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$M$ is a cpt. Riemannian manifold without boundary,$f\in\mathrm{Diff}^{1+\beta}(M)$. In [Sarig13], for all $\chi>0$, for everysmall enough $\epsilon>0$, Sarig had first constructed a coding$\widehat{\pi}:\widehat{\Sigma}\rightarrow M$ which covers the set of allLyapunov regular $\chi$-hyperbolic points when $\mathrm{dim}M=2$, where$\widehat{\Sigma}$ is a topological Markov shift over a locally-finite andcountable directed graph. $\widehat{\pi}$ is H\"older continuous, and isfinite-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$\widehat{\pi}[\widehat{\Sigma}^\#]\supseteq \{\text{Lyapunov regular andtemperable }\chi\text{-hyperbolic points}\}$. We later extended Sarig's resultfor the case $\mathrm{dim}M\geq2$ in [BO18]. In this work, we offer an improvedconstruction for [BO18] such that ($\forall\epsilon>0$ small enough) we couldidentify canonically the set $\widehat{\pi}[\widehat{\Sigma}^\#]$. We introducethe notions of $\chi$-summable, and $\epsilon$-weakly temperable points. In[BCS], the authors show that for each homoclinic class of a periodic hyperbolicpoint $p$, there exists a maximal irreducible component$\widetilde{\Sigma}\subseteq\widehat{\Sigma}$ s.t. all invariant ergodicprobability $\chi$-hyperbolic measures which are carried by the homoclinicclass of $p$ can be lifted to $\widetilde{\Sigma}$. We use their constructionin the context of ergodic homoclinic classes, to show the stronger claim,$\widehat{\pi}[\widetilde{\Sigma}\cap\widehat{\Sigma}^\#]=H(p)$ modulo allconservative (possibly infinite) measures ($\mathrm{dim}M\geq2$); where $H(p)$is the ergodic homoclinic class of $p$, as defined in [RHRHTU11], with the(canonically identified) recurrently-codable points replacing the Lyapunovregular points in the definition in [RHRHTU11].

Canonically Codable Points and Irreducible Codings