Shadowing property and transitivity of a set-valued map and its inverse limit

Lidong Wang; Yingcui Zhao

REVIEW 3 major objections 4 minor 20 references

Shadowing property and transitivity of a set-valued map and its inverse limit

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Shadowing of a surjective upper semi-continuous set-valued map on a compact metric space is equivalent to shadowing of its inverse set-valued map and of the shift maps on its two generalized inverse limits.

desk verdict The paper's central claim is not proven: Lemma 3.1 is false, and the Section 4 proof has a directional error; the interesting witness-reversal idea is salvageable only with a different argument. read the letter →

arxiv 2607.17325 v1 pith:BGFQLCZ7 submitted 2026-07-19 math.DS

Yingcui Zhao , Lidong Wang This is my paper · ORCID

classification math.DS MSC 37B6537B0554C60

keywords shadowingpropertyset-valuedmapsgeneralizedinverselimitsmaptransitivitymixingchainuppersemi-continuous

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 asks when the shadowing property—the ability to approximate every approximate orbit by a true orbit—survives passage from a set-valued map to its inverse set-valued map and to the shift map on the generalized inverse limit. The central result is that for a surjective upper semi-continuous set-valued map on a compact metric space, shadowing of the map, shadowing of its inverse set-valued map, and shadowing of the shift on both induced inverse limits are all equivalent. This removes continuity and openness assumptions that earlier work needed. The authors also show that transitivity, weak mixing, mixing, chain transitivity and chain mixing pass from the inverse-limit shift to the set-valued map, and that for set-valued maps with shadowing, several strong mixing notions collapse into one.

What carries the argument

The key objects are the inverse set-valued map F←(x)={y:x∈F(y)} and the generalized inverse limit lim←F={(x_0,x_1,...): x_i∈F(x_{i+1})} with shift σ. The one-sided inverse-limit metric ρ(x,y)=sup_i d(x_i,y_i)/(i+1) discounts tails by diam X=1, letting errors accumulate only finitely. The load-bearing step is the 'reverse the witnesses' construction: instead of reversing a pseudo-orbit (which would require openness or lower semi-continuity), the proof reverses the points y_i that witness approximate membership, for which x_i∈F(y_i) holds exactly, so only upper semi-continuity (closed graph) is needed.

What would settle it

Look for a compact metric space X and a surjective upper semi-continuous F such that F has shadowing but F← does not (or vice versa). The theorem asserts no such F exists, so finding any concrete example would refute Theorem 3.6. A natural test case is the set-valued map F(x)=[0,x] on [0,1]: the paper shows F fails shadowing via the slowly increasing pseudo-orbit x_i=iδ/2, and predicts F←(x)=[x,1] also fails; one can verify directly whether the symmetric slowly decreasing pseudo-orbit of F← is shadowable, which settles the prediction.

Watch

Extended reading notes

Core claim

The paper's core claim is Theorem 3.6: for a compact metric space X and an upper semi-continuous surjective set-valued map F:X→2^X, F has shadowing if and only if its inverse set-valued map F← has shadowing. The proof works by reversing the witnesses y_i of a pseudo-orbit of F← instead of the pseudo-orbit itself, using exact membership x_i∈F(y_i), then appending the initial point x_0 as the end of a finite forward pseudo-orbit and passing to a diagonal limit; only closedness of the graph of F is used. Corollary 3.7 then strings this together with the two inverse-limit equivalences, so all four systems—F, F←, (lim←F←, σ), and (lim←F, σ)—share shadowing simultaneously. In particular, F has sha

Load-bearing premise

The proof depends on F being upper semi-continuous (so its graph is closed, allowing diagonal limits to remain orbits) and surjective (so F← has nonempty values); if either fails, the orbit-reconstruction argument collapses.

Editorial extensions

If this is right

For any surjective upper semi-continuous set-valued map on a compact metric space, the shadowing property of the map and of its inverse set-valued map are equivalent, with no continuity or openness requirement.
The shift map on the generalized inverse limit lim←F has shadowing exactly when the original set-valued map F does, answering the motivating question directly.
The four systems F, F←, (lim←F←, σ), and (lim←F, σ) all have shadowing simultaneously, so shadowing can be studied in whichever of these representations is most convenient.
If the shift on the generalized inverse limit is transitive, weakly mixing, mixing, chain transitive, or chain mixing, then the set-valued map itself has the corresponding property.
For set-valued maps with shadowing, total transitivity, weak mixing, mixing, the specification property, and chain mixing are mutually equivalent.

Reading between the lines

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

Because the proof uses only closed-graph regularity, the same 'reverse the witnesses' technique may extend to related shadowing variants (e.g., asymptotic or ergodic shadowing) where pseudo-orbit reversal had previously been a bottleneck; this is not shown in the paper.
The equivalence in Corollary 3.7 suggests shadowing is a property of the whole set-valued relation rather than of a chosen direction; one could test whether other directional invariants, such as specification or topological entropy, also pass unchanged between F and F←.
The theorem converts a question about a potentially discontinuous multivalued map into a question about a single-valued shift on an inverse limit, which may make numerical or computational checks of shadowing more tractable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

Before you invest time in this paper: the central shadowing equivalence is not proven. Lemma 3.1, which Theorem 3.2 and Corollaries 3.4–3.7 all lean on, is false. The stress-test counterexample is correct. Let me spell it out. Take X = {0} ∪ {a_n: n∈N} ∪ {b_n: n∈N} ∪ {A,B} with a_n,b_n → 0 and d(A,B)=1. Define F(0)={0,A,B}, F(a_n)={b_n}, F(b_n)={B}, F(A)={A}, F(B)={B}. The graph is closed, so F is u.s.c.; adding a chain of preimages for the a_n makes it onto without breaking anything. For any δ>0, choose n with d(b_n,0)<δ; then (a_n,0,A) is a δ-chain. But every orbit of F starting at a_n goes a_n → b_n → B → B → …, so its second point is B, distance 1 from A. Hence no orbit ε-shadows this chain for ε<1/2, no matter how small δ is. Lemma 3.1 fails. The proof's induction erroneously uses u.s.c. as though it gave two-sided distance control; u.s.c. only gives F(u) ⊂ B(F(x),η) when u is close to x, not the reverse. Since Theorem 3.2 constructs its inverse-limit pseudo-orbit by applying Lemma 3.1 to finite segments, that construction is unavailable. The claimed F ⇔ σ-on-lim←F← equivalence is thus unsupported.

I want to give credit where it's due. The question is well motivated, and the idea of reversing the witnesses of a pseudo-orbit rather than the pseudo-orbit itself is genuinely different from Khan–Kumar–Das [20]. It might be the right route under stronger hypotheses. The three examples are helpful, and the paper's engagement with the set-valued inverse-limit literature looks honest.

The Section 4 issue the reader flagged is real but secondary. In Theorem 4.5, the u.s.c. step indeed goes in the wrong direction: from d(z^{i+1}_0, z^i_1)<δ1 you get F(z^{i+1}_0) ⊂ B(F(z^i_1),δ/2), which doesn't tell you F(z^{i+1}_0) is within δ/2 of z^i_0. You'd need the reverse inclusion — that's l.s.c. territory, not assumed. Probably repairable, but not as written.

There are also minor problems: Definition 2.4 has a dangling '[?]' citation, and the text has many typos.

Bottom line: this should not be accepted in its current form. The false lemma is load-bearing, not cosmetic. I would still send it to a serious referee, because the conjecture is plausible and the witness-reversal idea deserves a proper test — a referee might find the correct proof or a full counterexample. But I wouldn't cite it as a theorem in the meantime.

Referee Report

3 major / 4 minor

Summary. The paper studies a surjective upper semi-continuous set-valued map F on a compact metric space X and the shift map σ on the generalized inverse limits lim←F and lim←F←. It claims: F has shadowing iff σ on lim←F← has shadowing; F← has shadowing iff σ on lim←F has shadowing; and via Theorem 3.6, F and F← always have equivalent shadowing, so that F, F← and the two shift systems have shadowing simultaneously. Sections 4–5 prove transitivity/mixing/chain-transitivity/chain-mixing implications from the shift to F, and an equivalence of mixing and chain mixing under shadowing. The main shadowing equivalence relies on Lemma 3.1, a finite-chain tracking lemma for u.s.c. maps.

Significance. The advertised conclusion would be a real strengthening of Khan–Kumar–Das [20], removing continuity and openness. Theorem 3.6 is elegant and appears to be correct, and the three examples illustrate the scope clearly. But the central Theorem 3.2 rests on Lemma 3.1, which is false as stated. Since the inverse-limit-to-map direction and the resulting equivalences (Corollaries 3.4–3.7) depend on that lemma, the paper's principal contribution is not established. The Section 4 chain-transitivity/mixing proof has a separate, also load-bearing, u.s.c.-versus-l.s.c. error.

major comments (3)

[§3, Lemma 3.1] Lemma 3.1 is false. Take X={0}∪{a_n}∪{b_n}∪{A,B} with a_n,b_n→0 and d(A,B)=1. Define F(0)={0,A,B}, F(a_n)={b_n}, F(b_n)={B}, F(A)={A}, F(B)={B}. The graph is closed, so F is u.s.c. For ε=1/2 and k=2, any δ>0 choose n with d(b_n,0)<δ. Then (a_n,0,A) is a δ-chain: d(F(a_n),0)=d(b_n,0)<δ and d(F(0),A)=0. But every orbit from a_n is a_n→b_n→B→⋯, so z_2=B, d(z_2,A)=1>1/2; no δ works. The proof reverses the u.s.c. inclusion after (a): u.s.c. gives F(x_0^1)⊂B(F(x_1),δ_{k−1}), not d(F(x_0^1),x_1')<δ_{k−1} for a fixed x_1'∈F(x_1). This is a genuine failure of finite shadowing, not a typo.
[§3, Theorem 3.2] The proof of Theorem 3.2 is built on the false Lemma 3.1. The line "By Lemma 3.1, there exists 0<η<δ′..." is what supplies the finite orbit segments x^i_j with d(x^i_j,x^{i+j}_0)<δ′ for 0≤j≤M; those segments are then used to prove that (x^i) is a δ-pseudo-orbit of σ. Without Lemma 3.1 the constructed (x^i) is not shown to satisfy ρ(σ(x^i),x^{i+1})<δ, and the conclusion that σ having shadowing implies F having shadowing does not follow. Corollaries 3.4, 3.5 and 3.7, and the abstract's four-way equivalence, depend on this direction.
[§4, Theorem 4.5] In the chain-transitivity and chain-mixing proofs, the implication from u.s.c. is used in the wrong direction. From ρ(z^{i+1},σ(z^i))<δ_1 one gets d(z^{i+1}_0,z^i_1)<δ_1. The stated u.s.c. condition yields only F(z^{i+1}_0)⊂B(F(z^i_1),δ/2); this says each u'∈F(z^{i+1}_0) is within δ/2 of some element of F(z^i_1), not necessarily of the specific z^i_0∈F(z^i_1). If F(z^i_1) has several separated components, d(F(z^{i+1}_0),z^i_0) can remain large. The needed inclusion F(z^i_1)⊂B(F(z^{i+1}_0),δ/2) is a lower semi-continuity property and is not available. Hence the construction of the δ-chain (u_i) of F is invalid in both parts of Theorem 4.5.

minor comments (4)

[Throughout] There are numerous typos and formatting artifacts: "valud" in the abstract, the overline/arrow notation for inverse limits is inconsistently typeset, and the reference list has incomplete entries (e.g., [20] is a preprint without a journal source).
[§2, Definition 2.4] The symbols N and Z+ are used interchangeably in places; for example, N is used both as the nonnegative integers and as a threshold in Definition 2.4(3) and Lemma 4.1. Please standardize.
[§3, Proposition 4.3] In the chain-transitivity part of Proposition 4.3, the sentence "there is a δ′-chain (x_i)_{i=0}^n of F from y to x with x_n=y and x_0=x" has the endpoints reversed; the proof then uses x_0=x and x_n=y. This should be corrected for readability.
[§4, Corollary 4.2] Corollary 4.2 cites "Theorem 3.13 from [12]" without stating the result or verifying that its hypotheses hold in the present set-valued setting. Please include the statement or a proof so the chain of implications is self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shadowing equivalence is proved from definitions; comparison with [20] is contextual, not load-bearing.

full rationale

The derivation chain is self-contained. The main shadowing equivalences (Theorem 3.2, 3.3, 3.6, Corollaries 3.4, 3.5, 3.7) are proved directly from the stated hypotheses: compactness, upper semi-continuity, surjectivity, and the definition of shadowing. In particular, Theorem 3.6 constructs finite reversed-witness pseudo-orbits and obtains an orbit of F← by a diagonal limit using only the closed graph of F; the claimed conclusion is not assumed among the hypotheses. The comparison with [20] is used as context and contrast, not as an input to the proof; the self-citation [19] is merely a pointer to the authors' earlier related work and is not load-bearing. There are no fitted parameters, no quantities defined in terms of the target result, no renamed predictions, and no uniqueness theorem imported from the authors' own prior work. Any objection that Lemma 3.1 or a theorem is false would be a correctness concern, not a circularity concern. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. The only inputs are standard dynamical definitions and the standing compactness/regularity hypotheses on F; the proofs are deductive.

assumptions (4)

domain assumption X is a compact metric space and F : X → 2^X is upper semi-continuous and onto (F(X) = X).
Used throughout; on compact spaces u.s.c. gives closed graph, and surjectivity ensures F← maps into 2^X and inverse limits are nonempty (Remark 3.10).
standard math The metric ρ on X^N defined by ρ(x,y) = sup_i d(x_i,y_i)/(i+1), with diam X = 1.
The discounting metric controls tails in Theorems 3.2–3.3; rescaling the metric to diameter 1 is harmless (Remark 3.10).
standard math Standard definitions of shadowing, transitivity, mixing, chain transitivity, chain mixing, and generalized inverse limits.
Definitions 1.1 and 2.1–2.4; no extra mathematical content is assumed beyond the standard framework.
standard math Upper semi-continuity of a set-valued map on a compact space implies closed graph.
Invoked in the diagonal argument at the end of Theorem 3.6 and in Theorem 4.5.

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Cite this review

Pith. "Pith review of Shadowing property and transitivity of a set-valued map and its inverse limit." pith.science (2026). https://pith.science/paper/BGFQLCZ7

@misc{pith2026260717325,
  author       = {Pith},
  title        = {Pith review of: Shadowing property and transitivity of a set-valued map and its inverse limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BGFQLCZ7}},
  note         = {Machine review of arXiv:2607.17325}
}

abstract

We study the properties of shadowing, transitivity, weakly mixing, mixing, chain transitivity and chain mixing of a set-valued map and its generalized inverse limit. Concerning shadowing, we prove that for a surjective upper semi-continuous set-valued map $F$ on a compact metric space, $F$ has shadowing if and only if the shift map on the generalized inverse limit $\underleftarrow{\lim}\,\underleftarrow{F}$ of its inverse set-valued map has shadowing; dually, $\underleftarrow{F}$ has shadowing if and only if the shift map on $\underleftarrow{\lim}F$ has shadowing. We further show that the shadowing of $F$ and that of $\underleftarrow{F}$ are always equivalent; consequently $F$, $\underleftarrow{F}$ and the shift maps on $\underleftarrow{\lim}\,\underleftarrow{F}$ and on $\underleftarrow{\lim}F$ all have shadowing simultaneously. In particular, $F$ has shadowing if and only if the shift map on its directly induced generalized inverse limit $\underleftarrow{\lim}F$ has shadowing. This strengthens a recent theorem established under continuity and openness assumptions. We show that if the shift map on the generalized inverse limit is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing), then the set-valued map is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing). For a set-valued map with shadowing, the properties of total transitivity, weak mixing, mixing, specification and chain mixing are mutually equivalent.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 1 linked inside Pith

[20]

A. Khan, R. Kumar and T. Das, Shadowing property for set-valued map and its inverse limit, preprint, vol. arXiv:2412.11221, 2024. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, China Email address:zycchaos@126.com School of Data Science, Zhuhai College of Science and Technology, zhuhai, China Email address:wld0707@...

arXiv 2024
[1]

Akin,Topological Dynamics, American Mathematical Society, 1997

E. Akin,Topological Dynamics, American Mathematical Society, 1997

1997
[2]

Aubin and H

J. Aubin and H. Frankowska,Set-valued analysis, Birkhauser, Boston, 1990

1990
[3]

Baniˇ c and S

I. Baniˇ c and S. Greenwood, A characterisation of the dimension of inverse limits of set-valued functions on intervals,Fundamenta Mathematicae, vol. 249, no. 1, pp. 1–19, 2020

2020
[4]

Charatonik and S

W. Charatonik and S. Sahan, Menger curve as inverse limit,Topology and its Applications, vol. 304, p. 107788, 2021

2021
[5]

Davies, S

G. Davies, S. Greenwood, S. Lockyer and Y. Maehara, Inverse limits of up- per semicontinuous functions and indecomposable continua,Topology and its Applications, vol. 288, p. 107471, 2021. SHADOWING PROPERTY AND TRANSITIVITY OF A SET-V ALUED MAP AND ITS INVERSE LIMIT 19

2021
[6]

Erceg and J

G. Erceg and J. Kennedy, Topological entropy on closed sets in [0,1]2,Topology and its Applications, vol. 246, pp. 106–136, 2018

2018
[7]

Fakhari and F

A. Fakhari and F. Ghane, On shadowing: Ordinary and ergodic,Journal Of Mathematical Analysis And Applications, vol. 364, pp. 151–155, 2010

