# Verification report — OWR-14298367-003 (shadowing and generalized hyperbolicity without bounded distortion)

Verification date: 2026-09-29 (revised the same day after a second independent referee report).

**Verdict.** Part (a) is answered affirmatively. For every dissipative composition operator T_f on
L^p(X,B,μ), 1 ≤ p < ∞, bounded distortion is not needed: T_f has the shadowing property if and only if
it is generalized hyperbolic (GH). Real and complex scalars are both covered, and no separability is
assumed. Both properties are also equivalent to three further conditions:
- I − T_f is surjective;
- a uniform exponential "tent" condition holds for the densities h_k = d(μ∘f^k)/dμ on the wandering set;
- the fibre weighted shifts are equi-shadowing.

Part (b) is answered negatively: the characterization by the masses m_k = μ(f^k(W)) fails in both
directions.
- The direction "(HC) ⇒ shadowing" was already refuted by Bernardes, D'Aniello and Maiuriello
  (arXiv:2607.15831, Example 6.2).
- The note adds an elementary hyperbolic example for which none of (HC), (HD), (GH) holds.

For p = 2, complex scalars and separable L²(μ), Part (a) is a special case of Pituk's theorem
(arXiv:2608.19499, Theorem B). For a single atom W it reduces to the characterization of shadowing for
bilateral weighted shifts by Bernardes and Messaoudi (ETDS 41 (2021), Theorem 18). The note is
unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 19/2024 (doi:10.4171/OWR/2024/19). The source is the
  Mini-Workshop "New Horizons in Linear Dynamics, Universality, and the Invariant Subspace Problem",
  abstract of M. Maiuriello (joint with E. D'Aniello and U. B. Darji), pp. 1078–1081.
  - Open Problem (1), p. 1080, is set in a dissipative setting without bounded distortion. It asks
    "does the characterization provided in [1] for the shadowing property still hold?", and whether a
    composition operator is GH if and only if it has the shadowing property.
  - The report's setting: (X, B, μ) is σ-finite; f is a bijective bimeasurable map; f and f⁻¹ satisfy
    μ(f⁻¹(B)) ≤ cμ(B); and X is the disjoint union of the sets f^k(W) with 0 < μ(W) < ∞.
  - In the report, GH is defined with σ(T|_M) ⊂ D and σ(T⁻¹|_N) ⊂ D.
  - The second referee fetched the report again (anonymously) and confirmed that the heading "Open
    Problems" and all of Open Problem (1) are on p. 1080; Open Problem (2) is at the top of p. 1081.
- **Original source of the question.** D'Aniello–Darji–Maiuriello, J. Differential Equations 298 (2021)
  68–94 (arXiv:2009.11526), Section 5, Problem 5.0.1. The same two questions are asked there. The
  conditions (HC), (HD), (GH) and Corollaries SC and GH were checked against the arXiv source.
  Problem numbers follow arXiv v1 (the only arXiv version), as in [Pit26].
- **Corpus record.** ulamai/UnsolvedMath v1.6.0, record OWR-14298367-003 (status `open`). Its statement
  is the text of Open Problem (1).

## Readings
| Reading | Answer | Where |
|---|---|---|
| (a) shadowing ⇔ GH, with GH in the spectral form of the report (also used in [BCDMP18], [Pit26]) | yes, for all 1 ≤ p < ∞ | Theorem 1.2 (i)⇔(iii) |
| (a) with T\|_M, T⁻¹\|_N proper contractions for some equivalent norm ([DDM21], [CGP21]) | yes (same notion; shadowing is renorming-invariant) | Section 2.1 |
| (a) with proper contractions in the given L^p norm (no renorming) | not the intended reading: [DDM21]'s own proofs establish only spectral bounds, and under this reading [DDM21, Corollary GH] would already fail for a single weighted shift with r(T) < 1 < ‖T‖ (one atom, so bounded distortion holds) | Section 2.1 |
| (b) shadowing ⇒ (HC) or (HD) or (GH) | no | Example 6.1 (new, elementary) |
| (b) (HC) or (HD) or (GH) ⇒ shadowing | no | Example 6.2 = [BDM26, Example 6.2] (prior work) |
| (b) a characterization of shadowing by explicit growth conditions | yes, with densities h_k(x) in place of masses m_k | Theorem 1.2(iv); Remark 6.4 recovers [DDM21] under bounded distortion |

