Theorem: Basic Inequality/One-Sided Shift Space of Finite Type

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:
Let $x, y \in X_\mathbf A ^+$.
If $x_0 = y_0$, then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f w} x - \map {\LL_f w} y}
      | r = \cmod {\sum_{\map {\mathbf A} {k, x_0} = 1} \paren {e^{\map f {k x} } \map w {k x} - e^{\map f {k y} } \map w {k y} } }
}}
{{eqn | o = \le
      | r = \sum_{\map {\mathbf A} {k, x_0} = 1} \cmod { e^{\map f {k x} } \paren {\map w {k x} - \map w {k y} } } + \sum_{\map {\mathbf A} {k, x_0} = 1} \cmod{ \paren {e^{\map f {k x} } -  e^{\map f {k y} } } \map w {k y} }
}}
{{eqn | r = \sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map u {k x} } \cmod {\map w {k x} - \map w {k y} } + \sum_{\map {\mathbf A} {k, x_0} = 1} \cmod{ e^{\map f {k x} } -  e^{\map f {k y} } } \cmod {\map w {k y} }
}}
{{eqn | o = \le
      | r = \sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map u {k x} } \size w_\theta \map {d_\theta} {kx, ky} + \sum_{\map {\mathbf A} {k, x_0} = 1} \size {e^w}_\theta \map {d_\theta} {kx, ky} \norm w_\infty
}}
{{eqn | r = \sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map u {k x} } \size w_\theta \theta \map {d_\theta} {x, y} + \sum_{\map {\mathbf A} {k, x_0} = 1} \size {e^w}_\theta \theta \map {d_\theta} {x, y} \norm w_\infty
}}
{{eqn | r = 
\map {\LL_u 1} x \size w_\theta \theta \map {d_\theta} {x, y} + \sum_{\map {\mathbf A} {k, x_0} = 1} \size {e^w}_\theta \theta \map {d_\theta} {x, y} \norm w_\infty
}}
{{eqn | r = \size w_\theta \theta \map {d_\theta} {x, y} + \sum_{\map {\mathbf A} {k, x_0} = 1} \size {e^w}_\theta \theta \map {d_\theta} {x, y} \norm w_\infty
}}
{{eqn | o = \le
      | r = \map {d_\theta} {x, y} \paren { \size w_\theta \theta + \sum_{\map {\mathbf A} {k, x_0} = 1} \size {e^w}_\theta \theta \norm w_\infty }
}}
{{eqn | o = \le
      | r = \map {d_\theta} {x, y} \paren { \theta \size w_\theta + N \size {e^w}_\theta \theta \norm w_\infty }
}}
{{end-eqn}}
where $N$ is the size of the alphabet of $X_\mathbf A ^+$.
If $x_0 \ne y_0$, then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f w} x - \map {\LL_f w} y}
      | r = \cmod {\sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map f {k x} } \map w {k x} - \sum_{\map {\mathbf A} {k, y_0} = 1} e^{\map f {k y} } \map w {k y} }
}}
{{eqn | o = \le
      | r = \sum_{\map {\mathbf A} {k, x_0} = 1} \cmod { e^{\map f {k x} } \map w {k x} } + \sum_{\map {\mathbf A} {k, y_0} = 1} \cmod {e^{\map f {k y} } \map w {k y} }
}}
{{eqn | r = \sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map u {k x} } \cmod {\map w {k x} } + \sum_{\map {\mathbf A} {k, y_0} = 1} e^{\map u {k y} } \cmod {\map w {k y} }
}}
{{eqn | o = \le
      | r = \sum_{\map {\mathbf A} {k, x_0} = 1} e^{\map u {k x} } \norm w_\infty + \sum_{\map {\mathbf A} {k, y_0} = 1} e^{\map u {k y} } \norm w_\infty
}}
{{eqn | r = \map {\LL_u 1} x \norm w_\infty + \map {\LL_u 1} y \norm w_\infty
}}
{{eqn | r = 2 \norm w_\infty
}}
{{eqn | r = \map {d_\theta} {x, y} 2 \norm w_\infty
}}
{{eqn | o = \le
      | r = \map {d_\theta} {x, y} \paren { \theta \size w_\theta + 2 \norm w_\infty }
}}
{{end-eqn}}
Thus:
:$\size {\LL_f w}_\theta \le \theta \size w_\theta + \tilde C \norm w_\infty$
where:
:$\tilde C := \max \set { N \size {e^w}_\theta \theta, 2 }$
In particular, for all $n \ge 1$:
{{begin-eqn}}
{{eqn | l = \size {\LL_f^n w}_\theta
      | o = \le
      | r = \theta \size {\LL_f^{n-1} w}_\theta + \tilde C \norm {\LL_f^{n-1} w}_\infty
}}
{{eqn | o = \le
      | r = \theta \size {\LL_f^{n-1} w}_\theta + \tilde C \LL_u^{n-1} \norm w_\infty
}}
{{eqn | r = \theta \size {\LL_f^{n-1} w}_\theta + \tilde C \norm w_\infty
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \size {\LL_f^n w}_\theta
      | o = \le
      | r = \theta \size {\LL_f^{n-1} w}_\theta + \tilde C \norm w_\infty
}}
{{eqn | o = \le
      | r = \theta \paren { \theta \size {\LL_f^{n-2} w}_\theta + \tilde C \norm w_\infty } + \tilde C \norm w_\infty
}}
{{eqn | r = \theta^2 \size {\LL_f^{n-2} w}_\theta + \paren {1 + \theta} \tilde C \norm w_\infty
}}
{{eqn | o = \le
      | r = \theta^3 \size {\LL_f^{n-3} w}_\theta + \paren {1 + \theta + \theta^2} \tilde C \norm w_\infty
}}
{{eqn | o = \vdots
}}
{{eqn | o = \le
      | r = \theta^n \size w_\theta + \paren {1 + \theta + \cdots + \theta^n} \tilde C \norm w_\infty
}}
{{eqn | r = \theta^n \size w_\theta + \frac {1 - \theta^{n+1} }{1 - \theta} \tilde C \norm w_\infty
}}
{{eqn | o = \le
      | r = \theta^n \size w_\theta + \frac {\tilde C}{1 - \theta} \norm w_\infty
}}
{{end-eqn}}
{{Finish|details}}