DIMENSION SETS FOR INFINITE IFS’S: THE TEXAN CONJECTURE

ABSTRACT. We refere to the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system as the restricted dimension set and the corresponding set for all subsystems as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval. 1