Canonically Codable Points and Irreducible Codings

$M$ is a cpt. Riemannian manifold without boundary, $f\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\beta }(M)$. In [Sarig13], for all $\chi >0$, for every small enough $ϵ>0$, Sarig had first constructed a coding $\widehat{\pi }:\widehat{\mathrm{\Sigma }}\to M$ which covers the set of all Lyapunov regular $\chi$-hyperbolic points when $\mathrm{d}\mathrm{i}\mathrm{m}M=2$, where $\widehat{\mathrm{\Sigma }}$ is a topological Markov shift over a locally-finite and countable directed graph. $\widehat{\pi }$ is H"older continuous, and is finite-to-one on $\widehat{\Sigma}^#:={\underline{u}\in\widehat{\Sigma}:\exists v,w\text{ s.t. }#{i\geq0:u_i=v}=\infty, #{i\leq0:u_i=w}=\infty}$;and$\widehat{\pi}[\widehat{\Sigma}^#]\supseteq {\text{Lyapunov regular and temperable }\chi$\text{-hyperbolic points}$}$. We later extended Sarig’s result for the case $\mathrm{d}\mathrm{i}\mathrm{m}M\ge 2$ in [BO18]. In this work, we offer an improved construction for [BO18] such that ($\mathrm{\forall }ϵ>0$ small enough) we could identify canonically the set $\widehat{\pi }[{\widehat{\mathrm{\Sigma }}}^{\mathrm{\#}}]$. We introduce the notions of $\chi$-summable, and $ϵ$-weakly temperable points. In [BCS], the authors show that for each homoclinic class of a periodic hyperbolic point $p$, there exists a maximal irreducible component $\tilde{\mathrm{\Sigma }}\subseteq \widehat{\mathrm{\Sigma }}$ s.t. all invariant ergodic probability $\chi$-hyperbolic measures which are carried by the homoclinic class of $p$ can be lifted to $\tilde{\mathrm{\Sigma }}$. We use their construction in the context of ergodic homoclinic classes, to show the stronger claim, $\widehat{\pi }[\tilde{\mathrm{\Sigma }}\cap {\widehat{\mathrm{\Sigma }}}^{\mathrm{\#}}]=H(p)$ modulo all conservative (possibly infinite) measures ($\mathrm{d}\mathrm{i}\mathrm{m}M\ge 2$); where $H(p)$ is the ergodic homoclinic class of $p$, as defined in [RHRHTU11], with the (canonically identified) recurrently-codable points replacing the Lyapunov regular points in the definition in [RHRHTU11].