Pith. sign in

REVIEW 5 major objections 6 minor 24 references

A note on multi-transitivity in non-autonomous discrete systems

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In non-autonomous discrete systems, mild mixing automatically forces full mixing, reversing the order these properties have for single maps.

desk verdict New mild-mixing concept and correct counterexamples, but the headline open-problem results rest on an unproven theorem the author himself undermines, and one counterexample is inconsistent. read the letter →

arxiv 2505.24657 v1 pith:B6XQGELT submitted 2025-05-30 math.DS

Hongbo Zeng This is my paper
classification math.DS MSC 37B55
keywords non-autonomousdiscretesystemsmildmixingmulti-transitivityLi-Yorkechaostopologicaltransitivitycollectiveconvergencecounterexamples
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 studies stronger forms of transitivity for non-autonomous discrete systems: sequences $(f_1,f_2,\dots)$ of continuous self-maps of a compact metric space, where the orbit of a point is obtained by applying the maps in order. Its central claim is that mild mixing—the requirement that the product of the system with every transitive non-autonomous system is again transitive—actually implies ordinary mixing for such time-dependent systems, the opposite of what happens for autonomous systems where mixing is strictly stronger than mild mixing. From this it derives that mild mixing implies multi-transitivity—transitivity of every finite product of the system with its iterates—answering an open problem of [15], and, under minimality and uniform-convergence hypotheses, that multi-transitivity implies Li-Yorke chaos, answering a second open problem. The paper also gives counterexamples showing that the multi-transitivity theorem of [15] and the minimality/transitivity theorems of [18] are false, and reproves the latter two under added 'collective convergence' and no-isolated-point assumptions.

What carries the argument

The load-bearing construction is a transitive non-autonomous system on the two-sided shift $\Sigma_2$ built from powers of the shift map $\sigma$: at times $n_k$ the map is $\sigma^{n_k}$, at times $n_k+1$ it is $\sigma^{-n_k}$, and elsewhere it is the identity. Paired with a non-mixing system whose return-time set for a pair of open sets has gaps $\{n_k\}$, this product has no time at which both coordinates return, so the product fails to be transitive, contradicting mild mixing. A second mechanism is collective convergence—the requirement that the shifted sequence of compositions $\{f_n^{n+k}\}_k$ approximates $\{f^k\}_k$ uniformly in the starting index—which lets the paper transfer weak mixing, total transitivity, syndetical transitivity and sensitivity between a non-autonomous system and its limiting map, and is the added hypothesis that repairs the false theorems of [15] and [18].

What would settle it

Build a compact metric space and a non-autonomous system $(X,f_{1,\infty})$ that is mildly mixing—its product with every transitive non-autonomous system is transitive—but has two nonempty open sets $U,V$ such that $f_1^n(U)\cap V$ is empty for infinitely many $n$; Theorem 5 says no such system exists, so one example would falsify the central claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the hierarchy of transitivity notions flips when the dynamics is allowed to vary in time. Theorem 5 states that for any non-autonomous system $(X,f_{1,\infty})$, mild mixing implies mixing; if $X$ has no isolated points the converse also holds, and in general mixing does not imply mild mixing. Consequently mild mixing is the strictly stronger property in this setting. Corollary 1 then gives mild mixing $\Rightarrow$ multi-transitivity, answering open problem 2 of [15], and Theorem 4 shows that under minimality, feeble openness, surjectivity and collective convergence, multi-transitivity $\Rightarrow$ Li-Yorke chaos, answering open problem 1. The paper further proves that the earlier theorem of [15] asserting equivalence of multi-transitivity of $(X,f_{1,\infty})$ with that of every tail $(X,f_{k,\infty})$ is false, via two shift-space counterexamples, and that Theorems 2.3 and 2.4 of [18] on transfer of minimality and transitivity from a uniformly convergent sequence to its limit map fail without extra hypotheses; corrected versions are proved under the no-isolated-point condition plus collective convergence.

Load-bearing premise

The load-bearing borrowed premise is that, in a minimal non-autonomous system, multi-transitivity already implies weak mixing of all orders; Theorems 4 and 13 use this premise without proof, and if it fails those two theorems collapse.

Editorial extensions

If this is right

  • Every mildly mixing non-autonomous discrete system is mixing, so a time-dependent system that mixes with every transitive partner cannot have any pair of nonempty open sets whose return times have arbitrarily large gaps.
  • Mildly mixing non-autonomous systems are multi-transitive, resolving open problem 2 of [15]; on spaces without isolated points, mild mixing and mixing coincide.
  • Under the paper's hypotheses (minimality, feeble openness, surjectivity, uniform and collective convergence), multi-transitive non-autonomous systems are Li-Yorke chaotic, resolving open problem 1 of [15].
  • The equivalence of multi-transitivity of an NDS with multi-transitivity of its tails $(X,f_{k,\infty})$ claimed in [15, Theorem 3.1] is false; the shift examples show the implication can fail in each direction.
  • For NDS that are feebly open, surjective and collectively convergent to their limit map, weak mixing, total transitivity and syndetical transitivity are preserved under passage to the limit and back, giving a transfer principle the earlier literature lacked.

Reading between the lines

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

  • Beyond the paper: if mild mixing implies mixing in every non-autonomous system, then on compact metric spaces the product-with-every-transitive-system condition is too strong to serve as an intermediate notion between mixing and weak mixing for time-varying dynamics; its role is mainly to expose how differently product transitivity behaves when the dynamics is not autonomous.
  • Beyond the paper: the counterexamples that interleave powers $\sigma^n$ and $\sigma^{-n}$ with the identity suggest a general obstruction—if a sequence of maps returns to the identity along a subsequence, multi-transitivity and weak mixing can diverge; the collective-convergence hypothesis appears to be exactly the condition that rules out such interlacing, so testing weaker 'eventual' forms of co
  • Beyond the paper: Theorem 4 inherits its force from the borrowed result [15, Theorem 4.1]; checking whether that result survives the paper's own counterexamples would determine whether Li-Yorke chaos follows from multi-transitivity alone or truly needs minimality.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper studies stronger forms of transitivity for non-autonomous discrete systems (NDS) generated by uniformly convergent sequences of continuous self-maps. It offers counterexamples to Theorem 3.1 of Salman-Das and to Theorems 2.3/2.4 of Sharma-Raghav, introduces a notion of mildly mixing for NDS, and claims that mild mixing implies mixing (Theorem 5), that mild mixing implies multi-transitivity (Corollary 1), and that minimal multi-transitive NDS are Li-Yorke chaotic (Theorem 4). It also proves transfer results between an NDS and its limit under collective convergence and presents further counterexamples showing that some autonomous-system theorems fail for NDS, together with sufficient conditions restoring them.

Significance. If the main claims were fully supported, the paper would make a useful contribution. Theorem 5 reverses the autonomous hierarchy (mild mixing implies mixing) with a clever sparse-shift construction, and Corollary 1 would answer open problem 2 of Salman-Das. The counterexamples to [15, Theorem 3.1] and to [18, Theorems 2.3/2.4] appear correct and are valuable. However, the advertised Theorems 4 and 13 rest on the unproved [15, Theorem 4.1], whose only published proof the paper itself undermines, and Example 7 is invalid. The significance is therefore conditional until these load-bearing gaps are addressed.

major comments (5)
  1. [Example 7 (Section 3)] The example is internally inconsistent. The case definition gives f_i = f^k for i=3k and f_i = f^{-k} for i=3k+1, so f^{3k}_1 = f^k and f^{3k+1}_1 = f^{3k+2}_1 = id; the claim that f^{2n}_1 = id for every n is false (for instance, f^6_1 = f^2). The displayed sequence does not match the case definition, and under either reading the conclusion that the system has dense 2-periodic points and is not syndetically transitive is not established. In fact, under the case definition, the return times to any pair of open sets are {3k : f^k(U) ∩ V ≠ ∅}, which is syndetic because f is a minimal rotation, so the system is syndetically transitive. This example therefore cannot support the asserted failure of `transitive plus dense periodic points implies syndetically transitive' for NDS.
  2. [Theorems 4 and 13] Both proofs invoke [15, Theorem 4.1] without proof. The paper itself states immediately after Example 4 that the proof of [15, Theorem 4.1] cited [18, Theorem 2.3], and Example 4 is a counterexample to [18, Theorem 2.3]. No replacement proof of [15, Theorem 4.1] is supplied. Consequently, the conclusion in Theorem 4 that a minimal multi-transitive NDS is weakly mixing of all orders, and hence the whole Li-Yorke chaos construction, is unsupported. Theorem 13 uses the same theorem to infer multi-transitivity of f_{1,∞} from multi-transitivity of the limit f; even if [15, Theorem 4.1] were true, no minimality of f_{1,∞} is established, so the application is not justified. In addition, the induction in Theorem 4 needs a uniform p_{k+1} for all x in S_k, but S_k is defined as an infinite set and Theorem 3 only supplies a p depending on the open pair; a finite-representative or compactness argument is missing.
  3. [Theorem 5] The converse assertion is false as stated. The proof of the converse uses Lemma 2 for the transitive system (Y, g_{1,∞}), so the no-isolated-point hypothesis is required for Y and not merely for X. Under Definition 1 of mild mixing, mixing does not imply mild mixing for all transitive NDS: the final paragraph of the proof itself constructs a mixing f_{1,∞} and a transitive g_{1,∞} on a two-point space for which the product is not transitive. The statement should either restrict the class of admissible transitive systems in the definition (for example, to spaces without isolated points) or the converse should be withdrawn. As written, Theorem 5 asserts an equivalence that is not supported.
  4. [Theorem 2] The proof does not establish the claimed contradiction. After applying multi-transitivity to the 2mp sets, the conclusion is f^{kh}_1(U'_h) ∩ V'_h ≠ ∅ for h = 2(i-1)p + j. This gives exponents kh = k(2(i-1)p + j), not the claimed 2kpi, and no choice of h produces 2kpi for each j simultaneously. Thus the assertion that a finite maximal hitting time forces a larger one is unsupported. Since Theorem 2 is stated as a main result on multi-transitivity, it needs a corrected proof or should be removed.
  5. [Theorem 12] The proof as written is invalid. The statement that `the set of times when any non-empty open set U×U1 visits V×V1 is infinite' is not justified, and the subsequent application of transitivity uses the set f^{-r0}_1(U1) in the Y-coordinate, where f^{-r0}_1 is a map on X. The notation and argument do not establish mild mixing of the limit f. Although Remark 2 suggests the result may follow from Theorem 5 and [18, Theorem 2.5], the proof in the text needs to be replaced. It should also be checked whether [18, Theorem 2.5] is affected by the paper's own corrections to [18].
minor comments (6)
  1. [Throughout] There are numerous typographical errors, including `an non-autonomous', `Cnovas', `Topologe and its Applications' in references [15] and [18], and `Topogogical' in reference [24]; these should be corrected.
  2. [Theorem 5, final paragraph] The sentence `it is obvious that f_{1,∞} × g_{1,∞} is not mildly mixing' should presumably read `is not transitive'; a product system is not itself a candidate for mild mixing in the sense used here.
  3. [Theorems 9 and 10] The notation `S(y, ε/2)' appears where an open ball `B(y, ε/2)' is intended.
  4. [Example 7] The displayed sequence repeats `f_{1,∞} =' and does not match the defining formula; this needs correction regardless of the mathematical issues raised in the major comments.
  5. [Theorem 19, Eq. (3.2)] The index range in Eq. (3.2) is suspect: the preceding union over i=1..M of f^i_1(f^{-k+1}_1(U)) naturally corresponds to a union over i=1..M of f^i_k(U), not i=1..M-k, unless a nontrivial truncation argument is supplied.
  6. [Example 5] The proof says `Since σ is weakly mixing' when σ is in fact mixing; this is harmless but should be corrected for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the paper's derivations are direct and its cited lemmas are independent of the conclusions.

full rationale

Walking the claimed derivation chain, I find no step in which a conclusion is equivalent by construction to an input, a fitted parameter is relabeled as a prediction, or a load-bearing premise is supplied only by a self-citation whose content is the target theorem. Theorem 5 is proved by construction: assuming non-mixing produces hitting-time gaps {n_k}, the paper builds a transitive non-autonomous system g1,infty supported on those gaps and shows the product f1,infty × g1,infty is non-transitive, which is exactly the negation of mild mixing. The counterexamples (Examples 1, 2, 4, 5, 6, 7, 8) directly verify the definitions and do not presuppose the statements they refute. The transfer theorems (8-18) use collective convergence and the cited limit-system results as hypotheses, not as conclusions. Theorem 4 invokes [15, Theorem 4.1] and Lemma 1 from [20]; although [20] is a self-citation and [15, Theorem 4.1] is not proved here, both are used as stated external lemmas whose assumptions do not include the target result, so this is a correctness risk rather than circularity. Similarly, Theorem 13's use of [15, Theorem 4.1] is a citation-dependency concern, not a reduction of the theorem to itself. No 'prediction' is fitted and no uniqueness claim is imported from the author's prior work.

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

No free parameters are fitted. The paper's axioms are standard results in topological dynamics plus several borrowed theorems from [15] and the autonomous literature. The most fragile is [15, Theorem 4.1], used as a black box.

assumptions (7)
  • domain assumption Multi-transitivity plus minimality of (X,f1,∞) implies weak mixing of all orders (Theorem 4.1 of [15]).
    Invoked without proof in Theorem 4 and Theorem 13. Since the paper corrects other results in [15], this cited theorem is load-bearing.
  • standard math For autonomous systems, syndetical transitivity plus weak mixing implies multi-transitivity (Moothathu [6]).
    Used in Theorem 13 and Theorem 14 to transfer conclusions from the limit map.
  • standard math For autonomous systems, total transitivity and dense periodic points imply weak mixing (Banks [21]).
    Used in Theorem 17.
  • standard math Multi-sensitivity implies thick sensitivity, and thick sensitivity plus transitivity implies multi-sensitivity for autonomous systems (Huang-Kolyada-Zhang [23]).
    Used in Theorem 16.
  • standard math A transitive system on a space without isolated points has infinite return time sets (Lemma 2, Miralles et al. [13]).
    Used in Theorem 5 converse and Theorem 9.
  • standard math The compact-limit-point criterion for Li-Yorke chaos (Lemma 1, Zeng-Huang-Liu [20]).
    Used in Theorem 4 to conclude Li-Yorke chaos.
  • domain assumption Uniform convergence plus collective convergence implies D(f^k_r, f^k) can be made small uniformly in k (Lemma 3 and [18]).
    The central technical hypothesis of Theorems 8-17.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A note on multi-transitivity in non-autonomous discrete systems." pith.science (2026). https://pith.science/paper/B6XQGELT

@misc{pith2026250524657,
  author       = {Pith},
  title        = {Pith review of: A note on multi-transitivity in non-autonomous discrete systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6XQGELT}},
  note         = {Machine review of arXiv:2505.24657}
}
abstract

This paper is concerned with some stronger forms of transitivity in non-autonomous discrete systems$(f_{ 1,\infty})$ generated by a uniformly convergent sequence of continuous self maps. Firstly, we present two counterexamples to show that Theorem 3.1 obtained by Salman and Das in [Multi-transitivity in nonautonomous discrete systems Topol. Appl. 278(2020)107237] is not true. Then, we introduce and study mildly mixing in non-autonomous discrete systems, which is stronger than mixing. We obtain that multi-transitivity implies Li-Yorke chaos and that mildly mixing implies multi-transitivity, which answer the open problems 1 and 2 in the paper above. Additionally, we give a counterexample which shows that Theorem 2.3 and Theorem 2.4 given by Sharma and Raghav in [On dynamics generated by a uniformly convergent sequence of maps Topol. Appl. 247 (2018)81-90] are both incorrect and give the correct proofs of them. Finally, some counterexamples are constructed justifying that some results related to stronger forms of transitivity which are true for autonomous systems but fail in non-autonomous systems, and establish a sufficient condition under which the results still hold in non-autonomous systems.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

24 extracted references · 21 canonical work pages

  1. [18]

    Sharma, M

    P. Sharma, M. Raghav, On dynamics generated by a uniformly convergent sequence of maps, Topologe and its Applications, 247 (2018)81-90

    work page 2018

  2. [1]

    Birkhoff, Dynamical Systems, American Mathematical Society, Washington,2008

    G. Birkhoff, Dynamical Systems, American Mathematical Society, Washington,2008

    work page 2008

  3. [2]

    T. Li, J. Yorke, Period three implies chaos. American Mathematical Monthly, 82(1975)985-992

    1975

  4. [3]

