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

Let $f \in F_\theta ^+$ be a real-valued function.

Let $\LL _f : F_\theta^+ \to F_\theta^+$ be the Ruelle-Perron-Frobenius Operator.

Then $\LL _f$ is normalized if and only if:

$\LL _f 1 = 1$

Sources

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