Non-normal numbers with respect to infinite Markov partitions

TL;DR

This paper studies special classes of non-normal numbers related to infinite Markov partitions, focusing on their distributional properties and Baire category, revealing their prevalence and complexity.

Contribution

It introduces and analyzes the concepts of extremely non-normal numbers and maximal oscillation frequency numbers within the context of infinite Markov partitions.

Findings

Characterizes the Baire category of these non-normal numbers.

Shows the abundance of such numbers in the space of all real numbers.

Provides insights into their distributional and oscillatory behaviors.

Abstract

In the present paper we investigate two special types of non-normal numbers. On the one hand we call a number extremely non-normal if the set of accumulation points of its frequency of blocks vector is the full set of shift invariant probability vectors. On the other hand we call a number having maximal oscillation frequency if for any fixed block the set of accumulation points of its frequency vector is the full set of possible probability vectors. The goal is to investigate the Baire category of these numbers for infinite Markov partitions.