Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
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Theorem
Let $\struct {X_\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.
Let $f : X_\mathbf A \to \C$ be a continuous mapping.
Let $\theta \in \openint 0 1$.
Let $C > 0$.
Then:
- $\forall x, y \in X_\mathbf A : \size {\map f x - \map f y} \le C \map {d_\theta} {x, y}$
- $\forall n \in \N : \map {\mathrm {var}_n} f \le C \theta ^n$
where:
That is:
- $\ds \size f_\theta = \sup \set { \dfrac {\cmod {\map f x - \map f y} }{\map {d_\theta} {x,y} } : x,y \in X_\mathbf A, x \ne y}$
where $\size \cdot_\theta $ is the Lipschitz seminorm.
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Proof
Sufficient Condition
Suppose:
- $\forall x, y \in X_\mathbf A : \size {\map f x - \map f y} \le C \map {d_\theta} {x, y}$
Let $n\in\N$.
By definition of the metric:
- $\forall x, y \in X_\mathbf A:$
- $\forall i \in \openint {-n} n : x_i = y_i \implies \map {d _\theta} {x, y} \le \theta ^n$
Thus, by the assumption:
- $\forall x, y \in X_\mathbf A:$
- $\forall i \in \openint {-n} n :x_i = y_i \implies \size {\map f x - \map f y} \le C \map {d_\theta} {x, y} \le C \theta ^n$
Therefore:
- $\map {\mathrm {var}_n} f \le C \theta ^n$
$\Box$
Necessary Condition
Suppose:
- $\forall n \in \N : \map {\mathrm {var}_n} f \le C \theta ^n$
Let $x, y \in X_\mathbf A$.
If $x = y$, then obviously:
- $\size {\map f x - \map f y} = 0 \le C \map {d_\theta} {x, y}$
So, let $x \ne y$.
Then:
- $\exists N \in \N : d_\theta (x,y) = \theta ^N$
In particular:
- $\forall i \in \openint {-N} N : x_i = y_i$
Therefore:
- $\size {\map f x - \map f y}\le \map {\mathrm {var}_N} f\le C \theta ^N = C \map {d_\theta} {x, y}$
$\blacksquare$