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[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?

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    $\begingroup$ What does "is a fractal" mean here? $\endgroup$
    – Qiaochu Yuan
    Commented Jun 1, 2013 at 5:44
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    $\begingroup$ @Qiaochu: Well, good point. I do not know and it seems to me - as non-expert!- that in general there is no such a uniquely agreed definition. Most probably, I am thinking at something like "if you cut a piece of the infinite fern into $n$ equal pieces, each looks like $1/m$-th of the original piece with $n/m\notin\mathbb{Z}$", but I am more than happy if the answer contains a precise definition ;-) $\endgroup$
    – Filippo Alberto Edoardo
    Commented Jun 1, 2013 at 7:22
  • $\begingroup$ A possibility for a definition of "fractal" is "metric space with fractionary Hausdorff dimension". $\endgroup$
    – Qfwfq
    Commented Jun 1, 2013 at 12:04
  • $\begingroup$ @Qfwfq - I don't think that's a good definition of fractal, there are metric spaces, even subspaces of $\mathbb{R}^2$, which have integer Hausdorff dimension and are quintessentially fractal. See my comment here for an example: mathoverflow.net/questions/56677/… (there's a typo, should be side length 1/4 rather than 1/2). $\endgroup$
    – Pablo Shmerkin
    Commented Jun 1, 2013 at 14:08
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    $\begingroup$ I think the question is interesting, but somehow is still not well-posed. There is the problem of what is a fractal, discussed in earlier comments, but also the problem of what is the "infinite fern". I mean, it is a rigid analytic varieties, but there are at least three ways to formalize this notion, Tate's and Berkovich's and Huber's. Each leads to a different space.. $\endgroup$
    – Joël
    Commented Jun 3, 2013 at 13:51

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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 )

  • “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.

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