arXiv subjects

D. Ahmadi Dastjerdi

Publications and source records attributed to D. Ahmadi Dastjerdi.

5 recordsLinked to original sources

arXiv2014

Intertwined Synchronized Systems

An asymmetric-RLL$(d_1,k_1,d_0,k_0)$ system is a subshift of $\{0,1\}^{\mathbb Z}$ with run of $ 1 $ and $ 0$ restricted to $S=[d_1,k_1]\subseteq\mathbb N_{0}=\mathbb N\cup\{0\}$ and $S'=[d_0,k_0]\subseteq\mathbb N_{0}$ respectively. We extend this concept to the case when $S$ and $S'$ are arbitrary subsets of $\mathbb N_{0}$ and we call it a $(S,S')$-gap shift. Moreover, for $i=1,2$, if $X_{i}$ is a synchronized system generated by $V_{i}=\{v^{i}α_{i}:α_i v^{i}α_i\in\mathcal B(X_i),α_i\not\subseteq v^{i}\}$ where $ α_i $ is a synchronizing word for $ X_i $, then a natural generalization of $(S,S')$-gap shifts is a coded system $Z$ generated by $\{v^{1}α_1 v^{2}α_2:v^{i}α_i\in V_{i}, i=1,2\}$ and called the intertwined system. We investigate the dynamical properties of $Z$ with respect to $X_1$ and $X_2$.

math.DS↗

arXiv2013

On synchronized non-sofic subshifts

We show that a synchronized coded system $X$ is intrinsically ergodic of full support if and only if $ h(Y)$, the topological entropy of $Y$, is less than $h(X) $ whenever $Y$ is a proper subsystem of $ X $. We also show that like systems with specification property, SVGL's the non-mixing extension of systems with specification property, are intrinsically ergodic of full support. Moreover, we compute the entropy of the underlying graph of the Fischer cover of a synchronized system.

math.DS↗

arXiv2012

Equivalencies between beta-shifts and S-gap shifts

Let $ X_β$ be a sofic $ β$-shift for $ β\in (1, 2] $. We show that there is an $ S $-gap shift $ X(S) $ such that $ X_β $ and $ X(S) $ are right-resolving almost conjugate. Conversely, a condition on $ S \subseteq \mathbb N\cup \{0\} $ is given such that for this $S$, there is a $ β$ such that $ X(S) $ and $ X_β $ have the same equivalency. We show that if $X_β$ is SFT, then there is an $S$-gap shift conjugate to this $X_β$; however, if $X_β$ is not SFT, then no $S$-gap shift is conjugate to $X_β$. Also we will investigate the existence of these sort of equivalencies for non-sofics.

math.DS↗

arXiv2011

Computations on Sofic S-gap Shifts

Let $S=\{s_{n}\}$ be an increasing finite or infinite subset of $\mathbb N \bigcup \{0\}$ and $X(S)$ the $S$-gap shift associated to $S$. Let $f_{S}(x)=1-\sum\frac{1}{x^{s_{n}+1}}$ be the entropy function which will be vanished at $2^{h(X(S))}$ where $h(X(S))$ is the entropy of the system. Suppose $X(S)$ is sofic with adjacency matrix $A$ and the characteristic polynomial $χ_{A}$. Then for some rational function $ Q_{S} $, $χ_{A}(x)=Q_{S}(x)f_{S}(x)$. This $ Q_{S} $ will be explicitly determined. We will show that $ζ(t)=\frac{1}{f_{S}(t^{-1})}$ or $ζ(t)=\frac{1}{(1-t)f_{S}(t^{-1})}$ when $|S|<\infty$ or $|S|=\infty$ respectively. Here $ζ$ is the zeta function of $X(S)$. We will also compute the Bowen-Franks groups of a sofic $S$-gap shift.

math.DS↗

arXiv2011

Dynamics and Topology of S-gap Shifts

Let $S=\{s_i\in\mathbb N\cup\{0\}:0\leq s_i<s_{i+1}\}$ and let $d_{0}=s_{0}$ and $Δ(S)=\{d_{n}\}_{n}$ where $d_{n}=s_{n}-s_{n-1}$. In this note, we show that an $S$-gap shift is subshift of finite type (SFT) if and only if $S$ is finite or cofinite, is almost-finite-type (AFT) if and only if $Δ(S)$ is eventually constant and is sofic if and only if $Δ(S)$ is eventually periodic. We also show that there is a one-to-one correspondence between the set of all $S$-gap shifts and $\{r \in \mathbb R: r \geq 0\}\backslash \{\frac{1}{n}: n \in {\mathbb N}\}$ up to conjugacy. This enables us to induce a topology and measure structure on the set of all $S$-gaps. By using this, we give the frequency of certain $S$-gap shifts with respect to their dynamical properties.

math.DS↗