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

2023; Birkhäuser; Volume: 29; Issue: 2 Linguagem: Inglês

10.1007/s00029-022-00820-z

1420-9020

Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański,

Quantum chaos and dynamical systems

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 \`a 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 Jarn\'ik and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.