slug: peng-2026-zero-dimensional-shadowing-extension
bibkey: peng2026shadowing
doi: null
url: https://arxiv.org/abs/2606.14435v3
triage: wall
motivation_gids:
  - D5/S1/Dynamics/Shadowing/MittagLefflerInverseLimit.shadowing_inverse_limit_of_mittag_leffler

Zero-dimensional shadowing extensions

Problem

Peng’s Question 5.1 asks whether every compact Hausdorff shadowing system (X,f) is a continuous onto equivariant factor of a compact zero-dimensional shadowing system (Y,g). Time is natural, including zero. Neither f nor g is assumed invertible or onto; no metrizability or countable-weight condition is imposed. Empty X has the solution Y empty. The general nonempty question is unresolved here.

Motivation

The reusable background bridge is preservation of finite-cover shadowing by an arbitrary-directed compact Hausdorff Mittag-Leffler inverse limit. It does not choose the symbolic stages or provide their bonding maps.

Gap

A sufficient symbolic construction still requires all of the following:

1. Finite graphs at every finite observation scope, with edge-preserving deletion maps satisfying identities and every composition exactly.
2. Complete-path lifting for every target infinite path and every finite prefix, or genuinely proved ML stabilization of complete-path images. Vertex or edge surjectivity alone is insufficient.
3. Nonempty closed decoding cells, nested under every deletion, refining every finite open cover of X at some stage.
4. Actual graph-of-f witnesses for every stage edge, or a proved buffered closed-graph replacement implying sound limit decoding.
5. Complete coding of each actual orbit of each x in X at every stage.

Separate face lifts need not share one tracing orbit, and independently constructed refinements need not commute around deletion squares. A one-face refinement therefore does not settle the all-face construction. A join-tree gluing theorem also requires its actual running-intersection hypotheses and cannot be applied to cyclic proper-face scopes.

Source feasibility criterion

For a once-fixed finite-face stage, suppose the vertical source relations R_G are closed and symmetric. For a column v define its closed source cell C(v) as the intersection of all constraints (v^G,x) in R_G. For a live finite word w=v_0…v_m, meaning a prefix of an entire old stage path, define

$$K(w)=\bigcap _{i=0}^m{f}^{-i}C(v_i).$$

Every old whole array has one common constrained source orbit if and only if K(w) is nonempty for every live finite word. The reverse implication uses nested compact source sets for the same prescribed whole array. The maximal source-array incidence is then compact and forward invariant; its projection onto the entire stage is exactly this common-orbit condition. These are ordinary mathematical reductions, not additional Lean theorems.

Arbitrarily fine proper-superface lifts of the same frozen old array force this condition: shadow each lifted top row and take one compact subnet of the source tracing points. Closedness preserves every old constraint in the limit. For infinite coordinate indexing, monotone cofinal controls provide the required proper-superface tail. Open relations give only their closures under this argument, so closing a relation changes the target and requires a separate construction choice.

The remaining existence obligation is to choose the controls and finite alphabets globally, before any context or word is supplied, so that every stage’s live finite word has nonempty K(w). Source compactness does not supply these finite feasibility facts. A failed word blocks every sufficiently fine whole lift of arrays having that prefix, but does not refute the existential question for other global choices.

A proposed finite visible/hidden graph can be checked against its entire old path target using reachable hidden subsets. With N live visible and M live hidden vertices, failure has a live prefix of length at most N 2^M+1. This checks a fixed candidate; it does not prove a successful candidate exists. Closed multivalued source itinerary images need not be sofic even when the source has shadowing, so exact source-image pruning cannot be assumed to terminate in a finite graph.

Route

If this coherent family is constructed, compactness supplies its inverse limit and the same-orbit coding fibers. Nested arbitrarily small cells give unique decoding, edge soundness gives equivariance, the closed graph gives continuity, and same-orbit fibers give surjectivity. The inverse-limit shadowing theorem then supplies shadowing of Y. None of the missing construction clauses is supplied by treating it as a hypothesis and calling the original question solved.

Falsifier

A coherent all-face construction would answer the question positively. A scheme-specific lifting failure does not answer it negatively; a negative answer requires a compact Hausdorff shadowing system admitting no such extension by any construction.

Evidence

The inspected primary source is Peng arXiv:2606.14435v3, Question 5.1 on printed p.21. The independent source capsule also verifies the exact Mittag-Leffler quantifiers of Definition 1.10. The self-contained inverse-limit proof is in the cited Lean module; it does not assume that eventual-image restrictions inherit shadowing.

Triage

This is an unresolved construction question outside the metric case. The Mittag-Leffler shadowing bridge is known-background infrastructure, not an original question settlement.

