Canonically Codable Points and Irreducible Codings

Abstract

$M$ is a cpt. Riemannian manifold without boundary, $f\in\mathrm{Diff}^{1+\beta}(M)$. In [Sarig13], for all $\chi>0$, for every small enough $\epsilon>0$, Sarig had first constructed a coding $\widehat{\pi}:\widehat{\Sigma}\rightarrow M$ which covers the set of all Lyapunov regular $\chi$-hyperbolic points when $\mathrm{dim}M=2$, where $\widehat{\Sigma}$ 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 $\mathrm{dim}M\geq2$ in [BO18]. In this work, we offer an improved construction for [BO18] such that ($\forall\epsilon>0$ small enough) we could identify canonically the set $\widehat{\pi}[\widehat{\Sigma}^\#]$. We introduce the notions of $\chi$-summable, and $\epsilon$-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 $\widetilde{\Sigma}\subseteq\widehat{\Sigma}$ s.t. all invariant ergodic probability $\chi$-hyperbolic measures which are carried by the homoclinic class of $p$ can be lifted to $\widetilde{\Sigma}$. We use their construction in the context of ergodic homoclinic classes, to show the stronger claim, $\widehat{\pi}[\widetilde{\Sigma}\cap\widehat{\Sigma}^\#]=H(p)$ modulo all conservative (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 Lyapunov regular points in the definition in [RHRHTU11].