## Results in the paper
- **Lemma 2.1** ([BCDMP18, Theorem A]; self-contained proof). A GH splitting gives the explicit bounded
  solution y_n = Σ_{j≥1} T^{j−1}Pz_{n−j} − Σ_{j≥0} T^{−j−1}Qz_{n+j} of y_{n+1} = Ty_n + z_n. Hence
  shadowing. With z_n ≡ z this gives the bounded right inverse R = Σ_{j≥0} T^jP − Σ_{j≥1} T^{−j}Q of
  I − T (added in revision 2). So for dissipative T_f, surjectivity of I − T_f is equivalent to right
  invertibility.
- **Lemma 2.2 (fibre model).** T_f is isometrically conjugate to T on L^p(W × Z), where
  (Tu)(x,k) = a_k(x) u(x,k+1) and a_k = (h_k/h_{k+1})^{1/p}. The adjoint of I − T acts fibrewise as
  ρ_k(θ_k − θ_{k−1}), with ρ_k = h_k^{−1/p}. This is the conjugacy of [CDV24, Theorem 2.4], normalized
  differently.
- **Lemma 3.1 (averaging).** If the equation y_{n+1} = Ty_n + x has bounded solutions, then
  ‖(I−T)*φ‖ ≥ C⁻¹‖φ‖. Shadowing gives this with C = 1/δ₁.
- **Lemma 3.2 (rotation).** V_λ⁻¹TV_λ = λT for |λ| = 1, so I − T is surjective if and only if
  λ̄I − T is. This is a form of the classical rotation invariance of the spectrum of weighted shift
  operators over aperiodic maps (see [Ant21, Lemma 3.2]).
- **Lemma 3.3 (localization).** A lower bound for (I−T)* gives, for almost every x, the same lower bound
  on each fibre, tested on a countable family of boxes and tents. Measurability is spelled out.
- **Lemma 4.1 (one fibre, deterministic).** For S_k = −(1/p) log h_k(x):
  - boxes exclude humps;
  - tents exclude long flat stretches;
  - together these force a V-shape of S with exponential rates.

  The constants are explicit and depend only on δ ≤ 1.
- **Theorem 1.2.** For dissipative T_f, the following are equivalent: (i) shadowing; (ii) I − T_f
  surjective; (iii) GH; (iv) the uniform tent condition; (v) equi-shadowing of the fibre shifts.
  - The constants are K = (2e^{H₂})^p and r = 2^{−p/(2R)}, where H₂ = log(2/δ) + 1 and
    R = ⌊16e^{2H₂}/δ⌋ + 1.
  - K and r depend on p and δ. The fibre constants K^{1/p} and r^{1/p} depend on δ only.
  - For a single atom W, T_f is isometrically conjugate to a bilateral weighted backward shift on
    ℓ^p(Z). Then Theorem 1.2 (i)⇔(iv), with Remark 6.4, is the characterization of Bernardes and
    Messaoudi [BM21, Theorem 18], and Lemmas 2.1, 3.1 and 4.1 give another proof of it. The results of
    [DDM21] build on [BM21, Theorem 18].
- **Corollary 1.4.** T_f is hyperbolic if and only if J ∈ {±∞} almost everywhere. A finite peak on a
  set of positive measure produces eigenvalues on the unit circle.
- **Proposition 1.5.** This answers Part (b):
  - (a) is [BDM26, Example 6.2] (Example 6.2 in the paper; (⋆) holds there with c = 3);
  - (b) is Example 6.1: μ(α_j) = 2^j, μ(β_j) = 2^{−j}. Here T_f is 2^{−1/p}·isometry ⊕ 2^{1/p}·isometry,
    so it is hyperbolic, and m_k = 2^k + 2^{−k} violates (HC), (HD) and (GH).
- **Example 6.3.** Every fibre is hyperbolic, but T_f has neither property, so uniformity is needed. This
  is the composition-operator form of the diagonal example after [CDV24, Corollary 2.15].
- **Remark 6.4.** Under bounded distortion, (iv) reduces to (HC), (HD) or (GH). This recovers
  [DDM21, Corollaries SC and GH].

