4.4 Proving freedom of action
We say that a \( \beta \)-approximation \( \psi \) approximates a \( \beta \)-allowable permutation \( \rho \) if \( \psi _A^{\mathcal L}\leq \rho _A^{\mathcal L}\) and \( \psi _A^{\mathcal A}\leq \rho _A^{\mathcal A}\) for each path \( A : \beta \rightsquigarrow \bot \). If \( \psi \) approximates \( \rho \) then \( \psi ^n \) approximates \( \rho ^n \) for each \( n : \mathbb Z \). 1 A \( \beta \)-approximation \( \psi \) exactly approximates a \( \beta \)-allowable permutation \( \rho \) if \( \psi \) approximates \( \rho \), and in addition, if \( a \) is an atom and \( A : \beta \rightsquigarrow \bot \), then \( a \notin \operatorname {\mathsf{coim}}\psi _A^{\mathcal A}\) implies \( \rho (a)^\circ = \rho (a^\circ ) \) and \( \rho ^{-1}(a)^\circ = \rho ^{-1}(a^\circ ) \).
We say that freedom of action holds at a type index \( \delta \) if every coherent \( \delta \)-approximation exactly approximates some \( \delta \)-allowable permutation.
Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( L \) be \( A \)-flexible. Then there is a coherent extension \( \chi \geq \psi \) with \( L \in \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\).
Define \( L' : \mathbb Z \to {\mathcal L}\) by \( L'(n) = L \), then appeal to proposition 4.15 to obtain \( \chi \geq \psi \). All we must do is check that \( \psi \) is coherent at \( (L, L) \), which is trivial.
Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( L \) be \( A \)-inflexible with path \( (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). Suppose that \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). 2 Suppose that freedom of action holds at level \( \delta \). Then there is a coherent extension \( \chi \geq \psi \) with \( L \in \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\).
Let \( \rho \) be a \( \delta \)-allowable permutation that \( (\psi _B)_\delta \) approximates. Then for each \( n : \mathbb Z \), as \( (\psi ^n_B)_\delta \) approximates \( \rho ^n \), we obtain \( (\psi ^n_B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho ^n(\operatorname {\mathsf{supp}}(t)) \) as \( (\psi ^n_B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). 3 Define \( L : \mathbb Z \to {\mathcal L}\) by \( L(n) = f_{\delta ,\varepsilon }(\rho ^n(t)) \).
Suppose that there is some \( n \) such that \( L(n) \in \operatorname {\mathsf{coim}}\psi ^{\mathcal L}\). Note that
So as \( \psi ^{-n} \) is coherent, we obtain \( (L(n), f_{\delta ,\varepsilon }(t)) \in (\psi ^{-n}_A)^{\mathcal L}\). In particular, \( f_{\delta ,\varepsilon }(t) \in \operatorname {\mathsf{coim}}\psi _A^{\mathcal L}\) already, and no work needs to be done.
We first check the hypothesis of proposition 4.11 for adding orbits. If \( f_{\delta ,\varepsilon }(\rho ^m(t)) = f_{\delta ,\varepsilon }(\rho ^n(t)) \), then \( \rho ^m(t) = \rho ^n(t) \), so \( \rho ^{m+k}(t) = \rho ^{n+k}(t) \), giving \( f_{\delta ,\varepsilon }(\rho ^{m+k}(t)) = f_{\delta ,\varepsilon }(\rho ^{n+k}(t)) \) as required.
We now check the criterion of proposition 4.15 for adding orbits coherently. It suffices to show that \( \psi \) is coherent at \( (L(n), L(n+1)) \) for each \( n : \mathbb Z \). This is witnessed by \( \rho \), which satisfies
and
as required. 4
If \( (\psi _i)_{i : I} \) is a chain of coherent approximations where \( I \) is a linear order, then the supremum \( \psi \) is coherent.
Direct, using the same idea as the proof of proposition 4.15.
Freedom of action holds at all type indices \( \beta \leq \alpha \).
By induction, we may assume freedom of action holds at all \( \delta {\lt} \beta \). Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( \chi \) be a maximal coherent extension, which exists by Zorn’s lemma and proposition 4.23.
Suppose that there is a litter \( L \) such that there exists a path \( A \) where \( L \notin \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\). Let \( L \) have minimal position with this property, and let \( A \) be such a path.
Suppose that \( L \) is \( A \)-flexible. Then by proposition 4.21, there is an extension \( \varphi \) of \( \chi \) such that \( L \in \operatorname {\mathsf{coim}}\varphi _A^{\mathcal L}\), contradicting maximality of \( \chi \).
Suppose that \( L \) is \( A \)-inflexible, with path \( (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t \). Then \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). Indeed, by definition 2.25 (coherent data) and proposition 2.22 (fuzz maps), for each atom or near-litter \( y \) that appears in the range of \( \operatorname {\mathsf{supp}}(t)_C \), we have \( \iota (y) {\lt} \iota (t) {\lt} \iota (L) \), giving the desired conclusion by minimality of the position of \( L \) and the criteria of proposition 2.19. Thus, we obtain the same contradiction by proposition 4.22.
So \( \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\) is the set of all litters for each path \( A \). We then use the fact that our model data is coherent to recursively compute the allowable permutation \( \rho \) with the same action as \( \chi \). Then \( \chi \) exactly approximates \( \rho \), so \( \psi \) also exactly approximates \( \rho \). 5