I have a finite collection of diffeomorphisms $g_1,\cdots,g_n$ taking the unit interval $I$ to disjoint subintervals $I_1, I_2,\cdots,I_n$. If $G$ is the semigroup they generate, the limit set of $G$ (also called the attractor of the IFS) is a Cantor set, and under suitable hypotheses, Bowen (and under slightly weaker hypotheses, Urbanski) showed that the Hausdorff dimension of this Cantor set is the smallest zero of the pressure function $P$, defined by $$P(t) = \lim_{n \to \infty} \frac 1 n \log \sum_{w \in G_n} \|w'\|^t$$ where $G_n$ is the set of elements in $G$ of word length $n$, and $\|\cdot\|$ is the sup norm (the hypotheses for Bowen's theorem is that the $g_i$ are uniformly contracting; Urbanski proves the same theorem when the $g_i$ are allowed to have neutral fixed points; my examples have such points).

This is all well and good, but how do I actually estimate the least zero of the pressure function for an explicit example? (yes, I mean numerically) My $g_i$ are all given by the restrictions of explicit polynomial functions of low degree, but the computational bottleneck seems to be the large number of elements in $G_n$.

Or is there a better method to estimate the Hausdorff dimension in practice? Note that although I just want to estimate the dimension, I would like to be able to (computer-assisted if necessary) give rigorous bounds on the error.


Write $\Lambda_n(t) = \sum_{w\in G_n} |f'(w)|^{-t}$, where $f\colon \bigcup_j I_j \to I$ is the interval map whose inverse branches are the maps $g_j$, and where $|(f^n)'(w)|$ is the supremum of $|(f^n)'(x)|$ taken over all $x$ in the basic interval corresponding to the word $w$.

Suppose that $f$ is uniformly expanding (the $g_j$ are uniformly contracting) and $f'$ is Holder continuous. Then there exists $V\in \mathbb{R}$ such that $|(f^n)'(x)/(f^n)'(y)| \leq e^V$ whenever $x$ and $y$ are in the same basic interval (of any order). In this case standard estimates yield $$ (1) \qquad \qquad e^{nP(t)} \leq \Lambda_n(t) \leq e^V e^{nP(t)} $$ for every $n$. This doesn't get rid of the fact that there are a large number of elements in $G_n$, but it at least gives you the rigorous bound $|P(t) - \frac 1n \log \Lambda_n(t)| \leq \frac Vn$ for any given $n$. Versions of (1) can be found in (Rufus Bowen, "Some systems with unique equilibrium states", Math. Systems Theory 8 (1974/75), no. 3, 193–202).

In the non-uniformly expanding case (Urbanski's), it's not quite so easy, since in general there may not be any $V$ with the bounded distortion property described above. Nevertheless, certain partial bounded distortion results should still be available, and at least for $t$ less than the Hausdorff dimension of the limit set, I believe the techniques in (Climenhaga and Thompson, "Equilibrium states beyond specification and the Bowen property", arXiv:1106.3575), particularly Section 4.3 and Proposition 5.3, will suffice to show that there is a constant $V$ such that (1) holds. I'm not sure how easy that constant will be to compute, or how it may decay as $t$ approaches the Hausdorff dimension of the limit set -- however, if you can use that to get $P(t)>0$ for some $t$ that you suspect is a very good estimate, then you have the lower bound on Hausdorff dimension, which is typically the harder one, and the upper bound may be accessible by other means.

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    $\begingroup$ Reading through what I just wrote, I'm not sure how useful it will actually be, since to get within $\delta$ of the pressure you need $n=V/\delta$, which may be quite large. I suppose it depends on how big the bounded distortion constant $V$ ends up being, which is basically a measure of how far your IFS is from being linear. Oh well... I'll leave it up here anyway... $\endgroup$ – Vaughn Climenhaga Oct 1 '11 at 1:36
  • $\begingroup$ Thanks for the references and the suggestions. I am in fact interested in the non-uniformly contracting (OK, expanding) case, so I'll take a look at the Climenhaga-Thompson paper. $\endgroup$ – Danny Calegari Oct 1 '11 at 6:16
  • $\begingroup$ Just out of curiosity, what did you have in mind when you say "the upper bound may be accessible by other means"? $\endgroup$ – Danny Calegari Oct 1 '11 at 6:20
  • $\begingroup$ If you want to show that $t_1\leq dim_HZ\leq t_2$, getting $\dim_H Z\geq t_1$ requires you to prove something about every possible $\epsilon$-cover of $Z$, which is one reason Bowen's equation and the thermodynamic formalism is a useful tool for computing Hausdorff dimension, because it lets you sidestep that need. Proving $dim_H Z\leq t_2$, on the other hand, just requires proving something for particular good $\epsilon$-covers of $Z$, which for something like the limit set of an IFS can be chosen in a very canonical way, in this case as being composed of basic intervals. $\endgroup$ – Vaughn Climenhaga Oct 1 '11 at 16:43
  • $\begingroup$ So the "other means" I meant were to pick a $t$ that you think is an upper bound, and then compute an upper bound for the $t$-dimensional Hausdorff measure by using covers by basic intervals. In the end, though, I suppose that just comes down to computing the sum $\sum_{w\in G_n} \|w'\|^t$ and showing that it goes to $0$, which is more or less equivalent to showing that $P(t)\leq 0$. So this is probably another example of something that's pretty straightforward from the theoretical side but can be problematic numerically. $\endgroup$ – Vaughn Climenhaga Oct 1 '11 at 16:48

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