Lasota-Yorke Inequality/One-Sided Shift Space of Finite Type

Theorem
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 $\norm \cdot_\theta$ be the Lipschitz norm on $F_\theta ^+$.

Let $\norm \cdot_\infty$ be the supremum norm on $F_\theta ^+$.

Let $f \in F_\theta ^+$.

Let $u := \map \Re f$ be the real part of $f$.

Let $\LL_f$ and $\LL_u$ denote the transfer operators.

If $\LL_u 1 = 1$, then there is a $C > 0$ such that:
 * $\norm {\LL_f ^n w}_\theta \le C \norm w_\infty + \theta ^n \norm w_\theta$

for all $w \in F_\theta ^+$ and $n \in \N$.

Proof
Recall the basic inequality:
 * There exists a $C_0 > 0$ so that we have:


 * for all $w \in F_\theta ^+$ and $n \in \N$.

On the other hand, we have:

since for all $x \in X_\mathbf A ^+$:

Therefore: