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 $\map C {X _\mathbf A ^+, \C}$ be the continuous mapping space.

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

Let $B$ be either $\map C {X _\mathbf A ^+, \C}$ or $F_\theta^+$.

Let $f \in B$.

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

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

Also known as
This is also called:
 * the transfer operator
 * the Ruelle operator.

Also see

 * Ruelle-Perron-Frobenius Operator on One-Sided Shift Space of Finite Type is Linear Bounded Operator