Projective-cover obstruction

The proposed construction “take a projective compact Hausdorff cover and lift f” cannot supply a shadowing extension in general. This obstruction applies to every projective cover of the following original shadowing source, not only to one choice of lift. It does not refute Question 5.1.

Let X be the one-point compactification of discrete natural numbers, with f(n)=n+1 and f(infinity)=infinity. Every finite open cover is refined by the finite clopen partition

$$\{\{0\},\dots ,\{K-1\},\{K,K+1,\dots ,\infty \}\}.$$

Its quotient map commutes with the saturated successor map on {0,…,K}. Use this same partition as the control cover. The partition labels of any whole pseudo-orbit are an exact saturated-successor orbit. A sequence beginning below K is traced by its finite initial point, and one beginning in the tail is traced by infinity. Thus (X,f) has finite-cover shadowing. Time zero is included, and f is not onto. X is already zero-dimensional, so its identity map is a solution of Q5.1 for this source.

Suppose a compact extremally disconnected (Y,g) with continuous g and shadowing admitted a continuous onto equivariant pi:Y->X. Set U_n=pi^{-1}({n}). These are nonempty pairwise disjoint clopen sets and g(U_n) is contained in U_{n+1}. Extremal disconnectedness makes

$$W=\overline{\bigcup _{k\ge 0}{U}_{2^k}}$$

clopen. On U_n its indicator a is 1 exactly when n is a power of two: for unselected n, the open U_n is disjoint from the selected union and therefore from its closure. Let O be the union of all U_n. W is contained in closure(O).

Consider any finite a-itinerary beginning with 1. Its finite-cylinder preimage is a nonempty open set meeting closure(O), hence it meets some U_n. Forward containment then realizes that same finite itinerary on n,n+1,… in the discrete tube indices. In particular, if its first and next 1 are at times 0 and d>0, then n and n+d are consecutive powers of two. Thus d is itself a power of two. Conversely, every such distance 2^k occurs by starting on U_{2^k}.

The existing finite-observation theorem says that the entire a-itinerary image of a shadowing system is the complete-path label language of one finite graph. Suppose it has N vertices. Choose D=2^{N+2} and start an actual orbit on U_D. Its initial word has 1 at times 0 and D, with 0 at all times strictly between. Among graph vertices at times 1,…,N+1, two agree; write their times as i<j. Erase the intervening zero cycle and keep the original entire suffix. This remains an entire graph path and has consecutive 1s at distance

$$d=D-(j-i),2^{N+1}<d<2^{N+2}.$$

It must be an actual a-itinerary, while no power of two lies strictly between these consecutive powers. This is a contradiction. Therefore no extremally disconnected compact Hausdorff shadowing system admits such a factor onto (X,f). Gleason’s characterization identifies compact Hausdorff projective spaces with extremally disconnected spaces, so the same exclusion applies to all projective covers and all continuous lifts of these dynamics.

The irreducible Gleason cover here is beta(N): the continuous extension of N->X collapses the remainder to infinity. It is irreducible because the singleton fibers over the dense isolated natural points force every closed onto subset to contain their closure. Every continuous equivariant lift agrees with successor on these singleton fibers, hence is the unique continuous beta(N) successor extension by density. This particular lift fails shadowing by the same argument.

Every continuous binary observation on X is eventually constant on the natural points. Its clopen algebra is the finite/cofinite algebra. Extremal disconnectedness completes the cover’s clopen algebra far beyond these source observations: the power-of-two tube coloring cannot descend continuously to X. It is these extra cover observations that fail the necessary finite-graph condition. Sofic observation languages alone are only a necessary condition for shadowing; the obstruction does not replace the remaining tracing and coherent-graph construction obligations by that condition.

The general open-forward-tube obstruction is implemented in D5/S1/Dynamics/Shadowing/StoneanForwardTubeObstruction. The specialization to X, the shadowing proof for X, and the description of its irreducible cover are mathematical exposition, not separate new Lean declarations or an original open-problem settlement. The graph construction, projectivity characterization and closure machinery are reused; no worldwide novelty claim is made for their combination.

The remaining construction must use zero-dimensional covers with controlled clopen observations, rather than projectivity alone. It still must provide one globally chosen coherent finite-face family in which every live finite word is source-feasible, with complete-path deletion lifting or proved Mittag-Leffler stabilization. The projective obstruction neither supplies this finite feasibility nor disproves existence of such a family for arbitrary compact Hausdorff shadowing sources.

ASSUMED-UNVERIFIED

No worldwide open-status conclusion is inferred from a bounded search. The final Good–Meddaugh journal Theorem 8, pp.722-723, supplies the attributed Mittag-Leffler preservation result, including nonsurjective bonds. Its Lean formalization and subsequent freezing are separate from the still missing coherent symbolic construction in this problem dossier.