\subsection{Measure estimates for perturbed inverse branches} \label{sec:coding-measure} The exact return branches will be defined only on part of each phase interval. Some initial points may cease to admit a controlled return. We need to estimate both the distribution of the codes that continue indefinitely and the measure of the points that are lost. The relevant estimates depend on the errors in one branch and on the length excluded from its domain. They do not require a bound on the digits. \subsubsection{Compositions and distortion} Let $I_0\subset J$ be open. For every finite word $\mathbf b$ and every positive integer $k$, let $D_{\mathbf b,k}\subset J$ be open and let $h_{\mathbf b,k}$ be a differentiable, injective map on this domain, with nonzero derivative. For a fixed $\mathbf b$, assume that the images $h_{\mathbf b,k}(D_{\mathbf b,k})$ are pairwise disjoint as $k$ varies. The maps may depend on the whole preceding word. Define the admissible terminal domains $K_{\mathbf b}$ and compositions $H_{\mathbf b}$ recursively by \begin{align} K_\varnothing&=I_0,& H_\varnothing&=\mathrm{id},\notag\\ K_{\mathbf b k}&=D_{\mathbf b,k}\cap h_{\mathbf b,k}^{-1}(K_{\mathbf b}),& H_{\mathbf b k}&=H_{\mathbf b}\circ h_{\mathbf b,k}. \label{eq:actual-compositions} \end{align} Here $\mathbf b k$ denotes the word obtained by appending $k$. The set of initial points carrying a finite word is $C_{\mathbf b}=H_{\mathbf b}(K_{\mathbf b})$. Every intermediate point of this composition lies in its prescribed domain. In particular $K_{\mathbf b}\subset J$. The sets $K_{\mathbf b}$ are open, and $H_{\mathbf b}$ is continuous and injective on $K_{\mathbf b}$. The images $C_{\mathbf b}$ are Borel. Children lie in their parent and siblings are disjoint: \begin{equation} C_{\mathbf b k}\subset C_{\mathbf b},\qquad C_{\mathbf b k}\cap C_{\mathbf b l}=\varnothing\quad(k\ne l). \label{eq:actual-nesting} \end{equation} These assertions follow inductively from~\eqref{eq:actual-compositions}. For Borelness one may use that a continuous injection on a Borel subset of $\R$ has Borel image. For disjointness, apply injectivity of $H_{\mathbf b}$ to the disjoint branch images. Set \begin{equation} E_n=\bigcup_{|\mathbf b|=n}C_{\mathbf b},\qquad S=\bigcap_{n\geq0}E_n. \label{eq:surviving-phases} \end{equation} Thus $E_0=I_0$ and $E_{n+1}\subset E_n$. Each $x\in S$ lies in a unique cell at each depth. The resulting nested words define a sequence $\mathbf a(x)\in\mathcal D$, and \begin{equation} \{x\in S:(a_0(x),\ldots,a_{r-1}(x))=\mathbf b\} =S\cap C_{\mathbf b}. \label{eq:actual-prefix-fiber} \end{equation} It follows that $S$, the digit sequence, and the code $\psi(x)=c(\mathbf a(x))$ are Borel. For $x\in S$, let $y_n(x)$ be the unique point of $K_{(a_0(x),\ldots,a_{n-1}(x))}$ mapped to $x$ by its composition. Injectivity gives $y_0(x)=x$ and the exact recurrence \begin{equation} y_n(x)=h_{(a_0(x),\ldots,a_{n-1}(x)),a_n(x)}(y_{n+1}(x)). \label{eq:actual-inverse-recurrence} \end{equation} All these phases lie in $J$. Suppose that $d_n,e_n\geq0$ and that, for $|\mathbf b|=n$ and $t\in D_{\mathbf b,k}$, \begin{align} |h_{\mathbf b,k}(t)-g_k(t)|&\leq d_n,\notag\\ \left|\log\frac{|h_{\mathbf b,k}'(t)|}{|g_k'(t)|}\right|&\leq e_n. \label{eq:local-branch-errors} \end{align} Put \begin{equation} L_n=\sum_{j=0}^{n-1}e_j+2\sum_{j=1}^{n-1}d_j,\qquad L_0=0. \label{eq:prefix-distortion-budget} \end{equation} \begin{proposition}[Distortion on the actual domains] \label{prop:actual-distortion} For $|\mathbf b|=n$ and $t\in K_{\mathbf b}$, \begin{equation} \left|\log\frac{|H_{\mathbf b}'(t)|}{|G_{\mathbf b}'(t)|}\right| \leq L_n. \label{eq:actual-distortion} \end{equation} In particular, for every Borel set $A\subset K_{\mathbf b}$, \begin{equation} \lambda(H_{\mathbf b}(A)) \leq e^{L_n}\lambda(G_{\mathbf b}(A)) \leq2e^{L_n}\lambda(I_{\mathbf b})\lambda(A). \label{eq:actual-image-bound} \end{equation} \end{proposition} \begin{proof} Write $\ell_{\mathbf b}=\log|G_{\mathbf b}'|$ and $R_{\mathbf b}=\log|H_{\mathbf b}'|-\ell_{\mathbf b}$. All derivatives are nonzero on the domains under consideration, by the chain rule. For a word $\mathbf b$ of length $n$, abbreviate $h=h_{\mathbf b,k}$ and $g=g_k$. At $t\in K_{\mathbf b k}$, \begin{align*} R_{\mathbf b k}(t) &=R_{\mathbf b}(h(t)) +\ell_{\mathbf b}(h(t))-\ell_{\mathbf b}(g(t)) +\log\frac{|h'(t)|}{|g'(t)|}. \end{align*} The first term is evaluated at $h(t)\in K_{\mathbf b}$. Both $h(t)$ and $g(t)$ are positive, so \eqref{eq:log-prefix-lipschitz} bounds the middle difference by $2d_n$. When $\mathbf b$ is empty this difference is zero, since $\ell_\varnothing=0$. The last term is bounded by $e_n$. Induction proves~\eqref{eq:actual-distortion}, including the absence of a $d_0$ term. Exponentiating gives $|H_{\mathbf b}'|\leq e^{L_n}|G_{\mathbf b}'|$. The one-dimensional change of variables formula for injective differentiable maps, applied on $A$, gives the first inequality in~\eqref{eq:actual-image-bound}. Integrating the upper bound in \eqref{eq:cylinder-density} gives the second. \end{proof} \subsubsection{The coding measure and the mass that remains} Assume from now on that \begin{equation} L=\sum_{j=0}^\infty e_j+2\sum_{j=1}^\infty d_j<\infty. \label{eq:total-distortion-budget} \end{equation} Then $L_n\leq L$ and $d_n\to0$. Define the excluded part of the terminal domain by \begin{equation} A_{\mathbf b}=K_{\mathbf b}\setminus \bigcup_{k\geq1}h_{\mathbf b,k}(D_{\mathbf b,k}). \label{eq:stopped-domain} \end{equation} This is the set from which the next controlled branch cannot be taken. \begin{theorem} \label{thm:branch-measure} Under~\eqref{eq:local-branch-errors} and \eqref{eq:total-distortion-budget}, \begin{equation} \psi_*(\lambda|_S)\leq e^L\lambda|_J. \label{eq:survivor-code-domination} \end{equation} If $\lambda(A_{\mathbf b})\leq\ell_n$ whenever $|\mathbf b|=n$, where $\ell_n\geq0$ and $\sum_n\ell_n<\infty$, then \begin{equation} \lambda(I_0\setminus S)\leq2e^L\sum_{n=0}^\infty\ell_n. \label{eq:total-stopping-loss} \end{equation} For every Borel $I\subset\R$ the restricted coding measure has a nonnegative integrable density $\rho_I$ such that \begin{equation} \psi_*(\lambda|_{I\cap S})=\rho_I\lambda, \qquad 0\leq\rho_I\leq e^L\quad\lambda\text{-almost everywhere}, \qquad \int_\R\rho_I\dd\lambda=\lambda(I\cap S). \label{eq:coding-density-mass} \end{equation} \end{theorem} \begin{proof} Apply the first inequality of~\eqref{eq:actual-image-bound} to $A=K_{\mathbf b}\subset J$. It gives \[ \lambda(C_{\mathbf b})\leq e^L\lambda(I_{\mathbf b}). \] Together with~\eqref{eq:actual-prefix-fiber}, these are the hypotheses of Corollary~\ref{cor:code-domination}. This proves \eqref{eq:survivor-code-domination}. For the loss estimate, injectivity of $H_{\mathbf b}$ gives the exact identity \[ H_{\mathbf b}(A_{\mathbf b}) =C_{\mathbf b}\setminus\bigcup_{k\geq1}C_{\mathbf b k}. \] Every point of $I_0\setminus S$ has a first depth at which it fails to continue. Consequently, \[ I_0\setminus S =\bigcup_{n\geq0}\ \bigcup_{|\mathbf b|=n} H_{\mathbf b}(A_{\mathbf b}). \] At a fixed depth, the second inequality of \eqref{eq:actual-image-bound} yields \[ \sum_{|\mathbf b|=n}\lambda(H_{\mathbf b}(A_{\mathbf b})) \leq2e^L\ell_n\sum_{|\mathbf b|=n}\lambda(I_{\mathbf b}) =2e^L\ell_n. \] Summing over depths proves~\eqref{eq:total-stopping-loss}. Restriction of the initial measure can only decrease its pushforward. Thus the