(non)-matching and (non)-periodicity for (N,α)(N,\alpha)-expansions

Recently a new class of continued fraction algorithms, the $(N,\alpha$)-expansions, was introduced for each $N\in\mathbb{N}$, $N\geq 2$ and $\alpha \in (0,\sqrt{N}-1]$. Each of these continued fraction algorithms has only finitely many possible digits. These $(N,\alpha)$-expansions `behave' very different from many other (classical) continued fraction algorithms. In this paper we will show that when all digits in the digit set are co-prime with $N$, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for $\alpha$ small enough, all rationals have an eventually periodic expansion with period 1. This happens for all $\alpha$ when $N=2$. We also find infinitely many matching intervals for $N=2$, as well as rationals that are not contained in any matching interval

Tanaka-Ito α{\alpha}-continued fractions and matching

Two closely related families of ${\alpha}$-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter ${\alpha}$ has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito ${\alpha}$-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada's ${\alpha}$-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka-Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients

Generalizations of Sturmian sequences associated with NN-continued fraction algorithms

Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired by Sturmian sequences, we introduce the $N$-continued fraction sequences $\omega(x,N)$ and $\hat{\omega}(x,N)$, which are related to the $N$-continued fraction expansion of $x$. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are $S$-adic sequences. When $N=1$, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that $\omega(x,N)$ and $\hat{\omega}(x,N)$ are $C$-balanced for some explicit values of $C$ and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for $N$-continued fraction expansions, which provides an additive version of $N$-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.Comment: 23 pages, 2 figure

The β\beta-transformation with a hole at 0

For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by $$K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\not \in (0, t) \textrm{ for all }n\ge 0\}.$$ In this paper we characterise the bifurcation set $E_\beta$ of all parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_\beta(t)$ is not locally constant. We show that $E_\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\beta\in(1,2)$. We prove that for Lebesgue almost every $\beta\in(1,2)$ the bifurcation set $E_\beta$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\beta\in(1,2)$ for which $E_\beta$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $\beta \in (1,2)$ a lower and upper bound for the value $\tau_\beta$, such that the Hausdorff dimension of $K_\beta(t)$ is positive if and only if $t< \tau_\beta$. We show that $\tau_\beta \le 1-\frac1{\beta}$ for all $\beta \in (1,2)$.Comment: 32 pages, 4 figure

Characterisation of transcriptionally active and inactive chromatin domains in neurons

The tandemly organised ribosomal DNA (rDNA) repeats are transcribed by a dedicated RNA polymerase in a specialised nuclear compartment, the nucleolus. There appears to be an intimate link between the maintenance of nucleolar structure and the presence of heterochromatic chromatin domains. This is particularly evident in many large neurons, where a single nucleolus is present, which is separated from the remainder of the nucleus by a characteristic shell of heterochromatin. Using a combined fluorescence in situ hybridisation and immunocytochemistry approach, we have analysed the molecular composition of this highly organised neuronal chromatin, to investigate its functional significance. We find that clusters of inactive, methylated rDNA repeats are present inside large neuronal nucleoli, which are often attached to the shell of heterochromatic DNA. Surprisingly, the methylated DNA-binding protein MeCP2, which is abundantly present in the centromeric and perinucleolar heterochromatin, does not associate significantly with the methylated rDNA repeats, whereas histone H1 does overlap partially with these clusters. Histone H1 also defines other, centromere-associated chromatin subdomains, together with the mammalian Polycomb group factor Eed. These dat

Continued fraction expansions with variable numerators

A new continued fraction expansion algorithm, the so-called -expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's -expansions, Schweiger's odd- and even-continued fraction expansions, and the Rosen fractions), these -expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension