by} \quad \phi_{n,n+1}(x) =  x + h_{n}(x),$$ and $ \phi_{k,n} : \mathcal{C}_{k} \to \mathcal{C}_{n} $ for $ k \le n $ defined as $$\phi_{k,n} = \phi_{n-1,n} \circ \phi_{n-2,n-1} \circ \cdots \circ \phi_{k,k+1}$$ with the agreement that $ \phi_{n,n} : \mathcal{C}_{n} \to \mathcal{C}_{n} $ is the identity map. Note that $$\phi_{m,n} \circ \phi_{k,m} = \phi_{k,n} \quad \textrm{for all } k \le m \le n.$$

When a flow $ {\mathfrak{F}}$ on $ X $ is free, one has a *cocycle* $ \rho : \cber[X] \to \mathbb{R}^{d} $ that assigns to a pair $ (x,y) \in \cber[X] $ the unique vector $ \vec{r} \in \mathbb{R}^{d} $ such that $ x + \vec{r} = y $. If $ (\mathcal{C}_{n}, h_{n}) $ is a spiral of cross sections for $ {\mathfrak{F}}$, then for all $ x \in \mathcal{C}_{0} $ $$\rho\bigl(x,\phi_{0,n}(x)\bigr) = h_{0}(x) + h_{1}\bigl
   (\phi_{0,1}(x)\bigr) + \cdots +
   h_{n-2}\bigl(\phi_{0,n-2}(x)\bigr) + h_{n-1}\bigl(\phi_{0,n-1}(x)\bigr).
   \tag{\dag}
   \label{eq:cocycle-estimate}$$

A spiral of cross sections $ (\mathcal{C}_{n}, h_{n}) $, $ n \in \mathbb{N} $, of a free flow is said to be *convergent* if for all $ x \in \mathcal{C}_{0} $ the limit $ \lim_{n} \rho\bigl(x, \phi_{0,n}(x)\bigr) $ exists. For a convergent spiral we define the *limit shift* maps $ H_{k} : 