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 $\LL_f : \map C {X_\mathbf A ^+} \to \map C {X_\mathbf A ^+}$ be the transfer operator.

Let $\beta $ be the spectral radius of $\LL_f$.

If $\mathbf A$ is irreducible, then:

$(1): \quad \beta$ is a simple eigenvalue of $\LL_f$ with a strictly positive eigenfunction $h \in F_\theta ^+$, i.e.:

$\set {g \in \map C {X_\mathbf A ^+} : \LL_f g = \beta g} = \C \cdot h$

Furthermore, if $\mathbf A$ is irreducible and aperiodic, then:

$(2): \quad$ There is an $r \in \openint 0 \beta$ such that:

$\map \sigma {\LL_f} \setminus \set \beta \subseteq \map B {0, r}$

where:

the left hand side denotes a closed disk

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

$(3): \quad$ There is a Borel probability measure $\mu$ such that:

$\ds \forall v \in \map C {X_\mathbf A ^+} : \int \LL_f v \rd \mu = \beta \int v \rd \mu$

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

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

Also known as

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$(2)$ is said that $\LL_f : F_\theta ^+ \to F_\theta ^+$:

has a spectral gap, or

is quasi-compact.

Also see

• Definition:Essential Spectral Radius of Bounded Linear Operator
• Definition:Quasi-Compact Operator

Proof

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Source of Name

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