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

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ 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 $\eta$. The transformations $S_\eta$ have a natural ergodic invariant measure $\mu_\eta$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure $\mu_{\eta}([-\frac12,\frac12])$ by the Ergodic Theorem. We show that the density of $\mu_\eta$ 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 $\eta$. 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.Comment: 30 pages, 4 figure

On the points without universal expansions

Let $1<\beta<2$. Given any $x\in[0, (\beta-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $\beta$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ For any $k\geq 1$ and any $(b_1b_2\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$, then we call $(a_n)$ a universal $\beta$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<\beta<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_{\beta}$ of points without universal expansions. For any $n\geq 2$, let $\beta_n$ be the $n$-bonacci number satisfying the following equation: $\beta^n=\beta^{n-1}+\beta^{n-2}+\cdots +\beta+1.$ Then we have $\dim_{H}(V_{\beta_n})=1$, where $\dim_{H}$ denotes the Hausdorff dimension. Similar results are still available for some other algebraic numbers. As a corollary, we give some results of the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper \cite{KarmaKan}.Comment: 15page

Assessing the future of air freight

The role of air cargo in the current transportation system in the United States is explored. Methods for assessing the future role of this mode of transportation include the use of continuous-time recursive systems modeling for the simulation of different components of the air freight system, as well as for the development of alternative future scenarios which may result from different policy actions. A basic conceptual framework for conducting such a dynamic simulation is presented within the context of the air freight industry. Some research needs are identified and recommended for further research. The benefits, limitations, pitfalls, and problems usually associated with large scale systems models are examined

Invariant densities for random β\beta-expansions

Let $\beta >1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i \beta^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor \beta \rfloor /(\beta -1)]$. We show existence and uniqueness of an absolutely continuous $K_{\beta}$-invariant probability measure w.r.t. $m_p \otimes \lambda$, where $m_p$ is the Bernoulli measure on $\{0,1\}^{\N}$ with parameter $p$ $(0 < p < 1)$ and $\lambda$ is the normalized Lebesgue measure on $[0 ,\lfloor \beta \rfloor /(\beta -1)]$. Furthermore, this measure is of the form $m_p \otimes \mu_{\beta,p}$, where $\mu_{\beta,p}$ is equivalent with $\lambda$. We establish the fact that the measure of maximal entropy and $m_p \otimes \lambda$ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure $m_p \otimes \mu_{\beta,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK]

Metrical theory for α\alpha-Rosen fractions

The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the $\alpha$-Rosen fractions. The metrical properties of these $\alpha$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature