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REVIEW 4 major objections 3 minor 17 references

$J$-class weighted translations on locally compact groups

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Weighted translations on locally compact groups are J-class exactly when one backward weight product vanishes on almost all of every compact set, and this is strictly weaker than hypercyclicity.

desk verdict A meaningful extension of J-class theory to weighted translations on locally compact groups, with solid necessary conditions but sufficiency statements that need quantifier and subset fixes before the main theorems are airtight. read the letter →

arxiv 2506.04730 v1 pith:QCRWNERJ submitted 2025-06-05 math.FA math.DS

M. R. Azimi , I. Akbarbaglu , A. R. Imanzadeh Fard This is my paper
classification math.FAmath.DS MSC 47A1647B38
keywords J-classoperatorextendedlimitsetweightedtranslationlocallycompactgrouphypercyclicL^pspaceHaarmeasurelineardynamics
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 a weighted translation $T_{a,\omega}$ on $L^p(G)$ of a locally compact group is $J$-class, meaning that for some vector the extended limit set—the set of all limits reachable by slightly perturbing the vector and letting time go to infinity—is the whole space. The authors show that, for translation elements that eventually move every compact set off itself, $J$-class behavior is controlled by one asymptotic condition: a backward weight product must tend to zero on larger and larger subsets of each compact set. That condition alone makes the extended limit set at the zero vector equal to $L^p(G)$, while hypercyclicity requires a second weight product to vanish as well. The paper constructs concrete weights for which $T_{a,\omega}$ is $J$-class but not hypercyclic, and it shows that torsion elements, which never have dense orbits, can still satisfy $J_{T_{a,\omega}}(0)=L^p(G)$.

What carries the argument

The machinery is the pair of weight products $\tilde{\omega}_m(x)=\prod_{i=0}^{m-1}\omega(xa^{-i})^{-1}$ and $\omega_m(x)=\prod_{i=1}^{m}\omega(xa^i)$, which appear in the iterates $T_{a,\omega}^m f(x)=\omega(x)\omega(xa^{-1})\cdots\omega(xa^{-(m-1)})f(xa^{-m})$. The extended limit set $J_T(x)$ collects the targets $y$ for which a sequence $x_n\to x$ satisfies $T^{n_k}x_n\to y$. The proofs use the change-of-variable formula for Haar measure to trade smallness of these products on large subsets of a compact set for approximation of arbitrary target functions, while the assumption that $a$ passes through compact subsets keeps the support of the target disjoint from the shifted support of the perturbation, so the two products can be controlled independently.

What would settle it

A single concrete counterexample would decide the boundary claim: exhibit a weighted translation $T_{a,\omega}$ on some $L^p(G)$ that is $J$-class but for which there is a compact set $\Delta$ of positive measure with no sequence $E_k\subseteq\Delta$ satisfying $\lambda(E_k)\to\lambda(\Delta)$ and $\operatorname{ess\,sup}_{E_k}\tilde{\omega}_{n_k}\to 0$; Theorem 2.1 would then be false. One could also test a locally compact non-second-countable group with an aperiodic element $a$ that fails to pass through compact subsets and check whether the equivalence in Theorem 2.6 still holds there.

Watch

Extended reading notes

Core claim

The core claim is a sharp boundary between $J$-class and hypercyclicity for weighted translations on locally compact groups. Theorem 2.6 states that if $a$ passes through compact subsets, then $J_{T_{a,\omega}}(0)=L^p(G)$ holds if and only if for every compact set $\Delta$ of positive measure there are Borel sets $E_k\subseteq \Delta$ with $\lambda(E_k)\to\lambda(\Delta)$ and, along a subsequence $n_k$, $\operatorname{ess\,sup}_{x\in E_k}\tilde{\omega}_{n_k}(x)\to 0$. For a nonzero $J$-vector, the same condition is necessary (Theorem 2.1) and becomes sufficient in Theorem 2.3 when joined with a second condition, $\operatorname{ess\,sup}_{x\in K}\omega_{n_k}(x)\to 0$ on some compact set $K$ of positive measure. Since the known hypercyclicity criterion for these operators requires both weight products to vanish on large sets, the paper's examples show the $J$-class condition is genuinely weaker and can even occur on non-separable $L^p(G)$, where hypercyclicity is impossible.

Load-bearing premise

The results assume the translation element moves every compact set completely away from itself after finitely many steps; if this support-separation property fails, the proof's key disjointness step collapses and no characterization is given.

Editorial extensions

If this is right

  • If the conditions of Theorem 2.3 hold, then the indicator function of any suitable compact set $K$ is a $J$-vector, so compactly supported functions can serve as starting points for the extended limit set.
  • For elements passing through compact subsets, $J_{T_{a,\omega}}(0)=L^p(G)$ is equivalent to the single backward-product condition, so the zero vector can be a $J$-vector in non-separable spaces where no dense orbit exists.
  • Hypercyclicity and $J$-class coincide only when both $\tilde{\omega}_n$ and $\omega_n$ vanish on large sets; the examples on $\mathbb{R}$ and $\mathbb{R}^+$ show the gap is inhabited.
  • For a torsion element of order $\gamma$, $J_{T_{a,\omega}}(0)$ fills the whole space exactly when $\omega_n^{-1}$ becomes arbitrarily small on almost all of every compact set, even though orbits are never dense.
  • Interior points of $J_{T_{a,\omega}}(0)$ force the whole extended limit set to be $L^p(G)$, giving a binary alternative: either $J_{T_{a,\omega}}(0)$ has empty interior or it is everything.

Reading between the lines

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

  • The asymmetry identified here suggests a testable recipe for other group actions: if an operator admits local perturbations supported away from the target, the zero-vector case should be characterized by one-sided decay of the cocycle, and two-sided decay should be required only for a dense orbit.
  • Because Theorem 2.1's necessary condition only involves $\tilde{\omega}_n$, a natural next question is whether condition (i) alone is already sufficient for a nonzero $J$-vector; the paper's sufficient condition (ii) may be an artifact of the particular proof using $\chi_K$.
  • The torsion result points toward constructing J-class operators on compact groups by choosing weights whose period-$\gamma$ product drops below 1 on large pieces of every compact set; the actual orbit remains bounded, but the extended limit set can still be everything.
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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

4 major / 3 minor

Summary. The paper studies, for a locally compact group G and a continuous weight ω, the weighted translation operator T_{a,ω} on L^p(G), and asks when it is a J-class operator, i.e. when some extended limit set J_T(x) equals the whole space. Theorem 2.1 gives a necessary condition: if T_{a,ω} is J-class, then for every compact Δ of positive measure there are sets E_k with λ(E_k)→λ(Δ) and a subsequence n_k along which ess sup_{E_k} tildeω_{n_k}→0. Theorem 2.3 states a sufficient condition involving, in addition, a fixed compact K on which ω_{n_k}→0 in essential supremum, and identifies χ_K as a J-vector. Theorem 2.6 characterizes J_{T_{a,ω}}(0)=L^p(G) by the first condition alone when a passes through compact subsets. Theorem 2.4 addresses torsion a and claims an equivalence between J_{T_{a,ω}}(0)=L^p(G) and a local weighted-product condition. Three examples on R and R_+ are claimed to satisfy the conditions while failing hypercyclicity, supporting the paper's headline that J-class behavior is strictly weaker than hypercyclicity.