## Computations (sanity checks; scripts and outputs in reproducibility/)
The proofs are by hand and do not depend on these computations. Tested with Python 3.13.5, numpy 2.3.1
and scipy 1.16.0.
- **Finder** (`claimant/`).
  - `verify_model.py` (0.1 s) checks Lemma 2.2 on 20 random atomic systems for each of p = 1 (exact
    Fractions), p = 2 and p = 3. All checks pass.
  - `fibre_lemma_check.py` (2.5 s) has three parts:
    - Lemma 4.1(a),(b) for q ∈ {1, 1.5, 2, 3, ∞}, on 120 random windows each;
    - 210 comparisons of the test-family minimum with the true fibre lower bound (tridiagonal eigenvalue
      problem for q = 2, linear programming for q = ∞);
    - Lemma 4.1(c),(d) with the explicit constants, on 12 windows of length 3–6·10⁴.

    There are 0 violations.
  - `examples.py` (0.3 s) computes the masses of Examples 6.1 and 6.2 exactly and the fibre lower bounds
    of Examples 6.1–6.3. For Example 6.3 the bound is 1/(n+1); for the flat fibre of Example 6.2 it is
    about π/(L+1).
  - All three outputs were re-run on 2026-09-29 and are byte-identical to the stored `.out` files.
- **Referee** (`referee/indep_adversarial.py`, about 70 s, written by the independent verifier).
  - Hill climbing against the true window lower bound, for q = 2 (singular values) and q = ∞ (linear
    programming). The maximal violations of Lemma 4.1(a),(b) stay negative: −0.35 and −1.68 for q = 2,
    −0.69 and −2.16 for q = ∞.
  - The full chain (d) was checked on 40 windows of length 2–8·10⁴, with 0 violations.
- **Lead** (`lead/lead_check.py`, under 1 s, written during the audit).
  - On 30 random V-shaped fibres (p ∈ {1, 1.5, 2, 3}), it checks that formula (1) solves
    y_{n+1} = B_x y_n + z_n, with residual ≤ 9·10⁻¹⁶, and that the bound K^{1/p}(1+s)/(1−s) holds
    (worst ratio 0.18).
  - It checks the tent estimates of Examples 6.2 and 6.3 (ratios ≤ 8/R and ≤ 8/R + 1/(n+1)) for
    q ∈ {1, 2, ∞}.
  - It checks the inequalities of Example 6.1 exactly for n ≤ 200.
- **Referee 2** (`referee2/`, written by the second independent referee from the text of the paper,
  before the other scripts were read; about 1 min in total).
  - `r2_model_check.py` (0.1 s) checks Lemma 2.2(a)–(d), the formula for T^n with |n| ≤ 5, and
    Lemma 3.2. It uses 25 random atomic dissipative systems (1–5 atoms, c ∈ [2, 5]) in each of five
    settings: p = 1 in exact Fractions, p = 1.5, p = 2, and p = 2, 3 with complex scalars. 0 failures;
    maximal error 0 (exact) and ≤ 5·10⁻¹³ (floating point).
  - `r2_fibre_check.py` (about 17 s with argument 2) checks window versions of Lemma 4.1(a)–(d) with the
    explicit constants, in log space, for q ∈ {1, 1.5, 2, 3, ∞}. δ is the minimum over all boxes of the
    window and over the tents of the prescribed R, iterated until δ and R are stable. The recorded
    output covers 36 windows (V-shaped, monotone, bumpy, random-walk and plateau) of length up to 9000:
    0 failures. In the steep windows δ ≈ 0.62–1 and R ≈ 473–1955, so (b) and (d) are exercised; for
    small δ only (a) and (c) can be tested. The default run (3 repetitions, 54 windows), done in
    revision 2, also gives 0 failures (`r2_fibre_check_default_run.out`).
  - `r2_examples_check.py` (8 s) checks Example 6.1 exactly ((⋆) with c = 2, the displayed inequalities
    for n ≤ 300, the n-th-root quantities 2 and 1/2, failure of bounded distortion, the norm identities
    for p = 1, 2, 3, the tent data K = 1, r = 1/2). It checks Example 6.2 (m_j = 2^j for |j| ≤ 60,
    h_k(β₀) = 2, (⋆) with c = 3) and the tent bound ‖Dτ_{b,R}‖_q ≤ (8/R)‖τ_{b,R}‖_q for
    q ∈ {1, 1.5, 2, 3, 7, ∞} and R ≤ 5000 (worst ratio to 8/R: 0.25). For Example 6.3 it checks the
    identity for (I − T)*ψ, the bound 8/R + 1/(n+1) (worst ratio 0.985) and the fibre lower bound
    1/(n+1). 0 failures.
  - `r2_shift_check.py` (2.3 s) checks formula (1) for the splitting of (iv)⇒(v) on V-shaped and
    monotone fibres, for p ∈ {1, 2, 3}: residual ≤ 1.1·10⁻¹⁴, and sup‖y‖/sup‖z‖ well below
    C′ = K^{1/p}(1 + r^{1/p})/(1 − r^{1/p}). For p = 2 the window lower bound of (I − B)* is ≥ 1/C′, as
    Lemma 3.1 predicts. It also checks the eigenvectors of Corollary 1.4. 0 failures.
  - The second referee also re-ran the five scripts in `claimant/`, `lead/` and `referee/`; all outputs
    were byte-identical to the recorded ones. In revision 2 the four referee-2 scripts were re-run: the
    outputs of `r2_model_check.py`, `r2_examples_check.py` and `r2_shift_check.py` (no argument) and of
    `r2_fibre_check.py 2` are byte-identical to the recorded ones.

