A new class of α -Farey maps and an application to normal numbers

Karma Dajani; Cornelis Kraaikamp; Hitoshi Nakada; Rie Natsui

doi:10.1017/prm.2025.10068

A new class of α -Farey maps and an application to normal numbers

Karma Dajani
, Cornelis Kraaikamp^*
, Hitoshi Nakada
, Rie Natsui

^*Corresponding author for this work

Applied Probability

Research output: Contribution to journal › Article › Scientific › peer-review

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Abstract

We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

Original language	English
Number of pages	37
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs	https://doi.org/10.1017/prm.2025.10068
Publication status	Published - 2025

Keywords

Farey map
natural extension
normal numbers
α-continued fraction expansions

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10.1017/prm.2025.10068

a-new-class-of-a-farey-maps-and-an-application-to-normal-numbersFinal published version, 907 KBLicence: CC BY

Cite this

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abstract = "We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).",

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AU - Dajani, Karma

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AU - Nakada, Hitoshi

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PY - 2025

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AB - We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

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KW - natural extension

KW - normal numbers

KW - α-continued fraction expansions

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JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ER -

A new class of α -Farey maps and an application to normal numbers

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