arXiv subjects

Karma Dajani

Publications and source records attributed to Karma Dajani.

At least 19 recordsLinked to original sources

arXiv2026

A strange continued fraction associated with the Romik map

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the ``Romik system''), is ergodic and has a $\sigma$-finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the $\sigma$-finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every $x$ asymptotically half of the regular continued fraction (RCF) convergents of $x$ are among the Romik convergents. We also show that related to the Romik map a ``strange'' continued fraction can be given. ``Strange,'' as the set of possible partial quotients (i.e., digits) for any $x\in [0,1]$ in this expansion is $\{ 0, \pm 2\}$. Various properties of this ``Romik expansion'' are given.

math.DS↗

arXiv2026

Coincidence of invariant measure for the alternate base transformations

We characterize all pairs $(\beta,n),(\beta^\prime,m)$ such that the alternate $(\beta,n)$ and $(\beta^\prime,m)$-transformations $K_{(\beta,n)}$ and $K_{(\beta^\prime,m)}$ have the same absolutely continuous invariant measure, where $K_{(\beta,n)}(i,x)=(i+1 \mod 2 ,T_i(x))$ with $i\in\{0,1\}$, $T_0(x)=T_\beta (x)=\beta x \mod 1$, $T_1(x)=T_n(x)=nx\mod 1$ with $\beta>1$ real and $n\geq 2$ an integer.

math.DS↗

arXiv2025

Inducing contractions of the mother of all continued fractions

We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques -- induced transformations and contraction -- in the setting of Shunji Ito's natural extension of the Farey tent map, which generates `slow' continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's $S$-expansions, and Nakada's parameterised family of $\alpha$-continued fractions for all $0<\alpha\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $\alpha$-continued fraction transformations as an explicit induced transformation of Ito's natural extension.

math.NT↗

arXiv2024

Approximation properties of the intermediate $\beta$-expansions

Given $\beta>1$ and $\alpha\in[0,1)$, let $T_{\beta, \alpha}(x)=\beta x+\alpha\pmod 1$. Then under the map $T_{\beta,\alpha}$ each $x\in[0,1]$ has an \emph{intermediate $\beta$-expansion} of the form $x=\sum_{i=1}^\infty\frac{c_i-\alpha}{\beta^i}$ {with each $c_i\in\{0,1,\ldots,\lf \beta+\alpha\rf\}$}. In this paper we study the approximation properties of $T_{\beta,\alpha}$ by considering the expected value $M_\beta(\alpha)$ of the \emph{normalized errors} $(\theta_{\beta,\alpha}^n(x))_{n\geq 1}$, where $$\theta_{\beta,\alpha}^n(x):=\beta^n\left(x-\sum_{i=1}^n\frac{c_i-\alpha}{\beta^i}\right),\quad n\in\mathbb{N}.$$ We prove that $M_\beta(\cdot)$ is continuous on $[0,1)$. As a result, $\mathcal{M_\beta}:=\{M_\beta(\alpha):\alpha\in[0,1)\}$ is a closed interval. In particular, if $\beta$ is a multinacci number, the map $T_{\beta,\alpha}$ has matching for Lebesgue almost every $\alpha\in[0,1)$, and then $M_\beta(\cdot)$ is locally linear almost everywhere on $[0,1)$.

math.DS↗

arXiv2024

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

We define two types of the $α$-Farey maps $F_α$ and $F_{α, \flat}$ for $0 < α< \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le α\le 1$ by R.~Natsui (2004). Then, for each $0 < α< \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{α, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associted with $α$-continued fractions does not vary by the choice of $α$, $0 < α< 1$. This extends the result by C.~Kraaikamp and H.~Nakada (2000).

math.DS↗

arXiv2023

A unifying theory for metrical results on regular continued fraction convergents and mediants

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of L\'evy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory.

math.DS↗

arXiv2023

Multifractal Properties of Tribonacci Chains

We introduce two 1D tight-binding models based on the Tribonacci substitution, the hopping and on-site Tribonacci chains, which generalize the Fibonacci chain. For both hopping and on-site models, a perturbative real-space renormalization procedure is developed. We show that the two models are equivalent at the fixed point of the renormalization group flow, and that the renormalization procedure naturally gives the Local Resonator Modes. Additionally, the Rauzy fractal, inherent to the Tribonacci substitution, is shown to serve as the analog of conumbering for the Tribonacci chain. The renormalization procedure is used to repeatedly subdivide the Rauzy fractal into copies of itself, which can be used to describe the eigenstates in terms of Local Resonator Modes. Finally, the multifractal dimensions of the energy spectrum and eigenstates of the hopping Tribonacci chain are computed, from which it can be concluded that the Tribonacci chains are critical.

cond-mat.dis-nn↗

arXiv2023

Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.

math.NT↗

arXiv2023

Ergodic properties of a parameterised family of symmetric golden maps: the matching phenomenon revisited

We study a one-parameter family of interval maps $\{T_α\}_{α\in[1,β]}$, with $β$ the golden mean, defined on $[-1,1]$ by $T_α(x)=β^{1+|t|}x-tβα$ where $t\in\{-1,0,1\}$. For each $T_α,\ α>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $α$, the corresponding density is a step function with finitely many jumps. We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps. A main tool in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity points meet after a finite number of steps. Each $T_α$ generates signed expansions of numbers in base $1/β$; via Birkhoff's ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic $T_α$-expansions. In particular, the frequency of $0$ is shown to vary continuously as a function of $α$ and to attain its maximum $3/4$ on the maximal interval $[1/2+1/β,1+1/β^2]$.

