On the derivative of the $\alpha$-Farey-Minkowski function

Munday, Sara

doi:10.3934/dcds.2014.34.709

Cited by 6 publications

(8 citation statements)

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“…A similar problem has also been studied in the case of singular functions which are increasing but not strictly increasing, such as for several variants of the Cantor ternary function, see [6,19,10,16,32] for example. Moreover, similar results have been considered for topological conjugacies (called α-Farey-Minkowski functions) between α-Lüroth maps by Munday [25] (an example is shown in Figure 2) and later by Arroyo [1], where he considers the conjugacy maps between the Gauss map and any α-Lüroth map.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 56%

Stability and perturbations of countable Markov maps

2018

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We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let T k and T be expanding countable Markov maps such that the inverse branches of T k converge pointwise to the inverse branches of T as k → ∞. Then under suitable regularity assumptions on the maps T k and T the following limit exists:where θ k is the topological conjugacy between T k and T and dim H stands for the Hausdorff dimension. This is in contrast with the fact that other natural quantities measuring the singularity of θ k fail to be continuous in this manner under pointwise convergence such as the Hölder exponent of θ k or the Hausdorff dimension dim H (µ•θ k ) for the preimage of the absolutely continuous invariant measure µ for T . As an application we obtain a perturbation theorem in non-uniformly hyperbolic dynamics for conjugacies between intermittent Manneville-Pomeau maps x → x + x 1+α mod 1 when varying the parameter α.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 56%

Stability and perturbations of countable Markov maps

2018

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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

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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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“…In this way, Lür α (x) is called the α-Lüroth expansion of x with respect to the given partition α. This expression can be found in [5] and [9]. Moreover, except for a countable set in I, this expression is unique.…”

mentioning

confidence: 92%

“…After Minkowski several generalisations have been constructed, beginning with Denjoy and Salem; see [3], [4] and [10]. Much more recently, Kessebohmer, Munday and Stratmann in [5], and Munday in [9], study this map from a dynamical systems point of view, and the relation with the generalised Lüroth expansion of numbers. In this note we exploit further this relation to obtain a family of functions which are generalisations of Minkowski's Question Mark function.…”

mentioning

confidence: 99%

“…They where studied in [1] and are a generalisation of the classical Lüroth expansion of [7]. In this note, we will restrict our attention to the class of α-Lüroth expansions which were introduced in [5] (see also [9]). Each of these representations depends on a countable partition of I defined by a sequence with the following properties: Let α = {t j } j≥1 ⊂ I be a strictly decreasing sequence such that t 1 = 1, and t j → 0 when j → ∞.…”

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confidence: 99%

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Generalised Lüroth expansions and a family of Minkowski's question-mark functions

Arroyo

2015

Comptes Rendus. Mathématique

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The Minkowski's Question-Mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the twentieth century. Using different representations of real numbers by infinite sequences of integers, called α-Lüroth expansions, we obtain different instances of the standard shift map on infinite symbols, all of them topologically conjugated to the Gauss Map. In this note we prove that each of these conjugations share properties with the Minkowski's Question-Mark function: all of them are singular homeomorphisms of the interval, and in the "rational" cases, they map the set of quadratic surds into the set of rational numbers. In this sense, this family is a natural generalisation of the Minkowski's Question-Mark function.

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On the derivative of the $\alpha$-Farey-Minkowski function

Cited by 6 publications

References 11 publications

Stability and perturbations of countable Markov maps

Stability and perturbations of countable Markov maps

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

Generalised Lüroth expansions and a family of Minkowski's question-mark functions

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