On new fractal phenomena connected with infinite linear IFS

TL;DR

This paper explores new fractal and number theoretical phenomena related to infinite linear iterated function systems, revealing non-faithfulness of certain cylinder systems and demonstrating the superfractality of non-normal number sets.

Contribution

It introduces a novel approach to analyze subsets of non-normal numbers in generalized L"uroth expansions, proving their superfractality without restrictions on the stochastic vector.

Findings

Cylinders of generalized L"uroth expansions are generally not faithful for Hausdorff dimension.

Conditions for faithfulness and non-faithfulness of cylinder systems are established.

The set of non-normal numbers in this context is shown to be superfractal.

Abstract

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems $\Phi$ of cylinders of generalized L\"uroth 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 of our main results, we obtain the non-faithfullness of the family of cylinders generated by the classical L\"uroth expansion. Possible infinite entropy of the stochastic vector $Q_\infty$ which determines the metric relations for partitions of the generalized L\"uroth expansions, possible non-faithfulness of the family $\Phi(Q_\infty)$ and the absence of general formulae for the calculation of the Hausdorff dimension even for probability measures with independent identically distributed symbols of generalized L\"uroth expansions are all facts that do not allow to apply usual probabilistic methods, as well as methods of dynamical systems to study fractal properties of corresponding subsets of non-normal numbers. Despite of the above mentioned difficulties, we develop new approach to the study of subsets of $Q_\infty$-essentially non-normal numbers and prove (without any additional restrictions on the stochastic vector $Q_{\infty}$) that this set is superfractal. This result answers the open problem mentioned in \cite{AKNT2} and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized L\"uroth expansion.