## Independent adversarial audit
Verdicts (2026-09-29):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (OWR PDF fetched anonymously; question also checked in the arXiv source of [DDM21]) |
| Proofs | CONFIRMED (every step checked by hand, including Lemma 3.3 measurability, the V-shape case analysis, the splitting and the bounded-solution formula) |
| Computations | CONFIRMED (all scripts re-run byte-identically; independent adversarial script) |
| Answer as posed | CONFIRMED (Part (a) yes; Part (b) no in its literal form) |
| Novelty | CONFIRMED with scope restrictions (p = 2 complex separable is in [Pit26]; Part (b) sufficiency direction is in [BDM26]) |
| Presentation | PAPER_CANDIDATE, with six required fixes |

All six required fixes were applied:
1. **Constants.** The dependence of the constants on p and δ is stated correctly; only K^{1/p} and
   r^{1/p} depend on δ alone.
2. **Counterexamples on ℓ^p.** The sentence about L^p with p ≠ 2 was replaced by the precise statement
   of [MNST26, Theorem 3.9]: for every 1 < p < ∞, p ≠ 2, there is an operator on ℓ^p(N) with shadowing
   that is not GH.
3. **Prior coverage.** The paper states that [Pit26, Theorem B] covers p = 2, complex scalars and
   separable L². The new cases are p ≠ 2, real scalars, non-separable L², and conditions (ii), (iv), (v).
   (This wording was made more modest in revision 2; see below.)
4. **Part (b).** Part (b) is labelled as already answered in one direction by [BDM26, Example 6.2].
   Example 6.1 is presented as elementary. Example 6.3 is credited to [CDV24].
5. **Theorem numbers, measurability and δ.**
   - GH ⇒ shadowing is cited as [BCDMP18, Theorem A], with the numbering used in [Pit26]; it is
     Theorem 33 in arXiv:1612.02921v1. See also [CGP21, Theorem 1] (arXiv version). The paper also
     gives a self-contained proof.
   - Measurability of x ↦ ‖ρ(x)Dθ‖ and of J is spelled out.
   - The case δ > 1 is handled by replacing δ with min(δ, 1).
6. **Literature re-check.** arXiv was re-checked on 2026-09-29, and nothing new was found. A specialist
   was not contacted. Before any journal submission, the author may wish to re-check arXiv and consult
   a specialist, since the area is very active.

The audit also simplified Lemma 3.3: the level-set step of the original write-up was replaced by a
direct monotonicity argument, valid for all q ∈ [1, ∞].

