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 $\hat{\pi }:\hat{\Sigma }\to M$ which covers the set of all Lyapunov regular $\chi$-hyperbolic points when $\dim M=2$, where $\hat{\Sigma }$ is a topological Markov shift over a locally-finite and countable directed graph. $\hat{\pi }$ is Hölder continuous, and is finite-to-one on ${\hat{\Sigma }}^{\#}:=\{\underline{u}\in \hat{\Sigma }:\exists v,w\text{ s.t. }\#\{i\ge 0:u_i=v\}=\infty ,\#\{i\le 0:u_i=w\}=\infty \}$; and $\hat{\pi }[{\hat{\Sigma }}^{\#}]\supseteq \{\text{Lyapunov regular and temperable }\chi \text{-hyperbolic points}\}$. We later extended Sarig's result for the case $\dim M\ge 2$ 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 $\hat{\pi }[{\hat{\Sigma }}^{\#}]$. 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 $\tilde{\Sigma }\subseteq \hat{\Sigma }$ s.t. all invariant ergodic probability $\chi$-hyperbolic measures which are carried by the homoclinic class of $p$ can be lifted to $\tilde{\Sigma }$. We use their construction in the context of ergodic homoclinic classes, to show the stronger claim, $\hat{\pi }[\tilde{\Sigma }\cap {\hat{\Sigma }}^{\#}]=H(p)$ modulo all conservative (possibly infinite) measures ($\dim M\ge 2$); 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].

A few typos corrected, clarified abstract