measure in~\eqref{eq:coding-density-mass} is also dominated by $e^L\lambda$ and is finite, since $S\subset J$. The Radon--Nikodym theorem gives its density. The domination implies $\rho_I\leq e^L$ almost everywhere by integration on measurable sets. Its integral is the total mass of the pushforward, which equals $\lambda(I\cap S)$. \end{proof} The last equality preserves the mass of the initial data that remain. It will allow statistical estimates along different curves to be integrated without renormalizing each curve separately. \begin{corollary}[Statistics and restricted loss] \label{cor:branch-statistics} Under the hypotheses and stopping bounds of Theorem~\ref{thm:branch-measure}, there is a Borel set $S_{\mathrm{stat}}\subset S$ with $\lambda(S\setminus S_{\mathrm{stat}})=0$ such that, for every $x\in S_{\mathrm{stat}}$, all six statistical limits in Theorem~\ref{thm:main} hold for the digits $a_m(x)$ and phases $y_m(x)$. For every Borel $I\subset\R$, \begin{equation} \lambda\bigl((I\cap I_0)\setminus S_{\mathrm{stat}}\bigr) \leq\min\left\{\lambda(I),\ 2e^L\sum_{n\geq0}\ell_n\right\}. \label{eq:restricted-statistical-loss} \end{equation} If $D\geq0$ and $\omega$ is a nonnegative measurable density with $\omega\leq D$ on $I_0$, the corresponding bound is \begin{equation} \int_{(I\cap I_0)\setminus S_{\mathrm{stat}}}\omega\dd\lambda \leq\min\left\{\int_I\omega\dd\lambda, \ 2De^L\sum_{n\geq0}\ell_n\right\}. \label{eq:weighted-statistical-loss} \end{equation} \end{corollary} \begin{proof} Take $S_{\mathrm{stat}}=S\cap\psi^{-1}(\mathcal B)$, with the inverse image understood on $S$. It is Borel. The coding domination and the full Lebesgue measure of $\mathcal B$ in $J$ show that its complement in $S$ is null. Equation~\eqref{eq:actual-inverse-recurrence} and the value bound in~\eqref{eq:local-branch-errors} give \[ |y_m(x)-g_{a_m(x)}(y_{m+1}(x))|\leq d_m\longrightarrow0. \] Apply Proposition~\ref{prop:typical-statistics} and Corollary~\ref{cor:continuous-transfer}. The loss in a restricted initial set is bounded both by its own measure and by the total loss in \eqref{eq:total-stopping-loss}. Adding the null set $S\setminus S_{\mathrm{stat}}$ does not change either bound. For the weighted version, integrate $\omega\leq D$ on the lost part of $I_0$; the other bound follows from inclusion in $I$. \end{proof} \subsubsection{Errors controlled by transverse amplitude} The bounds needed from the dynamics can now be stated quantitatively. The terminal connection will decrease the transverse amplitude enough to use the sequence \begin{equation} B_0=B,\qquad B_{m+1}=B_m^{4/3},\qquad B_m=B^{(4/3)^m}. \label{eq:era-amplitude-sequence} \end{equation} The branch value error will be of order $B_m$, its logarithmic derivative error of order $\sqrt{B_m}$, and the excluded length of order $B_m^\eta$, for a fixed $\eta>0$. The following consequence of Theorem~\ref{thm:branch-measure} records how these estimates will be used. \begin{lemma} \label{lem:amplitude-sums} Suppose $00$, and $B^\eta<1/160$. For every $p\geq\eta$ and $k\geq0$, \begin{equation} \sum_{m=k}^\infty B_m^p\leq2B_k^p. \label{eq:amplitude-tail-sum} \end{equation} \end{lemma} \begin{proof} The sequence $B_m$ is positive and at most $B$. Also $B^{\eta/3}<1/2$, since its cube is less than $1/160<1/8$. Using $p\geq\eta$ and $B\leq1$ gives \[ B_{m+1}^p=B_m^p B_m^{p/3} \leq B_m^p B^{p/3}\leq\tfrac12 B_m^p. \] Iteration from $k$ and summation of the geometric series prove the claim. \end{proof} \begin{corollary} \label{cor:amplitude-statistics} Fix $0<\eta\leq1/2$ and $C_v,C_i\geq0$. There is $B_*>0$, with $B_*<1$, such that the following holds for $01/2$. \end{corollary} \begin{proof} Choose $B_*<1$ so that, for $0