### Second independent referee (referee 2)
A second independent adversarial referee checked the released version (12 pages) on 2026-09-29. The
report is kept in the problem folder (`audit/REFEREE_REPORT_2.md`); the referee's scripts and outputs are
in `reproducibility/referee2/`. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (OWR PDF fetched again anonymously; Open Problem (1) is on p. 1080; [DDM21, Problem 5.0.1] checked in the arXiv v1 source; the spectral reading of GH is the intended one) |
| Proofs | CONFIRMED (every proof checked line by line; no error, and no gap needing more than a sentence) |
| Computations | CONFIRMED (four independent scripts, 0 failures; all five release scripts reproduce byte-identically) |
| Novelty | CONFIRMED with scope restrictions; two missing credits: Bernardes–Messaoudi [BM21, Theorem 18] and Antonevich's graded dichotomy [Ant21] |
| Presentation / house style | Good; minor fixes to references, one overstatement, release metadata |
| Fatal | No (PAPER_CANDIDATE after the required fixes) |

All five required fixes of the second referee were applied (revision 2, 2026-09-29):
1. **Bernardes–Messaoudi.** [BM21, Theorem 18] is now cited in "Relation to other work" and in "Scope
   and priority". The paper states that for a single atom W, T_f is (isometrically conjugate to) a
   bilateral weighted backward shift on ℓ^p(Z); that Theorem 1.2 with Remark 6.4 reduces to the
   Bernardes–Messaoudi characterization; that Lemmas 2.1, 3.1 and 4.1 give another proof of it for
   ℓ^p(Z) (they also cover c₀(Z)); and that [DDM21] builds on it.
2. **Antonevich.** [Ant21] (Sovrem. Mat. Fundam. Napravl. 67(2) (2021) 208–236,
   doi:10.22363/2413-3639-2021-67-2-208-236) and its English translation [Ant24] (J. Math. Sci. 278(1)
   (2024) 12–38, doi:10.1007/s10958-024-06903-w) are cited and compared in a new paragraph of "Relation
   to other work".
   - (a) The splitting by multiplication with an indicator function in (iv)⇒(iii), and the resulting
     bounded right inverse of I − T_f, parallel his graded dichotomy and right resolvent
     ([Ant21, Theorem 3.4]; compact X, a homeomorphism α, L² and, as remarked there, L^p).
   - (b) Lemma 3.2 is described as a form of the classical rotation invariance for aperiodic maps
     ([Ant21, Lemma 3.2]).
   - (c) The paper states that the necessity direction (ii)⇒(iv) and shadowing are not treated in
     [Ant21] in the dissipative measurable setting.
   - The novelty of (ii), (iv), (v) is now stated as "not found in this setting in the literature", in
     the paper and in this report.
3. **"New cases" sentence harmonized.** "Relation to other work" and "Scope and priority" now say the
   same thing: the cases p ≠ 2, real scalars and non-separable L²(μ) are not covered by
   [Pit26, Theorem B]; for complex scalars (i)⇔(ii) also follows from [DP26, Theorem 2.2]; the
   characterizations (ii), (iv), (v) were not found in this setting.
4. **Bibliography.**
   - (a) OWR24 note: Open Problem (1) on p. 1080.
   - (b) Mai22: Banach J. Math. Anal. 16(4), Paper No. 51 (2022).
   - (c) BBP26: Mediterr. J. Math. 23(5), Paper No. 146 (2026).
   - (d) Est22: the published version is cited instead, M. S. Al Ghafri, Y. Estaremi and Z. Huang,
     Mathematics 12(18) (2024), Paper No. 2809, doi:10.3390/math12182809. We identified it by title
     and abstract (Crossref and OpenAlex); the MDPI full text returned HTTP 403 to an anonymous
     request, so the bib note says that we read the arXiv version (arXiv:2209.11930).
   - (e) BCDMP18: the note is kept, and it now adds that [DDM21] cites the same result both as
     Theorem A (introduction) and as Corollary 8. The numbering of the published version could not be
     checked (ScienceDirect returned HTTP 403 to an anonymous request).
   - (f) DDM21: the note says that problem numbers follow arXiv v1.
5. **Release regenerated.** paper.pdf (now 14 pages), main.tex, references.bib, this report,
   `reproducibility/referee2/` (with a README row), source.zip, the three zenodo/ copies and the
   sha256/size/md5 lines of ZENODO_METADATA.md. The paper was rebuilt with tectonic (0 overfull or
   underfull boxes, 0 warnings), and every page was rendered and inspected.

Optional suggestions of the second referee that were also applied:
- the right inverse remark after Lemma 2.1 (suggestion 6);
- wording: "the contraction condition (HC)" in the abstract, "Two recent preprints settle it
  negatively", and "The claims were checked as follows" in the Verification paragraph (suggestion 7);
- a one-paragraph justification of the spectral reading of GH in Section 2.1 (suggestion 8);
- c = 3 stated in Example 6.2, and c^{L/p} in Lemma 3.3 (suggestion 10).

