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Shadowing in CR-Dynamical Systems

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Pith's one-line read This paper introduces four (i,j)-shadowing properties for CR-dynamical systems, characterises the finite-branch case, and connects the strictest variant to unique trajectories and to shadowing of the Mahavier shift.

desk verdict A careful, incremental taxonomy of four shadowing notions for CR-dynamical systems; the apparent gap in Proposition 6.5 closes once you recall constant trajectories are legal. read the letter →

arxiv 2505.23089 v1 pith:G7666KKA submitted 2025-05-29 math.DS math.GN

Andrew Wood This is my paper
classification math.DSmath.GN MSC 37B6537B0537B1054H2054E15
keywords CR-dynamicalsystemsshadowingpropertypseudo-orbitclosedrelationsset-valueddynamicalMahavierproductuniformspacestotaldisconnectedness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the classical shadowing property—approximate orbits being mirrored by real orbits—to CR-dynamical systems, where the dynamics is a closed relation G on a compact Hausdorff space rather than a continuous map. Because a relation can send a point to several or no successors, the paper separates two choices in the definition: how strictly a pseudo-orbit must match the relation (all successors versus at least one successor, i=1 versus i=2) and how strictly a shadowing point must track the pseudo-orbit (all of its trajectories versus at least one, j=1 versus j=2). This yields four (i,j)-shadowing properties, of which (2,2) recovers the established shadowing notion for set-valued dynamical systems, while (2,1) is a genuinely new, more restrictive notion. The paper proves that the four properties are distinct, gives complete characterisations when the set of non-degenerate points is finite, and relates the properties to total disconnectedness and to the shift map on the infinite Mahavier product. A sympathetic reader would take the contribution to be a coherent framework in which branching dynamics has several non-equivalent shadowing notions, each with its own rigidity strength.

What carries the argument

A (δ,i)-pseudo-orbit in (X,G) is a sequence in the non-degenerate points whose next term is within δ of all (i=1) or some (i=2) point in G(x_n), and a point y (ε,j)-shadows the sequence if all (j=1) or some (j=2) trajectory of y stays within ε of it. The carrying object is the relation G together with its sets of legal points and trajectories inside the infinite Mahavier product ⋆^∞_{i=0}G, the collection of infinite sequences ⟨x_0,x_1,…⟩ with (x_n,x_{n+1})∈G. The machinery includes the implication diagram (2,1)⇒(1,1),(2,2)⇒(1,2), the finite-nondegenerate characterisation via separation of finitely many relation values by entourages, and the passage to the Mahavier shift (X^+_G,σ^+_G) when G contains the graph of an isometry.

What would settle it

The paper's own two-point example is a decisive test: take X={0,1} with the discrete uniformity and G=X×X. The paper predicts (2,1)-shadowing fails because both points have two trajectories; indeed the constant pseudo-orbit 0,0,0,… is a (V,2)-pseudo-orbit for every entourage V, yet no point (U,1)-shadows it for the diagonal entourage U, since every point has a trajectory that visits 1.

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Extended reading notes

Core claim

The paper's central claim is that the (i,j)-shadowing properties form the correct generalisation of the shadowing property to closed relations, with (2,2) generalising set-valued shadowing and the remaining three variants offering genuinely new distinctions. Its main structural results are: with nondegenerate(G) finite, (X,G) always has (1,1)-, (2,2)- and (1,2)-shadowing, and it has (2,1)-shadowing exactly when every legal point has a single trajectory; if G contains the diagonal then (2,1)-shadowing forces X to be totally disconnected (indeed G=Δ_X), while total disconnectedness of X guarantees (1,2)-shadowing; and for relations containing the graph of an isometry, (2,1)-shadowing of (X,G) implies shadowing of the shift on the infinite Mahavier product, while shadowing of that shift implies (1,2)-shadowing of (X,G). The paper also shows through explicit examples that the four properties are mutually distinct, and that the metric-space and uniform-space formulations coincide.

Load-bearing premise

The framework assumes every positive iterate G^n is non-empty, which is equivalent to assuming the infinite Mahavier product is non-empty; if a relation only admits finite paths, the set of legal points can be empty and all shadowing properties become vacuous.

Editorial extensions

If this is right

  • The four (i,j)-shadowing properties are distinct: examples in the paper show (1,1), (1,2) and (2,2) can hold without (2,1), and (2,2) can fail while (1,1) holds.
  • When the branching set is finite, shadowing is almost automatic: only the (2,1) variant is restrictive, and its restriction is precisely uniqueness of trajectories for legal points.
  • If the relation contains the diagonal, the strictest shadowing variant (2,1) collapses the relation to the diagonal, so it cannot tolerate any actual branching.
  • For relations containing the graph of an isometry, (2,1)-shadowing of the relation implies classical shadowing of the associated Mahavier shift, while shadowing of that shift yields the weaker (1,2)-shadowing property.
  • In the Mahavier product, total disconnectedness of X is equivalent to total disconnectedness of X^+_G for relations containing an isometry's graph, but shadowing of the shift does not automatically pass back to (2,1)-shadowing of (X,G).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether (2,1)-shadowing, without isometry assumptions, forces the relation to be single-valued on a large set; a natural test is to seek branching relations with (2,1)-shadowing, which would show the strict variant does not merely encode function-like behaviour.
  • The finite-nondegenerate results suggest that finite-state CR-systems behave like shifts of finite type, so the open Mahavier-product questions could be probed by exhaustive symbolic computations on finite alphabets.
  • The uniform-space formulation means the framework transfers to non-metrizable compact Hausdorff spaces; a next step would be to check whether the (i,j) distinctions survive passage to inverse limits with a shift relation, extending the set-valued inverse-limit results cited in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces a family of four shadowing-type properties, called (i,j)-shadowing, for CR-dynamical systems (X,G), where X is a compact Hausdorff space with a uniformity and G is a closed relation. The parameter i controls whether the pseudo-orbit condition is required for all or for some choices in G(x_n), and j controls whether shadowing is required for all or for some trajectories of the shadowing point; the (2,2) case recovers the existing set-valued shadowing property. The paper proves basic permanence properties, including invariance under topological conjugacy and stability under taking powers for (2,j), gives a characterization when nondegenerate(G) is finite (Proposition 5.3), treats the case Δ_X ⊆ G (Theorems 6.2 and 6.10, Proposition 6.5), relates (i,j)-shadowing to shadowing in the infinite Mahavier product under an isometry hypothesis (Theorems 7.4 and 7.9), and studies the inverse relation G^{-1}. Several examples separate the four notions, and three open questions are formulated.

Significance. If the results hold, this is a natural and useful extension of the shadowing property from single-valued and set-valued dynamics to closed relations. The paper gives a systematic four-parameter framework, explicitly identifies (2,2)-shadowing with the existing set-valued shadowing notion, and obtains characterizations in the finite-nondegenerate and diagonal cases that generalize classical results. The concrete separating examples, the permanence lemmas, and the clearly stated open questions are assets. The main framework is sound, but the written proof of Proposition 6.5 contains an unjustified inference and the argument in Example 4.15 is incorrect as written; both need to be repaired because they are load-bearing for the inverse-relation results and for the claimed distinctness of the four properties, respectively.

major comments (2)
  1. [Section 6, Proposition 6.5] The proof is not valid as written. The assertion that (y,u), (u,x), (x,v), (v,z) ∈ U is not justified by (U,1)-shadowing: from (U,1)-shadowing of ⟨x,y,y,…⟩ by u one obtains (u,x) ∈ U and, for every trajectory of u, its first entry is U-close to y, but this does not give (y,u) ∈ U; similarly, (v,z) does not follow from shadowing of ⟨x,z,z,…⟩ by v. The statement itself is true and can be proved by a simpler constant-trajectory argument: for any (x,y) ∈ G, the sequence ⟨x,y,y,…⟩ is a (V,2)-pseudo-orbit for the V supplied by (2,1)-shadowing, and since Δ_X ⊆ G the constant sequence u,u,… is a trajectory of the shadowing point u, giving (u,x),(u,y) ∈ U and hence (x,y) ∈ U^2 for every sufficiently small symmetric U. This proof must be rewritten; Corollaries 6.6 and 8.5 depend on Proposition 6.5.
  2. [Section 4, Example 4.15] The proof that (X,G) does not have the (2,2)-shadowing property contains a false step. From (ϵ,2)-shadowing by y one only knows that some trajectory of y is close to the pseudo-orbit; one cannot conclude both d(0,y) < 1/2 and d(1,y) < 1/2, because a trajectory starting near 0 may pass through 0 and later be near 1. The example's conclusion is nevertheless correct, but a different argument is needed, for instance using the fact that a nonzero trajectory of G = (X×{0})∪Δ_X can only stay fixed or drop to 0, so it cannot track the monotone sequence from 0 to 1 across the value 1/2. Since this example supports the assertion that the four (i,j)-shadowing properties are distinct, the proof should be corrected.
minor comments (5)
  1. [Section 6, Theorem 6.10] In part (2), the entourage D should be chosen symmetric from the outset so that D^{-1}(F_{y_i}) ⊆ F_{y_i} follows from the fact that {D(x)} refines the pairwise disjoint cover F; as written, the proof uses this preimage containment implicitly.
  2. [Section 5, Example 5.4] In the case x0 = 1, the shadowing point is not 1 itself but a point 1 - 1/n near 1 whose unique trajectory follows the branch taken by the pseudo-orbit; the text should state this explicitly for clarity.
  3. [Section 6, Proposition 6.4(4)] The sentence invoking 0,1 ∈ G^2(y) is imprecise: only the trajectory that visits 1 is needed to obtain the contradiction, since the pseudo-orbit coordinate 2/3^{n+2} is far from 1.
  4. [Throughout] There are minor typographical issues, including 'Defintion' in Proposition 4.10, 'propety' in Example 5.5, and the notation T U for the intersection of all entourages, which should be defined or replaced by ∩_{U∈U} U.
  5. [Section 2.1] The equivalence between 'G^n is non-empty for each positive integer n' and '⋆∞_{i=0}G ≠ ∅' is stated without proof; it follows from a compactness/finite-intersection argument and should be indicated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the (i,j)-shadowing framework is built explicitly on prior definitions and external theorems, with no fitted parameters or conclusions assumed as inputs.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The (2,2)-shadowing property is explicitly identified (Observation 4.6 and the Note before it) with the existing set-valued shadowing notion, and this identification is definitional rather than a predicted consequence; the paper then builds the new (1,1), (1,2), and (2,1) properties on top of that definition and proves their relationships (Proposition 4.12, Examples 4.13-4.15). The main characterizations either follow from direct uniformity/epsilon-delta arguments (Propositions 5.1 and 5.3) or reduce to external theorems such as Theorem 3.8 (classical identity-map shadowing iff totally disconnected), Theorem 3.9, and Theorem 7.1 (isometry shadowing), none of which are authored by the present author or assumed to contain the target result. Lemmas 4.17, 4.19, and 4.21 are proven in the text; Theorems 6.2 and 7.2 apply them to known results. There are no fitted parameters, no quantity is defined in terms of the quantity it is used to predict, and no load-bearing premise rests on a self-citation. Some written proofs contain gaps (e.g., Proposition 6.5's inference of (y,u)∈U from (U,1)-shadowing does not follow as written, though the claim is salvageable by a constant-trajectory argument), but a proof gap is a correctness issue, not circularity. The paper also openly poses open questions (6.11, 7.6, 8.7, 9.1-9.3), confirming that its framework does not force the answers by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

Pure mathematics paper with no free parameters or invented physical entities. The axioms listed are the background mathematical facts and explicit structural assumptions the central results depend on.

assumptions (7)
  • standard math ZFC set theory and standard compactness arguments
    Used throughout, e.g., Theorem 2.3's compactness argument and Theorem 7.4.
  • standard math Every compact Hausdorff space has a unique compatible uniformity (the neighborhood system of the diagonal)
    Invoked in Section 2.2 to fix the uniformity used for shadowing definitions.
  • standard math Theorem 2.1: closed graph characterization of continuity on compact Hausdorff spaces
    Used to justify closed relations as generalizations and in Proposition 6.1.
  • domain assumption Theorem 3.8 and 3.9: identity map (or periodic map) has shadowing iff the space is totally disconnected
    External results cited from [23] and used as black boxes in Theorem 6.2, 6.10, and 7.1.
  • domain assumption Standing assumption: G^n is non-empty for every positive integer n
    Section 2.1; equivalent to infinite Mahavier product non-empty; used throughout to ensure legal points exist.
  • domain assumption Existence of an isometry f with Γ(f)⊆G in Section 7
    Section 7; restricts the Mahavier-product theorems to this setting.
  • domain assumption Uniform upper semi-continuity of F_k in Theorem 6.10(2)
    Definition 6.9 and Theorem 6.10; this technical condition is not always satisfied, leaving Question 6.11 open.

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Pith. "Pith review of Shadowing in CR-Dynamical Systems." pith.science (2026). https://pith.science/paper/G7666KKA

@misc{pith2026250523089,
  author       = {Pith},
  title        = {Pith review of: Shadowing in CR-Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7666KKA}},
  note         = {Machine review of arXiv:2505.23089}
}
abstract

A CR-dynamical system is a pair $(X, G)$, where $X$ is a non-empty compact Hausdorff space with uniformity $\mathscr{U}$ and $G$ is a closed relation on $X$. In this paper we introduce the $(i, j)$-shadowing properties in CR-dynamical systems, which generalises the shadowing property from topological dynamical systems $(X, f)$. This extends previous work on shadowing in set-valued dynamical systems.

Figures

Figures reproduced from arXiv: 2505.23089 by the authors.

Figure 1
Figure 1. The Comb Space from Example 4.14 Example 4.14. Let (X, G) be a CR-dynamical system, where X = [0, 1] and G is the Comb Space, G = ([0, 1] × {0} ∪ {0} × [0, 1]) ∪ [∞ n=1  1 n  × [0, 1], depicted in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The relation G from Example 4.15 Example 4.15. Let (X, G) be a CR-dynamical system, where X = [0, 1] and G = (X × {0}) ∪ ∆X, depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The relations G and G2 from Example 4.22 To see (X, G) has the (1, j)-shadowing property, we show (X, G) has exactly one (δ, 1)-pseudo-orbit for δ ∈ [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The relation G in Example 5.4 Example 5.4. We show that Proposition 5.3 does not hold in general when we remove the assumption nondegenerate (G) is finite. Take X = [0, 1] and ∆X. Then, each x ∈ X has exactly one trajectory, but it follows from Theorem 3.8 and Propo￾si…
Figure 5
Figure 5. Figure 5: The relation G in Example 5.5 Example 5.5. Consider the CR-dynamical system (X, G), where X = [0, 1] and G is as shown in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The relations G and G−1 in Example 8.6 Suppose ⟨xn | n ∈ ω⟩ is a (V, 2)-pseudo-orbit in X, G−1  . For each n ∈ ω, there exists yn+1 ∈ G−1 (xn) such that (yn+1, xn+1) ∈ V . Let y0 = x0. We claim ⟨yn | n ∈ ω⟩ (U, 2)-shadows ⟨xn | n ∈ ω⟩ in X, G−1  . We need only show ⟨…

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