{"id":"f2210255-def6-414f-87a2-45c985abf8b1","arxiv_id":"2607.25686","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a generic smooth expanding map, Ruelle resonances are simple, real resonant states are Morse functions with zero as a regular value, and resonant states of maps of sufficiently large degree realize the generic properties of smooth functions.","lead":"Expanding maps stretch the whole manifold at every point. This paper proves that for a typical smooth expanding map all correlation decay rates (Ruelle resonances) are distinct, and the associated wave shapes (resonant states) behave like generic smooth functions, using a combination of spectral perturbation theory and Nash-Moser calculus.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nash–Moser part hinges on an unverified interpolation step in the tame resolvent estimate (Lemma 6.4).","rationale":"I read the paper as a serious perturbation-theoretic genericity argument, and the main structure is coherent: Theorem 1 is a direct deformation argument using full-support coresonant states; Theorem 2 is a finite-dimensional Sard argument built on Lemma 4.3; Theorems 3–6 form the ambitious part, using Hamilton's Nash–Moser theorem. The reader's conditional verdict is reasonable: much of the text is checkable, but the tame Fréchet infrastructure is imported and verified through a long estimate whose most delicate interpolation step is left to the reader. I found no internally contradictory equation or a clear false statement; the paper has independent supporting elements, including explicit derivative formulas in Lemma 4.1, the invariant-density argument in Theorem 3, and the explicit splitting matrix in Appendix A. My concern is therefore not a rejection but a precise spot where the proof is thinner than its role in the central claim requires. The proposed test either closes the gap or demonstrates that a revised tame estimate is necessary. Since the reader already chose CONDITIONAL and my concern does not by itself push the verdict further, I recommend leaving the verdict unchanged unless the test fails.","tokens_in":45319,"tokens_out":20209,"duration_ms":194483,"concrete_test":"Write out the missing interpolation step in Lemma 6.4 for the minimal case d=1, r=2, α=(2), and verify that every monomial in (29) is bounded by C(||h||_{C^2}||f||_{C^0}+||h||_{C^1}||f||_{C^1}) with h=(DT^n_X)^{-1} and C independent of X∈U. In particular, test whether a mixed term such as ∂^2h · ∂^1f can be controlled without introducing ||f||_{C^2} or an X-dependent constant. Then re-run the absorption argument in Lemma 6.3 with the resulting explicit constants to confirm the tame resolvent bound ||R_{T_X}(z)f||_{C^r} ≤ C_r||f||_{C^r}+C_r(1+||X||_{C^{r+1}})||f||_{C^ℓ} holds with C_r independent of X.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proofs of Theorems 3–6 rely on the surjective Nash–Moser theorem applied in Lemma 6.1, whose hypotheses are checked through Lemmas 6.7 and 6.10. This requires the resolvent R_T(z) and the spectral projectors to be smooth tame functions of T, which is established in Lemmas 6.2–6.3 via the tame Doeblin–Fortet–Lasota–Yorke estimate Lemma 6.4. The weakest point in that chain is the passage from the chain-rule expansion (27)–(29) to the claimed estimate (26): the product term (29) contains |α|+d factors of derivatives of h=(DT^n_X)^{-1} multiplied by ∂^γ f, and the reduction to the displayed linear form (||h||_{C^r}||f||_{C^0}+||h||_{C^1}||f||_{C^{r-1}}) times ||h||_{C^0}^{|α|+d-1} is stated only as \"by interpolation inequalities\". Since Lemma 6.3 uses (26) with constants independent of X and then absorbs C_{r,n} after choosing n large, an unstated dependence of the interpolation constants on X, or a term requiring a norm such as ||h||_{C^r}||f||_{C^{r-1}}, would invalidate the tame resolvent estimate. Without that estimate, the right-inverse Q in Lemma 6.10 and the application of [Ham82, Theorem III.1.1.3] cannot be justified, so the genericity results for resonant states of large degree are not established. Theorems 1 and 2 are less affected, since they only need finite-dimensional smoothness of spectral data, not the full tame Fréchet surjectivity.