arXiv subjects

S. Jangjoo

Publications and source records attributed to S. Jangjoo.

2 recordsLinked to original sources

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↗