bibkey: peng2026shadowing authors: Dekui Peng year: 2026 title: ‘Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits’ doi: null url: https://arxiv.org/abs/2606.14435v3 claim: Question 5.1 asks whether every compact Hausdorff shadowing system is a factor of a compact zero-dimensional shadowing system. strata_touched: [] license: citation-only triage: anchor

Compact Hausdorff shadowing and zero-dimensional extensions

The inspected edition is arXiv:2606.14435v3, revised 2 August 2026. Natural time includes zero. Shadowing uses finite open covers: each error pair shares one member of a control cover, and every pseudo-orbit is traced at every time with respect to the prescribed output cover.

Question 5.1, PDF and printed p.21, reads:

Let (X,f) be a compact Hausdorff dynamical system with shadowing. Does there exist a compact zero-dimensional dynamical system (Y,g) with shadowing and a factor map π:(Y,g)→(X,f)?

The factor map is continuous, onto, and satisfies πg=fπ. There is no metrizability, countability, invertibility, or surjectivity assumption on f, and no nonempty-space guard. For empty X, empty Y gives the vacuous case. The nonempty general case remains unresolved here.

Definition 1.10, p.6, uses exactly the Mittag-Leffler condition

$$\forall i\exists k\ge i\forall j\ge k,{\phi }_i^j(X_j)={\phi }_i^k(X_k).$$

Theorem 3.1 addresses compact metric systems by an inverse sequence of SFTs with surjective bonds. Theorem 4.4 represents a compact shadowing system by metrizable shadowing factors. Neither supplies the arbitrary compact Hausdorff zero-dimensional extension of Question 5.1.

Section 5.1 indexes observations by finite subsets of an arbitrary uniformity base. Question 5.2 asks for surjectivity of row-deletion maps in the resulting symbolic arrays; coherent simultaneous-face lifts are the obstacle. This is a sufficient construction route to Question 5.1, not a separate solved problem.

The formal inverse-limit shadowing bridge is D5/S1/Dynamics/Shadowing/MittagLefflerInverseLimit. It assumes a genuine coherent ML diagram and therefore does not settle the construction question.

Verified locator

Versioned primary source: https://arxiv.org/abs/2606.14435v3. PDF: https://arxiv.org/pdf/2606.14435v3.

Question 5.1 is on printed/PDF p.21. Section 5.1 and Question 5.2 give the simultaneous finite-face row-deletion construction on pp.21–22. Definition 1.10 on p.6 supplies the Mittag-Leffler condition; Theorems 3.1 and 4.4 provide the metric-case and metrizable-factor results only.