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Three Maps and the Real Numbers

In: Numbers from all Angles

Author

Listed:
  • J. J. P. Veerman

    (Portland State University)

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.

Suggested Citation

  • J. J. P. Veerman, 2026. "Three Maps and the Real Numbers," Springer Books, in: Numbers from all Angles, chapter 0, pages 191-213, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-10000-9_10
    DOI: 10.1007/978-3-032-10000-9_10
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