Derivatives of slippery Devil's staircases

Miao, Junjie; Munday, Sara

doi:10.3934/dcdss.2017017

Cited by 1 publication

(1 citation statement)

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“…One of the possible interesting questions to ask, given the map S and the tree defined here, is whether or not a version of the Minkowski question mark function could be defined in this setting. We recall, briefly, that the original Minkowski question mark function was introduced as another way of demonstrating the Lagrange property of continued fractions, in that it maps every rational number to the subset of dyadic rationals (that is, those having denominators containing only powers of 2) and every quadratic irrational to the remaining rational numbers (see [15,10], and for other 1-dimensional analogues, [16,17]). These functions are now known as slippery Devil's staircases for the fact that they are strictly increasing but nevertheless singular with respect to the Lebesgue measure.…”

Section: (I)mentioning

confidence: 99%

A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs

Bonanno¹,

Vigna²,

Munday³

2019

Preprint

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We study the two-dimensional continued fraction algorithm introduced in [5] and the associated triangle map T , defined on a triangle △ ⊆ R 2 . We introduce a slow version of the triangle map, the map S, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We show that the two maps T and S share many properties with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle sequence, the expansion associated to T . Finally, we confirm the role of the map S as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of S, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation. CLAUDIO BONANNO, ALESSIO DEL VIGNA, AND SARA MUNDAYWe show that, similarly to the Farey map, S preserves an infinite Lebesgue-absolutely continuous measure and it is ergodic with respect to the Lebesgue measure on △. It follows that the statistical behaviour of summable observables along orbits of S is non-standard. This phenomenon, for the Farey map, makes it impossible to improve Khinchin's weak law for the coefficients of the regular continued fraction expansion to a strong law. However, we are able to exploit certain results from infinite ergodic theory to show that the system generated by S is pointwise dual ergodic and prove a weak law of large numbers for S, from which we obtain an analogue of Khinchin's weak law for the digits of the triangle sequences.The connection between the Gauss and the Farey maps and the regular continued fractions can be studied also through the Farey tree, a binary tree which contains all the rational numbers in (0, 1) (see e.g. [4]). The Farey tree is strongly related to the Farey map, but it can also be defined through the mediant operation on fractions. We recall the definition of the Farey tree and its basic properties in Section 5. Analogously, in this paper we define a tree of rational pairs, first by using a suitable modification of the map S limited to the set of indifferent fixed points, and then by using a generalised mediant operation defined on pairs of rational numbers. We prove that the two trees are in fact identical level by level, and that the tree is complete, that is, it contains every pair of rational numbers in s △. This last result improves on the results of [2], where the authors study different trees generated by the triangle map and its generalisations, but show that none of them are complete.The paper is organised as follows. In Section 2 we recall the definition of the triangle map T and the associated two-dimensional continued fraction algorithm. We also introduce the map S and study its basic ergodic properties. Lastly, we define a dynamical system on an infinite strip, which is isomorphic to the action of S on s △. This isomorphism gives a useful i...

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Section: (I)mentioning

confidence: 99%