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Normal Numbers and Symbolic Dynamics

  • Manfred Madritsch
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

The present chapter takes a dynamical point of view. The orbit of an element plays a central role in dynamics, and we can deduce several properties such as periodicity, uniqueness, randomness, etc. from the orbit. Starting with a description of the link between dynamical systems and numeration systems, we present the concept of normal and non-normal numbers providing different views on the dynamics of the system. Normal numbers are “normal” with respect to randomly chosen objects, whereas non-normal numbers and extreme variants thereof are examples of general objects from a topological point of view. In the following sections, we present how to obtain maximal randomness as well as constructing numbers with a given degree of chaos. Then we turn our attention to non-normal numbers. Since they are not completely random, we have to find a different measurement for analyzing their structure. The Hausdorff dimension will provide us with an interesting parameter in this context.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Élie Cartan de LorraineUniverstité de LorraineVandœuvre-Lès-Nancy CedexFrance

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