Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type

Definition
Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.

Let $\theta \in \openint 0 1$.

Let $F_\theta^+$ be the space of Lipschitz function on $X_\mathbf A ^+$.

Let $f \in F_\theta^+$.

The Ruelle-Perron-Frobenius operator $\LL_f : F_\theta^+ \to F_\theta^+$ is defined as:
 * $\ds \forall g \in F_\theta^+ : \map {\LL_f g} x := \sum_{y \in \sigma^{-1} \sqbrk x} e^{\map f y} \map g y$

where $\sigma^{-1} \sqbrk x$ denotes the preimage of $\set x$ under $\sigma$, i.e.
 * $\sigma^{-1} \sqbrk x = \set {\sequence {i, x_0, x_1, \ldots} : \map {\mathbf A} {i, x_0} = 1}$

Also known as
This is also called Ruelle operator.