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:

$\forall x, y \in X_\mathbf A : \size {\map f x - \map f y} \le C \map {d_\theta} {x, y}$

if and only if:

$\forall n \in \N : \map {\mathrm {var}_n} f \le C \theta ^n$

where:

$d_\theta$ denotes the metric

$\map {\mathrm {var}_n} f$ denotes the $n$th variation of $f$.

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:

$\size {\map f x - \map f y}\le \map {\mathrm {var}_N} f\le C \theta ^N = C \map {d_\theta} {x, y}$

$\blacksquare$