[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"](http://www.raco.cat/index.php/CollectaneaMathematica/article/download/56382/66675)  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?



Is Gouvêa-Mazur's "Infinite Fern" a fractal?
I post this as an auto-answer mainly not to leave the question open.


After googling a bit better, I discovered two recent works by M. Lapidus and L. Hung (both available on  [Lapidus' webpage](http://math.ucr.edu/~lapidus/#FFIRG) )


* “Nonarchimedean Cantor Set and
String”, Journal of Fixed Point
Theory and Applications 3 (2008), pp.
181-190, (Special issue
dedicated to Vladimir Arnold on the
occasion of his Jubilee. Vol. I.)
* “Self-Similar $p$-Adic Fractal Strings
and Their Complex Dimensions”, $p$-Adic
Numbers, Ultrametric Analysis and
Applications (Russian Academy of
Sciences, Moscow, and
Springer-Verlag), No. 2, 1 (2009),
pp. 167-180.


which seem to pose a "good" definition for a $p$-adic fractal. The definition follows the usual self-similarity one, attaching to each family $\{\Phi\_1,\dots,\Phi\_n\}$ of similarity contractions
$$
\Phi\_j:\mathbb{Z}\_p\longrightarrow\mathbb{Z}\_p
$$
the unique non-empy, compact, fixed subset $\mathcal{S}\subseteq \mathbb{Z}\_p$ such that $\mathcal{S}=\Phi\_j(\mathcal{S})$ for each $1\leq j\leq n$. They develop some theory for such objects and define its Minkowsy dimension, mainly following the box-counting dimension definition . In a closing remark of the second paper, they also say that "it would be interesting to generalize this theory from subspaces of $\mathbb{Q}\_p$ to Berkovich spaces" but I was unable to find anything more on the subject.


