Canonically Codable Points and Irreducible Codings

Abstract

$M$ is a cpt. Riemannian manifold without boundary, $f\in\mathrm{Diff}^{1+β}(M)$. In [Sarig13], for all $χ>0$, for every small enough $ε>0$, Sarig had first constructed a coding $\widehatπ:\widehatΣ\rightarrow M$ which covers the set of all Lyapunov regular $χ$-hyperbolic points when $\mathrm{dim}M=2$, where $\widehatΣ$ is a topological Markov shift over a locally-finite and countable directed graph. $\widehatπ$ is Hölder continuous, and is finite-to-one on $\widehatΣ^\#:=\{\underline{u}\in\widehatΣ:\exists v,w\text{ s.t. }\#\{i\geq0:u_i=v\}=\infty, \#\{i\leq0:u_i=w\}=\infty\}$; and $\widehatπ[\widehatΣ^\#]\supseteq \{\text{Lyapunov regular and temperable }χ\text{-hyperbolic points}\}$. 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 ($\forallε>0$ small enough) we could identify canonically the set $\widehatπ[\widehatΣ^\#]$. We introduce the notions of $χ$-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 $\widetildeΣ\subseteq\widehatΣ$ s.t. all invariant ergodic probability $χ$-hyperbolic measures which are carried by the homoclinic class of $p$ can be lifted to $\widetildeΣ$. We use their construction in the context of ergodic homoclinic classes, to show the stronger claim, $\widehatπ[\widetildeΣ\cap\widehatΣ^\#]=H(p)$ 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].