arXiv subjects
Publications and source records attributed to Charlene Kalle.
In this article we derive a formula for the Hausdorff dimension of Besicovitch-Eggleston level sets associated with non-autonomous dynamics constructed from families of countable affine iterated function systems. The formula obtained shows that the universal-lower-bound phenomenon present in the autonomous case studied by Fan et al. (2010) persists in this non-autonomous setting.
It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show that this condition holds for complex $\boldsymbolα$-Hurwitz algorithms with parameters $\boldsymbolα\in\mathbb{Q}^2$. This is equivalent to the existence of certain finite partitions related to these algorithms and lies at the root of explorations into their Diophantine properties. Our result provides a partial answer to a recent question formulated by Lukyanenko and Vandehey.
In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine IFS's. Also, we study Besicovitch-Eggleston sets for finite GLS number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.
Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the countably infinite alphabet $\mathbb N$ in which each possible block of digits $α_1, \ldots, α_k \in \mathbb N$, $k \in \mathbb N$, occurs with frequency $\prod_{d=1}^k L_{α_d}$. In other words, we construct $L$-normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers $x \in [0,1]$ that are normal in GLS number systems that correspond to the sequence $L$ or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion.
For a fixed $α$, each real number $x \in (0,1)$ can be represented by many different generalised $α$-Lüroth expansions. Each such expansion produces for the number $x$ a sequence of rational approximations $(\frac{p_n}{q_n})_{n \ge 1}$. In this paper we study the corresponding approximation coefficients $(θ_n(x))_{n \ge 1}$, which are given by \[ θ_n (x): = q_n \left|x-\frac{p_n}{q_n}\right|.\] We give the cumulative distribution function and the expected average value of the $θ_n$ and we identify which generalised $α$-Lüroth expansion gives the best approximation properties. We also analyse the structure of the set $\mathcal M_α$ of possible values that the expected average value of $θ_n$ can take, thus answering a question from \cite{Barrionuevo-Burton-Dajani-Kraaikamp-1994}.
The $β$-encoder is an analog circuit that converts an input signal $x \in [0,1]$ into a finite bit stream $\{b_i\}$. The bits $\{b_i\}$ are correlated and therefore are not immediately suitable for random number generation, but they can be used to generate bits $\{a_i\}$ that are (nearly) uniformly distributed. In this article we study two such methods. In the first part the bits $\{a_i\}$ are defined as the digits of the base-2 representation of the original input $x$. Under the assumption that there is no noise in the amplifier we then study a question posed by Jitsumatsu and Matsumura on how many bits $b_1, \ldots, b_m$ are needed to correctly determine the first $n$ bits $a_1,\ldots,a_n$. In the second part we show this method fails for random amplification factors. Nevertheless, even in this case, nearly uniformly distributed bits can still be generated from $b_1,\ldots,b_m$ using modern cryptographic techniques.
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.
We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences. For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point motions. These stationary measures are infinite $σ$-finite measures in the case of zero Lyapunov exponent. For varying Lyapunov exponent we find here a phase transition for the system of two-point motions, in which the support of the stationary measure explodes with intermittent dynamics and an infinite stationary measure at the transition point.
For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that the correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay. This contrasts other known results, where random systems adopt the best decay rate of the deterministic maps in the systems.
In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number $x$ that can be determined from just knowing its first $n$ decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs' Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin's formula for entropy.
We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.
Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for iterated function systems of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in $L^q$ for any $q > 1$. This provides a theory of critical intermittency alongside the theory for the well studied Manneville-Pomeau maps, where the intermittency is caused by a neutral fixed point.
We introduce a family of random $c$-Lüroth transformations $\{L_c\}_{c \in [0, \frac12]}$, obtained by randomly combining the standard and alternating Lüroth maps with probabilities $p$ and $1-p$, $0 < p < 1$, both defined on the interval $[c,1]$. We prove that the pseudo-skew product map $L_c$ produces for each $c \le \frac25$ and for Lebesgue almost all $x \in [c,1]$ uncountably many different generalised Lüroth expansions that can be investigated simultaneously. Moreover, for $c= \frac1{\ell}$, for $\ell \in \mathbb{N}_{\geq 3} \cup \{\infty\}$, Lebesgue almost all $x$ have uncountably many universal generalised Lüroth expansions with digits less than or equal to $\ell$. For $c=0$ we show that typically the speed of convergence to an irrational number $x$, of the sequence of Lüroth approximants generated by $L_0$, is equal to that of the standard Lüroth approximants; and that the quality of the approximation coefficients depends on $p$ and varies continuously between the values for the alternating and the standard Lüroth map. Furthermore, we show that for each $c \in \mathbb Q$ the map $L_c$ admits a Markov partition. For specific values of $c>0$, we compute the density of the stationary measure and we use it to study the typical speed of convergence of the approximants and the digit frequencies.
For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. In case the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $β$-transformations and random Lüroth maps with a hole.
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.
As a natural counterpart to Nakada's $α$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $σ$-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.
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.
Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by \[ \mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}. \] In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$, \[ \lim_{δ\rightarrow 0} \dim_H(\mathcal{B}\cap(q-δ, q+δ))=\dim_H\mathcal{U}_q, \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $ \dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$. As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[ \dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes.