bibkey: meddaugh2025neighborhoodshadowing
authors: Jonathan Meddaugh and Elyssa Stephens
year: 2025
title: Shadowing with small diameter sets of bounded cardinality
doi: null
url: https://arxiv.org/abs/2509.04312v1
claim: Question 5.3 asks for neighborhood two-shadowing without exact-length quasi-finite type; Question 5.6 asks for weak mixing and neighborhood N-shadowing without dense small periodic sets.
strata_touched:
  - D5/S1/Dynamics/Shadowing/PowersTwoReturnDistances
  - D5/S1/Dynamics/Shadowing/ProgressionExample
license: citation-only
triage: anchor

Neighborhood two-shadowing beyond quasi-finite type

Jonathan Meddaugh and Elyssa Stephens, Shadowing with small diameter sets of bounded cardinality, arXiv:2509.04312v1, Question 5.3:

Are there subshifts with neighborhood 2-shadowing which are not a subshift of quasi-finite type?

Definition 2.1 requires that for every $\epsilon >0$ there is $\delta >0$ such that every natural-time $\delta$-pseudo-orbit $(x_i)$ has a fixed nonempty finite tracing set $A$, with $|A|\le 2$, $\mathrm{diam}(A)<\epsilon$, and $\mathrm{dist}(x_i,{\sigma }^i(A))<\epsilon$ at every $i\ge 0$. The choice of a member of $A$ may vary with time; the set $A$ does not.

Definition 3.5 requires a threshold $M$ such that whenever $n\ge M$, $u,v$ are language words and some language word $w$ of length $n$ has $uw,wv$ in the language, there is a language word $z$ of length exactly $n$ with $uzv$ in the language. Ordinary $N$-shadowing without the initial diameter condition, density shadowing and entropy-based uses of “quasi-finite type” do not supply this statement.

The return-distance example is the binary two-sided subshift with consecutive-one distances $D=\{2^j:j\ge 0\}$. Its zero-run lengths are $2^j-1$, and distance one is allowed. Finite supports and the configuration with no ones remain in the space. The declaration D5/S1/Dynamics/Shadowing/PowersTwoReturnDistances.powers_two_return_distance_example establishes nonemptiness, compactness, continuity of the shift, neighborhood two-shadowing and failure of the exact-length condition. Its consumer exposition is in the corresponding Blueprint document; the original question is also retained in the problem entry.

The formal metric is Mathlib’s genuine compatible countable-product metric. It is not literally the source’s printed infimum over disagreeing windows: for distinct configurations, that printed expression has infimum zero. The finite-window arguments use the actual product uniformity, and the QFT obstruction uses nonempty words with $n=\max (M,1)$.

Weak mixing without nonempty small periodic sets

Question 5.6 in the same primary version asks:

Is there a weakly mixing system which has the neighborhood N-shadowing property (N>1) that does not have dense small periodic sets?

The source’s compact metric system uses a continuous map; it does not require invertibility. Its small-periodic-set definition omits explicit nonemptiness. The intended nonempty reading is necessary, since the empty set is compact, contained in every open set and periodic for every map.

D5/S1/Dynamics/Shadowing/ProgressionExample.quadratic_progression_example uses the one-sided binary carrier with a zero in every progression {s+jp : 0<=j<=p^2+2}, for every s>=0 and p>=1. It has the genuine first-disagreement metric, a continuous onto mixing shift, and original neighborhood two-shadowing. The nonempty clopen cylinder [1] contains no nonempty forward power-invariant subset, hence no nonempty compact periodic set. The exact construction and proof are exposed in its Blueprint source and problem entry.

The arithmetic-progression construction, two rotation masks and constants are attributed to the supplied completed Pro proposal. Native implementation and its verification are separate from its isolated ordinary mathematical review. The current arXiv metadata lookup returns v1; no exhaustive literature novelty or priority exclusion is claimed.

Literature boundary

The inspected primary version lists Question 5.3 as a question. A bounded independent search found no exact prior answer within its searched scope; this does not certify exhaustive priority or novelty. Carrasco-Olivera, Lee, Morales and Villavicencio, Finite-expansivity and N-shadowing (2022), DOI 10.1007/s00574-021-00253-w, has a 17-page author manuscript linked from Villavicencio’s research page. The inspected passages are Definition 1.5 and Theorem 3 (pp. 3–4) and Lemma 16 (pp. 11–12). Definition 1.5 uses a fixed nonempty tracing set of at most $N$ points and two-sided pseudo-orbit time, without a separate initial diameter bound. The set-distance definition on p. 3 is printed as a supremum; its literal definition is not identified with Definition 2.1 here. Theorem 3 states an $N$-shadowing hierarchy with interval and genus-two surface examples. Lemma 16 transfers finite-cardinality shadowing through the whole inverse image of a tracing set, without a separate initial diameter bound. These inspected statements do not give a direct answer to Question 5.3. This bounded comparison is not an exhaustive full-text exclusion or validation of the journal version, which remains unread. Dastjerdi and Hosseini, Sub-shadowings (2010), DOI 10.1016/j.na.2010.01.014, remains unread after full-text access failures (HTTP 403/401); no full-text exclusion is claimed for it. Dastjerdi and Jangjoo, arXiv:1108.3242, is withdrawn because of a flaw in Theorem 3.2 and is not used as proof.

Verified locator

Versioned primary: https://arxiv.org/abs/2509.04312v1. Full-text primary: https://arxiv.org/html/2509.04312v1. The relevant locators are Definition 2.1, Definition 3.5, Question 5.3, the small-periodic-set definition immediately before Proposition 4.3, and Question 5.6.