2010

Show all 20 references

[8]

Furstenberg,Recurrence in ergodic theroty and combinatorial number the- ory, Princeton University Press, Princeton, 1981

H. Furstenberg,Recurrence in ergodic theroty and combinatorial number the- ory, Princeton University Press, Princeton, 1981

1981
[9]

He and C

L. He and C. Liu, Invariant measures and uniform positive entropy property for inverse limits,Applied Mathematics-A Journal of Chinese Universities Series B, vol. 14, pp. 265–272, 1999

1999
[10]

Kwietniak and P

D. Kwietniak and P. Oprocha, A note on the average shadowing property for expansive maps,Topology and its Applications, vol. 159, pp. 19–27, 2012

2012
[11]

Liu and S

L. Liu and S. Zhao, Martelli’s chaos in inverse limit dynamical systems and hypersapce dynamical systems,Results In Mathematics, vol. 63, pp. 195–207, 2013

2013
[12]

X. Luo, X. Nie and J. Yin, On the shadowing property and shadowable point of set-valued dynamical systems,Acta Mathematica Sinica, English Series, vol. 36, pp. 1384–1394, 2020

2020
[13]

Mahavier, Inverse limits with subsets of [0,1]×[0,1],Topology and its Applications, vol

W. Mahavier, Inverse limits with subsets of [0,1]×[0,1],Topology and its Applications, vol. 141, pp. 225–231, 2004

2004
[14]

Nilson and J

C. Nilson and J. Bernardes, Inverse limits with the shadowing property,Topol- ogy and its Applications, vol. 325, p. 108384, 2023

2023
[15]

Raines and T

B. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit,Houston Journal of Mathematics, vol. 44, pp. 665–677, 2018

2018
[16]

Williams, One-dimensional nonwandering sets,Topology, vol

R. Williams, One-dimensional nonwandering sets,Topology, vol. 6, pp. 473– 487, 1967

1967
[17]

X. Wu, X. Wang and G. Chen,F-mixing property and (F 1,F 2)-everywhere chaos of inverse limit dynamical systems,Nonlinear Analysis, vol. 104, pp. 147–155, 2014

2014
[18]

Kong and Z

K. Kong and Z. Salleh, Topologically transitive and mixing properties of set-valued dynamical systems,Abstract and Applied Analysis, vol. 2021, p. 5541105, 2021

2021
[19]

Y. Zhao, L. Wang and N. Wang, Devaney chaos of a set-valued map and its inverse limit,Chaos, Solitons and Fractals, vol. 172, p. 113454, 2023

2023

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [20]

A. Khan, R. Kumar and T. Das, Shadowing property for set-valued map and its inverse limit, preprint, vol. arXiv:2412.11221, 2024. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, China Email address:zycchaos@126.com School of Data Science, Zhuhai College of Science and Technology, zhuhai, China Email address:wld0707@...

arXiv 2024

[2] [1]

Akin,Topological Dynamics, American Mathematical Society, 1997

E. Akin,Topological Dynamics, American Mathematical Society, 1997

1997

[3] [2]

Aubin and H

J. Aubin and H. Frankowska,Set-valued analysis, Birkhauser, Boston, 1990

1990

[4] [3]

Baniˇ c and S

I. Baniˇ c and S. Greenwood, A characterisation of the dimension of inverse limits of set-valued functions on intervals,Fundamenta Mathematicae, vol. 249, no. 1, pp. 1–19, 2020

2020

[5] [4]

Charatonik and S

W. Charatonik and S. Sahan, Menger curve as inverse limit,Topology and its Applications, vol. 304, p. 107788, 2021

2021

[6] [5]

Davies, S

G. Davies, S. Greenwood, S. Lockyer and Y. Maehara, Inverse limits of up- per semicontinuous functions and indecomposable continua,Topology and its Applications, vol. 288, p. 107471, 2021. SHADOWING PROPERTY AND TRANSITIVITY OF A SET-V ALUED MAP AND ITS INVERSE LIMIT 19

2021

[7] [6]

Erceg and J

G. Erceg and J. Kennedy, Topological entropy on closed sets in [0,1]2,Topology and its Applications, vol. 246, pp. 106–136, 2018

2018

[8] [7]

Fakhari and F

A. Fakhari and F. Ghane, On shadowing: Ordinary and ergodic,Journal Of Mathematical Analysis And Applications, vol. 364, pp. 151–155, 2010

2010