Basic 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 $\size \cdot_\theta$ be the Lipschitz seminorm 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 $\LL_u$ is normalized, then there is a $C > 0$ such that:

$\size {\LL_f ^n w}_\theta \le C \norm w_\infty + \theta ^n \size w_\theta$

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

Proof

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Sources

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