set itself.

Answer 2

In general no the dimension of the boundary isn’t always 1 less than the dimension of the set.

See this example: https://en.m.wikipedia.org/wiki/Rauzy_fractal

The dimension of this set is 2 but it’s boundary has dimension around 1.09