Ruelle-Perron-Frobenius Theorem

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

Let $\map C {X_\mathbf A ^+} := \map C {X_\mathbf A ^+, \C}$ be the continuous mapping space.

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

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

Let $\LL_f$ be the transfer operator.

Let $\map \sigma {\LL_f}$ be the spectrum of $\LL_f : F_\theta ^+ \to F_\theta ^+$.

If $\mathbf A$ is irreducible, then:
 * $(1)$ There is a simple eigenvalue $\beta \in \R_{>0}$ of $\LL_f : \map C {X_\mathbf A ^+} \to \map C {X_\mathbf A ^+}$ with a strictly positive eigenfunction $h \in F_\theta ^+$.

Furthermore, if $\mathbf A$ is irreducible and aperiodic, then:
 * $(2)$ There is an $r \in \openint 0 \beta$ such that:
 * $\map \sigma {\LL_f} \setminus \set \beta \subseteq \map B {0, r}$
 * where the denotes a closed disk.


 * $(3)$ There is a probability measure $\mu$ such that:
 * $\ds \forall v \in \map C {X_\mathbf A ^+} : \int \LL_f v \rd \mu = \beta \int v \rd \mu$


 * $(4) \ds \forall v \in \map C {X_\mathbf A ^+} : \lim _{n \to \infty} \norm {\beta^{-n} \LL_f ^n v - h \int v \rd \mu}_\infty = 0$


 * $(5) \ds \int h \rd \mu = 1$

Also known as
$(2)$ is called a spectral gap.