On new fractal phenomena connected with infinite linear IFS

Abstract

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non-faithfullness of the family of cylinders generated by the classical Lüroth expansion. We also develop new approach to the study of subsets of -essentially non-normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in 2 and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.