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

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## 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

## Source of Name

This entry was named for David Pierre Ruelle, Oskar Perron and Ferdinand Georg Frobenius.

## Sources

- 1990: William Parry and Mark Pollicott:
*Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics*: Chapter $2$: The Ruelle Operator