Definition:One-Sided Shift of Finite Type

Definition

Let $\mathbf A = \sqbrk a_k$ be a logical matrix for a $k \in \Z: k \ge 2$.

Let:

$X_\mathbf A ^+ = \set {x = \sequence {x_n}_{n \mathop \in \N} : x_n \in \set {1, 2, \ldots, k}, a_{x_n, x_{n + 1} } = 1}$

Let $\sigma_\mathbf A ^+ : X_\mathbf A ^+ \to X_\mathbf A ^+$ be the forward shift operator, that is:

$\map {\sigma _{\mathbf A} ^+ } x := y$

where $y_n = x_{n + 1}$ for all $n \in \N$.

Then the pair $\struct {X _\mathbf A ^+, \sigma_\mathbf A ^+}$ is called a one-sided shift of finite type.

This article is complete as far as it goes, but it could do with expansion. In particular: ... and then of course we have to do the one-sided shift of finite type going backwards, yeah? Currently, not planed to introduce $\Sigma ^-_\mathbf A$. But it may happen later. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: William Parry and Mark Pollicott: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics: Chapter $1$: Subshifts of Finite Type and Function Spaces