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

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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 $\LL_u$ is normalized, 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:
\(\text {(1)}: \quad\) \(\ds \size {\LL_f ^n w}_\theta\) \(\le\) \(\ds C_0 \norm w_\infty + \theta ^n \size w_\theta\)
for all $w \in F_\theta ^+$ and $n \in \N$.

On the other hand, we have:

\(\text {(2)}: \quad\) \(\ds \norm {\LL_f w}_\infty\) \(\le\) \(\ds \norm w_\infty\)

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

\(\ds \cmod {\map {\LL_f w } x}\) \(=\) \(\ds \cmod {\sum_{y \mathop \in \map {\sigma^{-1} } x} e^{\map f y} \map w y}\)
\(\ds \) \(\le\) \(\ds \sum_{y \mathop \in \map {\sigma^{-1} } x} \cmod {e^{\map f y} \map w y}\)
\(\ds \) \(=\) \(\ds \sum_{y \mathop \in \map {\sigma^{-1} } x} e^{\map u y} \cmod {\map w y}\)
\(\ds \) \(\le\) \(\ds \norm w_\infty \sum_{y \mathop \in \map {\sigma^{-1} } x} e^{\map u y}\)
\(\ds \) \(=\) \(\ds \norm w_\infty \map {\LL _u 1} x\)
\(\ds \) \(=\) \(\ds \norm w_\infty\)


Therefore:

\(\ds \norm {\LL_f ^n w}_\theta\) \(=\) \(\ds \norm {\LL_f ^n w}_\infty + \size {\LL_f ^n w}_\theta\) Definition of Lipschitz Norm
\(\ds \) \(\le\) \(\ds \norm w_\infty + \size {\LL_f ^n w}_\theta\) by $(2)$
\(\ds \) \(\le\) \(\ds \norm w_\infty + C_0 \norm w_\infty + \theta ^n \size w_\theta\) by $(1)$
\(\ds \) \(=\) \(\ds \paren {C_0 + 1} \norm w_\infty + \theta ^n \size w_\theta\)
\(\ds \) \(\le\) \(\ds \paren {C_0 + 1} \norm w_\infty + \theta ^n \norm w_\theta + \theta ^n \norm w_\infty\)
\(\ds \) \(=\) \(\ds \paren {C_0 + 1} \norm w_\infty + \theta ^n \norm w_\theta\) Definition of Lipschitz Norm


This needs considerable tedious hard slog to complete it.
In particular: Details
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Source of Name

This entry was named for Andrzej Lasota and James Alan Yorke.

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