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

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

Let $\map C {X _\mathbf A ^+}$ be the complex-valued continuous mapping space.

Let $f \in \map C {X _\mathbf A ^+}$.

Let $\LL_f$ be the Ruelle-Perron-Frobenius operator.

For all $w \in \map C {X _\mathbf A ^+}$, $n \in \N$ and $x \in X _\mathbf A ^+$, we have:
 * $\ds \map {\LL_f^n w} x = \sum_{y \mathop \in \map {\sigma^{-n} } x} e^{\map {f_n} y} \map w y $

where
 * $f_n := f + f \circ \sigma + \cdots + f \circ \sigma ^{n-1}$
 * $\map {\sigma ^{-n} } x$ denotes the preimage of $x$ under $\sigma^n$, i.e.:
 * $\map {\sigma^{-n} } x = \set {\sequence {i_1, i_2, \ldots, i_n, x_0, x_1, \ldots} : \map {\mathbf A} {i_1, i_2} = \cdots = \map {\mathbf A} {i_{n-1}, i_n} = \map {\mathbf A} {i_n, x_0} = 1}$