[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that *I do not know what the right definition of a fractal in the following question should be*. But a nice answer might contain such a definition. Moreover, a nice comment by Joël correctly points out that choosing which theory of rigid analytic spaces one wants to consider is crucial for my question which is otherwise ill-posed (but, again, I am more than happy if someone has an answer using *whichever theory* she prefers).

The *infinite fern* of Gouvêa-Mazur, introduced (I guess) in the paper by Barry Mazur "An "infinite fern" in the universal deformation space of Galois representations" is a subset of the space attached to a universal deformation ring to a certain $\mod{p}$ Galois representation and which consists of infinitely many paths crossing at infinitely many points (there is a picture on page 36 of Mazur's paper) and given that this infinitely many crossing are dense in each path (they roughly correspond to integers in a $p$-adic disk) it seems to me that the above set could naturally be regarded as having some fractal behaviour. So, my questions are: is this the case? If yes, are there any results or conjectures on its fractal dimension, or connections between its fractal structure and some arithmetic of $p$-adic families of modular forms?

`$n/m\notin\mathbb{Z}$`

", but I am more than happy if the answer contains a precise definition ;-) $\endgroup$5more comments