Canonically Codable Points and Irreducible Codings

• Published in: arXiv.org 2019-04
• Main Author: Snir Ben Ovadia
• Format: Article
• Language: English
• Subjects: Ergodic processes; Graph theory; Markov processes; Riemann manifold
• ISSN: 2331-8422

\(M\) 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].