Orbits of N-expansions with a finite set of digits

Jaap De Jonge,Hitoshi Nakada,Cor Kraaikamp

doi:10.1007/s00605-021-01658-x

Jaap De Jonge, Hitoshi Nakada + Show 1 more

Open Access

https://doi.org/10.1007/s00605-021-01658-x

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Abstract

For N in {mathbb {N}}_{ge 2} and alpha in {mathbb {R}} such that 0 < alpha le sqrt{N}-1, we define I_alpha :=[alpha ,alpha +1] and I_alpha ^-:=[alpha ,alpha +1) and investigate the continued fraction map T_{alpha }:I_{alpha }rightarrow I_{alpha }^-, which is defined as T_{alpha }(x):= frac{N}{x}-d(x), where d: I_{alpha }rightarrow {mathbb {N}} is defined by d(x):=leftlfloor frac{N}{x} -alpha rightrfloor . For Nin {mathbb {N}}_{ge 7}, for certain values of alpha , open intervals (a,b) subset I_{alpha } exist such that for almost every x in I_{alpha } there is an n_0 in {mathbb {N}} for which T_{alpha }^n(x)notin (a,b) for all nge n_0. These gaps (a, b) are investigated using the square varUpsilon _alpha :=I_{alpha }times I_{alpha }^-, where the orbitsT_{alpha }^k(x), k=0,1,2,ldots of numbers x in I_{alpha } are represented as cobwebs. The squares varUpsilon _alpha are the union of fundamental regions, which are related to the cylinder sets of the map T_{alpha }, according to the finitely many values of d in T_{alpha }. In this paper some clear conditions are found under which I_{alpha } is gapless. If I_{alpha } consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of I_{alpha } with regard to the fixed points of I_{alpha } under T_{alpha }.

Highlights

In 2008, Edward Burger and his co-authors introduced in [2] new continued fraction expansions, the so-called N -expansions, which are nice variations of the regular continued fraction (RCF) expansion
In a subsequent paper we will go into another very interesting property of orbits of N expansions that is hardly revealed by simulations such as Fig. 3: the existence of large numbers of gaps for large N and α close to αmax
In a forthcoming paper we show that the number of gaps is a function of N and α, and is growing larger and larger as N tends to infinity

Summary

Introduction

In 2008, Edward Burger and his co-authors introduced in [2] new continued fraction expansions, the so-called N -expansions, which are nice variations of the regular continued fraction (RCF) expansion. In order to get a better understanding of the orbits of N -expansions, it is useful to consider the graphs of Tα, which are drawn in the square Υα(= ΥN,α) := Iα × Iα− This square is divided in rectangular sets of points i := {(x, y) ∈ Υα : d(x) = i}, which are the two-dimensional fundamental regions associated with the one-dimensional cylinder sets we already use. In a subsequent paper we will go into another very interesting property of orbits of N expansions that is hardly revealed by simulations such as Fig. 3: the existence of large numbers of gaps for large N and α close to αmax. The proofs in this paper are often elementary, they require case distinctions and cumbersome calculations

Full arrangements and arrangements with more than four cylinders

Gaplessness if the branch number is large enough

Gaplessness if Iconsists of two cylinder sets

Gaplessness if Iconsists of three cylinder sets

Gaplessness in case Icontains two full cylinder sets