Many problems in several areas of Mathematics (including Dynamical Systems and Number Theory) can “reduced” to the analysis of *dynamical Cantor sets*: for instance, the theorems of Newhouse, Palis and Takens on homoclinic bifurcations of surfaces diffeomorphisms, and the theorems of Hall, Freiman and Moreira on the structure of the classical Lagrange and Markov spectra rely on the study of dynamical Cantor sets of the real line.

The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set is

where is .

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called *thermodynamical methods* are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence of fast-converging approximations for the Hausdorff dimension of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence converging to at *super-exponential speed*, i.e., for some constants and , where is the Cantor set of real numbers whose continued fraction expansions contain only and .

In particular, the thermodynamical methods give good *heuristics* for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list for and the first three digits of coincide for all , then it is likely that one has found the first three digits of ).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of .

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a *slowly* converging method producing two sequences and of *rigorous* bounds (i.e., for all ) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences converging slowly (e.g., for some constant and all ) towards , and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

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