Significance. The topic is a natural continuation of the Costakis--Manoussos theory of J-class operators, and the paper gives a concrete framework for comparing J-class behavior with hypercyclicity for weighted translations on locally compact groups. The main approximation mechanism, based on supports separated by the 'passes through compact subsets' property, is promising, and the examples are explicit and checkable. The paper also claims J-class behavior on nonseparable L^p spaces, which is of independent interest. However, several quantifier and proof issues in the main statements need to be repaired before the results can be accepted as proved.

major comments (4)
  1. [Theorem 2.3] In Theorem 2.3, condition (i) is quantified per compact subset Δ: for each Δ there is a subsequence (n_k), while condition (ii) refers to 'some subsequence (n_k)' without specifying that it is the same subsequence. This matters in the proof, where for a fixed f with Δ=σ(f) the authors use the subsequence supplied by (i) and then apply (ii) along that same subsequence (see the line 'our hypotheses imply ... ess sup ω_{n_k} < ... on K'). With the stated quantifiers there may be no single subsequence along which both (i) and (ii) hold. The theorem should be restated with an initial quantifier: 'there exists a strictly increasing sequence (n_k) such that for every compact Δ ... and for some fixed compact K ...'. This is load-bearing for the sufficiency claim.
  2. [Theorems 2.3 and 2.6] The condition λ(E_k)→λ(Δ) should be replaced by 'E_k⊆Δ and λ(Δ\E_k)→0'. As written, λ(E_k)→λ(Δ) does not imply λ(σ(f)\E_k)→0 when E_k is not contained in σ(f), yet the proofs use exactly that implication: in Theorem 2.3 the integral ∫|fχ_{E_k}−f|^p is estimated by λ(σ(f)−E_k)‖f‖_∞^p, and the same use appears in Theorem 2.6. The statements should make the inclusion explicit.
  3. [Theorem 2.6, proof of (i)⇒(ii)] In the proof of (i)⇒(ii), the inequalities λ(B_{η,k})<η^p and λ(C_{η,k})<η^p are not justified by the hypotheses ‖g_k‖_p<ε and ‖T^{n_k}g_k−χ_Δ‖_p<ε. These hypotheses only give λ(B)<(ε/η)^p and λ(C)<(ε/η)^p. One can repair the argument by choosing the norm bounds much smaller than η^2 before fixing η, but as written the proof is invalid.
  4. [Theorem 2.4] The proof of (i)⇒(ii) begins with 'Without loss of generality, assume λ(F)>3δ^p and δ/(1−δ)<ε'. These are not harmless normalizations for arbitrary compact F and arbitrary ε,δ: one cannot enlarge F or shrink δ. In addition the final estimate gives λ(F\E)<2δ^p, which is not the stated λ(F\E)<δ unless 2δ^p≤δ, an extra condition that is not assumed. The converse direction (ii)⇒(i) produces for each neighborhood pair some n, but the neighborhood definition of J_T(0) requires n exceeding any prescribed N, and it is not shown that the n supplied by (ii) can be taken arbitrarily large. The equivalence is therefore not proved as stated.
minor comments (3)
  1. [Throughout] The manuscript contains numerous typographical errors and broken line breaks ('lo cally', 'p aper', 'defined') and should be proofread carefully.
  2. [Theorem 2.1] In the proof of Theorem 2.1, the estimate λ(Δ\E)<3η^p is established for a fixed η; to obtain a sequence (E_k) with λ(E_k)→λ(Δ), the authors should explicitly let η→0 and diagonalize over k. This is routine but should be stated.
  3. [Example 2.11] In Example 2.11, the statement 'for each x∈[0,1/4] we have 1/4<ω(x)<11/16' is not correct at the endpoint x=1/4, where ω(x)=11/16. This does not affect the argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the J-class characterizations are derived by direct estimates from the definition, and the self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. Theorem 2.1 derives the necessary weight-smallness condition directly from J-class membership: if χ_Δ ∈ J_T(f), the approximating sequence (g_k) and times (n_k) supplied by the definition yield, by the change-of-variable formula, sets E_k with λ(Δ \ E_k) small and ess sup tildeω_{n_k} small on E_k. The sufficiency directions (Theorems 2.3 and 2.6) construct g_k(x) = tildeω_{n_k}(x a^{n_k}) f χ_{E_k}(x a^{n_k}); the product of the orbit weight and tildeω cancels exactly by the definition of tildeω, so the verification is a direct estimate rather than an assumption of the conclusion. The torsion case (Theorem 2.4) likewise uses the definition of J_T(0) and an explicit S_{a,ω} construction. External results are used only as stated: [10] supplies the 'passes through compact subsets' lemma and hypercyclicity criteria for weighted translations, and [12] supplies the power-bounded identity J_T(0) = L_T(0); both are independent published results, not self-citations. The authors' own works [5,8] appear only in a list of related literature and are not used to prove the main theorems. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported to force a choice. The only concern that arose is a possible quantifier mismatch between conditions (i) and (ii) in Theorem 2.3; that is a correctness or statement issue, not circularity, because fixing the subsequence before choosing Δ does not make the conclusion equal to the hypothesis.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim depends on standard Haar-measure tools and the explicit group-element assumptions (passes through compacts, or torsion). No free parameters are fitted; no new entities are postulated.

