Three Maps and the Real Numbers

J. J. P. Veerman ()
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J. J. P. Veerman: Portland State University

Chapter Chapter 10 in Numbers from all Angles, 2026, pp 191-213 from Springer

Abstract: Abstract In this chapter, we consider the three maps from [0, 1) to itself that are most important for our understanding of the statistical properties of real numbers. They are multiplication by an integer n modulo 1, rotation by an irrational number, and the Gauss map that we discussed in Chap. 6 . In doing this, we review three standard techniques to establish ergodicity. In this chapter we restrict all measures, transformations, and so on to live in one dimension ([0, 1) or $$\mathbb {R}/\mathbb {Z}$$R/Z). Furthermore, we indicate measures by a Latin letter ($$\ell $$ℓ for the Lebesgue measure) and their density by a Greek letter.

Date: 2026
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DOI: 10.1007/978-3-032-10000-9_10

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