Generalizing the Minkowski Question Mark Function to a Family of Multidimensional Continued Fractions
| Autor: | Peter Mcdonald, Thomas Garrity |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Mathematics - Number Theory 010102 general mathematics Zero (complex analysis) Derivative Function (mathematics) 01 natural sciences 010101 applied mathematics FOS: Mathematics Minkowski's question mark function Key (cryptography) Almost everywhere Number Theory (math.NT) 0101 mathematics Mathematics |
| Popis: | The Minkowski question mark function, maping the unit interval to itself, is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus the question mark function is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity. 40 pages, font problems fixed |
| Databáze: | OpenAIRE |
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