Transfer Operator with respect to One-Sided Shift Space of Finite Type is Linear Bounded Operator

Theorem

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

Let $\struct {B, \norm \cdot }$ be either:

$\struct {\map C {X ^+, \C}, \norm \cdot_\infty}$ the continuous mapping space with the supremum norm

or:

$\struct {\map {F_\theta^+} {X^+}, \norm \cdot_\theta}$ the space of Lipschitz functions with the Lipschitz norm

Let $f \in B$.

Then the transfer operator $\LL_f : B \to B$ is a bounded linear operator.

Proof

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