Derivatives of slippery Devil's staircases

Jun-Jie Miao; Sara Munday

doi:10.3934/dcdss.2017017

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2017, Volume 10, Issue 2: 353-365. Doi: 10.3934/dcdss.2017017

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Derivatives of slippery Devil's staircases

Jun-Jie Miao^1, and
Sara Munday^2,

1.
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
2.
Department of Mathematics, University of Bologna, Bologna, Italy

Received: December 2015

Revised: November 2016

Published: February 2018

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Abstract

In this paper we first give a survey of known results on the derivative of slippery Devil's staircase functions, that is, functions that are singular with respect to the Lebesgue measure and strictly increasing. The best known example of such a function is the Minkowski question-mark function, which was proved to be singular by Salem, in a paper which introduced some other constructions of singular functions. We describe all of these examples. Also we consider various generalisations of the Minkowski question-mark function, such as $α$-Farey-Minkowski functions. These examples all arise from one-dimensional dynamics. A few open questions and suggestions for filling minor gaps in the literature are proposed. Finally, we go back to ordinary Devil's staircases (i.e. non-decreasing singular functions) and discuss work done in that setting with the more general Hölder derivatives, and consider the outlook to extend those results to the strictly increasing situation.

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Mathematics Subject Classification: 37E05, 26A27, 28A80, 11K55.

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Figure 2.1. On the left, the functions $Q_1$, $Q_2$ and $Q_3$, and on the right, the Minkowski question-mark function, $Q:[0, 1]\to[0, 1]$

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