Diophantine approximation for conformal measures of one-dimensional iterated function systems

Abstract

Recall that a Borel measure $\mu$ on $\mathbb{R}$ is said to be extremal if $\mu$-almost every number in $\mathbb{R}$ is not very well approximable. In this paper, we investigate extremality (implied by the exponentially fast decay (efd) property) of conformal measures induced by regular infinite conformal iterated function systems. We then give particular attention to the class of such systems generated by the continued fractions algorithm with restricted entries. It is proved that if the index set of entries has bounded gaps, then the corresponding conformal measure satisfies the efd property and is extremal. Also a class of examples of index sets with unbounded gaps is provided for which the corresponding conformal measure also satisfies the efd property and is extremal.