Not applied: suggestion 9 (extension to weighted composition operators via the conjugation theorem of
[BBP26]). It is optional, and we did not check that theorem ourselves.

## Relation to the literature, novelty and scope
- **Searches (September 2026).** All requests were anonymous and are logged in the problem folder's
  `queries.log`.
  - arXiv API: title and abstract searches, e-print sources of the relevant preprints, and a
    date-sorted re-check on 2026-09-29.
  - OpenAlex: all 25 works it lists as citing [DDM21] were screened.
  - zbMATH Open and Crossref (used for DOIs).
  - Crossref and OpenAlex queries on one-sided invertibility of weighted shift operators and on graded
    dichotomy (second referee), which found [Ant21].
  - Two general web searches (one by the finder, one by the second referee).
- **Prior and concurrent work.**
  - [DDM21] proves the characterization under bounded distortion and poses the question.
  - [BDM26] (July 2026) refutes the sufficiency direction of Part (b) and does not treat Part (a).
  - [Pit26] (August 2026): Theorem A gives a counterexample in general Banach spaces; Theorem B proves
    the equivalence on separable complex Hilbert spaces, which covers Part (a) for p = 2 with complex
    scalars and separable L².
  - [MNST26] (August 2026) gives counterexamples on ℓ^p(N), 1 < p < ∞, p ≠ 2.
  - Dragičević–Pituk (Trans. AMS 2026, Theorem 2.2; known to us only as quoted in [Pit26] and [MNST26])
    prove shadowing ⇔ σ_s(T) ∩ T = ∅ on complex Banach spaces. Together with rotation invariance this
    gives (i) ⇔ (ii) for complex scalars.
  - [CDV24] contains the shift-operator model, the notion of equi-shadowing and the finite-dimensional
    case.
  - [BM21, Theorem 18] (Bernardes–Messaoudi, ETDS 41 (2021)) characterizes shadowing for bilateral
    weighted shifts on ℓ^p(Z) and c₀(Z). This is the single-atom case of Theorem 1.2, and [DDM21]
    builds on it.
  - [Ant21] (Antonevich, survey, 2021; English translation [Ant24], 2024) treats right invertibility of
    weighted shift operators B − I on L²(X, μ) (and L^p) over a homeomorphism of a compact space X. His
    graded dichotomy, a measurable projector-valued splitting whose rank may vary with the point, gives
    a right resolvent (Theorem 3.4 there). This parallels the splitting used in (iv)⇒(iii) and the
    right inverse after Lemma 2.1. His Lemma 3.2 is the classical rotation invariance of the spectrum
    for aperiodic maps. Necessity is proved there only under topological assumptions (maps of
    Morse–Smale type, maps with an attractor for scalar coefficients, model examples). The dissipative
    measurable setting, the direction (ii)⇒(iv) there, and shadowing are not treated.
  - The follow-ups [DDM22], [Mai22], [AEH24] (Mathematics 2024; preprint arXiv:2209.11930 by
    Estaremi) and [BBP26] (Mediterr. J. Math. 2026; arXiv:2512.06425) assume bounded distortion for
    their shadowing and hyperbolicity results. For [AEH24] this was checked in the arXiv version.
    [BBP26] was checked by both referees.
- **Novelty.** We did not find the following in the literature:
  - Part (a) for p ≠ 2, for real scalars and for non-separable L² (cases not covered by
    [Pit26, Theorem B]);
  - the characterizations (ii), (iv), (v) in this setting (for complex scalars (i)⇔(ii) also follows
    from [DP26, Theorem 2.2]; for a single atom the theorem is [BM21, Theorem 18]; the splitting in
    (iv)⇒(iii) parallels [Ant21]);
  - Example 6.1.

  This negative search is not a proof of priority. Four relevant preprints appeared in July–August
  2026, so concurrent work is possible.
- **Scope.** The results hold in the setting of the report: f is a bimeasurable bijection, (⋆) holds for
  f and f⁻¹, the wandering set has finite positive measure, and GH is taken in the spectral form. Two
  related questions are not addressed: the conservative case ([DDM21, Problem 5.0.2]) and Open Problem
  (2) of the report (structural stability).

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons
Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/