math.DS↗

arXiv2022

Random β-transformation on fat Sierpinski gasket

We consider the iterated function system (IFS) $$f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}β,\vec{q}\in\{(0,0),(1,0),(0,1)\}.$$ As is well known, for $β= 2$ the attractor, $S_β$, is a fractal called the Sierpiński gasket(or sieve) and for $β>2$ it is also a fractal. Our goal is to study greedy, lazy and random $β$-transformations on the attractor for this IFS with $1<β<2$. For $1<β\leq 3/2$, $S_β$ is a triangle and it is shown that the greedy transformation $T_β$ and the lazy transformation $L_β$ are isomorphic and they both admit an absolutely continuous invariant measure. We show that all $β$-expansions of a point $\vec{z}$ in $S_β$ can be generated by a random map $K_β$ defined on $\{0,1\}^\mathbb{N}\times\{0,1,2\}^\mathbb{N}\times S_β$ and $K_β$ has a unique invariant measure of maximal entropy when $1<β\leqβ_*$, where $β_*\approx 1.4656$ is the root of $x^3-x^2-1=0$. We also show existence of a $K_β$-invariant probability measure, absolutely continuous with respect to $m_1\otimes m_2 \otimes λ_2$, where $m_1, m_2$ are product measures on $\{0,1\}^\mathbb{N},\{0,1,2\}^\mathbb{N}$, respectively, and $λ_2$ is the normalized Lebesgue measure on $S_β$. For $3/2<β\leq β^*$, where $β^*\approx 1.5437$ is the root of $x^3-2x^2+2x=2$, there are radial holes in $S_β$. In this case, $K_β$ is defined on $\{0,1\}^\mathbb{N}\times S_β$. We also show that it has a unique invariant measure of maximal entropy.

math.DS↗

arXiv2021

Alternating $N$-expansions

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration.

math.DS↗

arXiv2021

Equilibrium States for the Random $β$-Transformation through $g$-Measures

We consider the random $β$-transformation $K_β$, defined on $\{0,1\}^{\mathbb N}\times[0, \frac{\lfloorβ\rfloor}{β-1}]$, that generates all possible expansions of the form $x=\sum_{i=0}^{\infty}\frac{a_i}{β^i}$, where $a_i\in \{0,1,\cdots,\lfloorβ\rfloor\}$. This transformation was first introduced by Dajani and Kraaikamp, and later studied by Dajani and de Vries, where two natural invariant ergodic measures were found. The first is the unique measure of maximal entropy, and the second is a measure of the form $m_p\times μ_β$, with $m_p$ the Bernoulli $(p,1-p)$ product measure and $μ_β$ is a measure equivalent to Lebesgue measure. In this paper, we give an uncountable family of $K_β$-invariant exact $g$-measures for a certain collection of algebraic $β$'s.

math.DS↗

arXiv2021

Dynamical behavior of alternate base expansions

We generalize the greedy and lazy $β$-transformations for a real base $β$ to the setting of alternate bases $\boldsymbolβ=(β_0,\ldots,β_{p-1})$, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_\boldsymbolβ$ and $L_\boldsymbolβ$ respectively, can be iterated in order to generate the digits of the greedy and lazy $\boldsymbolβ$-expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of $T_\boldsymbolβ$ and $L_\boldsymbolβ$. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the $p$-Lebesgue measure) $T_\boldsymbolβ$-invariant measure. We then show that this unique measure is in fact equivalent to the $p$-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $\frac{1}{p}\log(β_{p-1}\cdots β_0)$. We then express the density of this measure and compute the frequencies of letters in the greedy $\boldsymbolβ$-expansions. We obtain the dynamical properties of $L_\boldsymbolβ$ by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $β$-shift. Finally, we show that the $\boldsymbolβ$-expansions can be seen as $(β_{p-1}\cdots β_0)$-representations over general digit sets and we compare both frameworks.

math.DS↗

arXiv2020

Matching for random systems with an application to minimal weight expansions

We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [-1,1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated signed binary expansions never exceeds 1/2.

math.DS↗

arXiv2019

Multiple codings for self-similar sets with overlaps

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(Φ)$ is closed if and only if $\mathcal U_{\aleph_0}(Φ)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(Φ)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(Φ)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(Φ)$ and ${U_{\aleph_0}(Φ)}$.

math.DS↗

arXiv2019

Invariant measures, matching and the frequency of 0 for signed binary expansions

We introduce a parametrised family of maps $\{S_η\}_{η\in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_η(x)=2x-dη$, where $d\in \{-1,0,1 \}$. Each map $S_η$ generates binary expansions with digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter $η$. The transformations $S_η$ have a natural ergodic invariant measure $μ_η$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure $μ_η([-\frac12,\frac12])$ by the Ergodic Theorem. We show that the density of $μ_η$ is piecewise smooth except for a set of parameters of zero Lebesgue measure and full Hausdorff dimension and give a full description of the structure of the maximal parameter intervals on which the density is piecewise smooth. We give an explicit formula for the frequency of the digit 0 in typical signed binary expansions on each of these parameter intervals and show that this frequency depends continuously on the parameter $η$. Moreover, it takes the value $\frac23$ only on the interval $\big[ \frac65, \frac32\big]$ and it is strictly less than $\frac23$ on the remainder of the parameter space.

math.DS↗

arXiv2018

On the structure of $λ$-Cantor set with overlaps

Given $λ\in(0, 1)$, let $E_λ$ be the self-similar set generated by the iterated function system $\{x/3,(x+λ)/3,(x+2)/3\}$. Then $E_λ$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_λ$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_λ$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_λ$ and show that the spectrum of $E_λ$ vanishes if and only if $λ$ is irrational.

math.DS↗

arXiv2018

Multiple expansions of real numbers with digits set $\{0,1,q\}$

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \[ \mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, \] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}_{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., \[ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. \] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.

math.NT↗