Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

Abstract

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\le N$; as well as its counterpart for reals whose partial quotients are all $\ge N$ due to Good from 1941.

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Cited by in corpus (3)

• The bifurcation locus for numbers of bounded type
• Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence
• Dynamics of Ostrowski skew-product: I. Limit laws and Hausdorff dimensions