    Ruelle, F

    D. Ruelle, F. Takens, On the nature of turbulence, Communications in Mathematical Physics, 20(1971)167-192

    work page 1971

  5. [4]

    Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20(9)(2007)2115-2126

    T. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20(9)(2007)2115-2126

    2007

  6. [5]

    E. Akin, J. Auslander, A. Nagar, Variations on the concept of topological transitivity, Studia Mathematica, 235(3)(2016)225-249

    work page 2016

  7. [6]

    Moothathu

    T. Moothathu. Diagonal Points Having Dense Orbits[J]. Colloquium Mathematicum, 120(1)(2010)127-138

    work page 2010

  8. [7]

    Kwietniak, P

    D. Kwietniak, P. Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Ergodic Theory and Dynamical Systems, 32(5)(2012)1661-1672. 21

    work page 2012

Show all 24 references
  1. [8]

    Z. Chen, J. Li, J. Lv, On multi-transitivity with respect to a vector, Science China-mathematics, 57 (8) (2014) 1639-1648

    2014

  2. [9]

    H. Wang, H. Fu, S. Diao, P. Zeng, Chain mixing, shadowing properties and multi-transitivity, Bulletin of The Iranian Mathematical Society, 45(6)(2019)1605-1618

    2019

  3. [10]

    Kolyada, L

    S. Kolyada, L. Snoha, Topological entropy of nonautononous dynamical systems, Random Com- puter and Dynamics, 4(1996)205-233

    1996

  4. [11]

    C. Tian, G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28(2006)1067-1075

    2006

  5. [12]

    Salman, R

    M. Salman, R. Das, Sensitivity and property P in non-autonomous systems, Mediterranean Journal of Mathematics, 17(2020)pp.128

    2020

  6. [13]

    Miralles, M

    A. Miralles, M. Arcila, M. Sanchis, Sensitive dependence for nonau-tonomous discrete dynamical systems, Journal of Mathematical Analysis and Applications, 463(2018)268-275

    2018

  7. [14]

    Dvorakova, Chaos in nonautonomous discrete dynamical systems, Communications in Non- linear Science and Numerical Simulation, 17(2012)4649-4652

    J. Dvorakova, Chaos in nonautonomous discrete dynamical systems, Communications in Non- linear Science and Numerical Simulation, 17(2012)4649-4652

    2012

  8. [15]

    Salman, R

    M. Salman, R. Das, Multi-transitivity in nonautonomous discrete systems, Topologe and its Applications, 278(2020)107237

    2020

  9. [16]

    Balibrea, P

    F. Balibrea, P. Oprocha, Weak mixing and chaos in nonautonomous discrete systems, Applied Mathematics Letters, 25(2012)1135-1141

    2012

  10. [17]

    Canovas, Li-Yorke chaos in a class of nonautonomous discrete systems

    J. Canovas, Li-Yorke chaos in a class of nonautonomous discrete systems. Journal of Difference Equations and Applications, 17(4)(2011)479-486

    2011

  11. [19]

    On stronger forms of sensitivity in non-autonomous systems

    Vasisht, R.; Das, R. On stronger forms of sensitivity in non-autonomous systems. Taiwanese Journal of Mathematics, 22(2018)1139-1159

    2018

  12. [20]

    H. Zeng, C. Huang, B. Liu, Iteration problem for several chaos in non-autonomous discrete system, https://doi.org/10.48550/arXiv.2405.17173

    doi arXiv

  13. [21]

    Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory and Dynamical Systems, 17(3)(1997)505-529

    J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory and Dynamical Systems, 17(3)(1997)505-529

    1997

  14. [22]

    G. Liu, T. Lu, X. Yang, W. Anwar, Further discussion about transitivity and mixing of contin- uous maps on compact metric spaces. Journal of Mathematical Physics, 61(2020)112701

    2020

  15. [23]

    W. Huang,S. Kolyada, G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity. Ergodic Theory and Dynamical Systems, 2(2018)651-665

    2018

  16. [24]

    J. Pi, T. Lu, W. Anwar, Z. Mo, Further studies of topogogical transitiviey in nonautonomous discrete systems, Journal of Applied Analysis and Computation, 14(3)(2024)1508-1521. 22

    2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.