assumptions (5)
  • standard math Right Haar measure lambda is regular on Borel sets of finite measure and right-invariant under the change of variables used in the proofs.
    The proofs use regularity to find compact B subset A and the change of variable formula int_A f(xa^{-1}) d lambda(x) = int_{A a^{-1}} f(x) d lambda(x).
  • domain assumption The group element a passes through compact subsets: for every compact K, K intersect K a^{+/-m} = empty set eventually.
    Assumed in Theorems 2.1, 2.3, 2.6 to ensure disjointness of supports; only guaranteed for aperiodic elements in second countable groups ([10, Lemma 2.1]).
  • domain assumption In Theorem 2.4, a is a torsion element of finite order gamma, so a^gamma = e and the orbit F a^k is a finite union.
    Used to make the union union_{k=0}^{gamma-1} F a^k invariant under right translation by a, and to identify products omega_n with powers of omega_gamma.
  • standard math Known results from Costakis and Manoussos [12]: the equivalence between J_T(0)=X and the neighborhood condition, and J_T(x)=L_T(x) for power-bounded T.
    Quoted in the introduction and used in the converse directions of Theorems 2.4 and 2.6 to conclude J(0)=L^p(G) from local transitivity.
  • standard math Continuous compactly supported functions C_c(G) are norm-dense in L^p(G), 1 <= p < infinity.
    Used to reduce approximations to functions with compact support in Theorems 2.3, 2.4, and 2.6.

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Pith. "Pith review of $J$-class weighted translations on locally compact groups." pith.science (2026). https://pith.science/paper/QCRWNERJ

@misc{pith2026250604730,
  author       = {Pith},
  title        = {Pith review of: $J$-class weighted translations on locally compact groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCRWNERJ}},
  note         = {Machine review of arXiv:2506.04730}
}
abstract

A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,\omega}$ to be $J$-class on a locally compact group $G$, within the setting of $ L^p$-spaces for $ 1 \leq p < \infty $. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,\omega}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $ T_{a,\omega}$ is $J$-class but it fails to be hypercyclic.

Figures

Figures reproduced from arXiv: 2506.04730 by the authors.

Figure 1
Figure 1. Weight function Fix a nonzero number a ∈ R and consider the convolution (ω∗δa)(x) = ω(x−a), for each x ∈ R and f ∈ L p (R). Fix a = 2 and choose compact subset ∆ ⊆ R arbitrarily. Then for each x ∈ ∆ we have ω˜n(x) = 1 Qi=n−1 i=0 ω(xa−i) = 1 Qi=n−1 i=0 ω(x − 2i) → 0, as n → ∞. Also, for each x ∈ [0, 1 4 ] we have 1 4 < ω(x) < 11 16 . Hence, ωn(x) = i Y=n i=1 ω(xai ) = i Y=n i=1 ω(x + 2i) → 0, as n → ∞. Eventually, th… view at source ↗

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Works this paper leans on

17 extracted references · 17 canonical work pages

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