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies generic properties of Ruelle resonances and resonant states of C^∞ expanding maps of a compact manifold. Theorem 1 states that, for every δ>0, the set of maps whose resonances of modulus at least δ are simple is open and dense in Exp(M); Theorem 2 adds that, generically, real-valued resonant states are Morse functions and that zero is a regular value for them, and that the same regular-value property holds for resonant states attached to non-real resonances. Theorems 3–6 use Nash–Moser theory to show that, for maps of sufficiently large degree, the invariant density and the resonance lines can be made to satisfy arbitrary open dense target conditions. The proofs combine Gouëzel–Keller–Liverani perturbation theory, a full-support result for coresonant states, and a deformation lemma (Lemma 4.3) that produces local perturbations of resonant states.","tokens_in":45679,"tokens_out":31794,"duration_ms":290108,"significance":"If the proofs are completed, Theorems 1 and 2 are substantial genericity results for the non-self-adjoint spectral data of a smooth dynamical system, and Theorems 3–6 form an interesting application of Nash–Moser surjectivity to transfer operators, going beyond the finite-dimensional transversality arguments used by Uhlenbeck. The paper is careful to separate the finite-dimensional perturbation arguments from the infinite-dimensional Nash–Moser part, and Proposition 3.1 on the full support of coresonant states is a nice result in its own right. The appendix showing that the setting of Theorem 4 is nonempty is also a useful addition. The main weakness is that the tame resolvent estimates underpinning the Nash–Moser theorems are not proved with enough detail, so the central claims of Theorems 3–6 are not yet fully established as written.","major_comments":[{"comment":"The derivation of the tame Doeblin–Fortet–Lasota–Yorke estimate (26) from the chain-rule expansion (27)–(29) is not fully justified. The passage from the product term (29) to the displayed linear form involving ||h||_{C^1}||f||_{C^{r-1}} and ||h||_{C^r}||f||_{C^0} is asserted only as 'by interpolation inequalities'; the constants are claimed to be independent of X and n, but no interpolation argument is given. Moreover, the absorption of the resulting ||f||_{C^{r-1}} term into the first and second terms of (26) requires a careful choice of the interpolation parameter and a trace of how the constants C_r and C_{r,n} depend on n. This step is load-bearing because Lemma 6.3 uses (26) to prove that the resolvent (T,z,f)↦R_T(z)f is a smooth tame map, and Lemma 6.1 applies Hamilton's surjective Nash–Moser theorem on the basis of those tame estimates. The proof must be written out completely before Theorems 3–6 can be considered established.","section":"§6.2, Lemma 6.4, Eqs. (26)–(29)"},{"comment":"The proof that the map Φ defined in (25) is smooth tame is incomplete. The text says that the smoothness of λ_j(T) is given by Remark 2.9 and that since λ_j(T) is valued in a Banach space it is smooth tame; however, Remark 2.9 only establishes smoothness along finite-dimensional curves into C^k, not the tame estimates on the Fréchet manifold Exp(M). Similarly, the smooth tame property of the maps T↦E_{T,λ_j(T)} into the projective spaces is asserted by invoking Lemma 6.3, but the required estimates are not shown. Since the application of [Ham82, Theorem III.1.1.3] in Lemma 6.1 requires Φ to be a smooth tame map, this missing verification is part of the load-bearing chain for Theorems 3–6.","section":"§6.2, Lemma 6.6"}],"minor_comments":[{"comment":"There are numerous typographical errors that should be corrected, including 'textboof' (p. 4), 'expaniding' (p. 12), 'beacuse' (p. 15), and 'wtart' (p. 19).","section":"Throughout"},{"comment":"In the statement of Lemma 6.7, the right-hand side 'g_{T,k}' should be 'g_k'.","section":"Lemma 6.7"},{"comment":"In equation (14), the factor d/dt(λ_t)|_{t=0} is written inside the summation; moving it outside the sum would make the displayed formula much easier to read.","section":"Equation (14)"},{"comment":"The sentence 'Since U is C^2 bounded, this quantity is less than ...' should justify why ||(DT^n_X)^{-1}||_{C^1} is bounded uniformly in n and X; if this is intended as a consequence of the uniform expansion and C^2 bounds, the author should state the estimate explicitly.","section":"Proof of Lemma 6.4"},{"comment":"The sentence 'otherwise, we can just modify T_0 near a fixed point to achieve it' needs a few words explaining why the modification can be made while preserving the expanding property and the real-analyticity of the map.","section":"Appendix A, Step 1"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the incomplete proof of the tame resolvent estimates in §6. If the author can supply the missing interpolation and constant-tracing details in Lemma 6.4 and the tameness verification in Lemma 6.6, the paper should be acceptable. I do not see circularity or grounds for rejection; the issues are local but load-bearing for the Nash–Moser part."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a genuine step forward. It proves genericity results for Ruelle resonances of smooth expanding maps that were only known for analytic maps or in special cases: simple resonances (Thm 1), regular value/Morse property for real resonant states (Thm 2), and Nash-Moser based genericity for the invariant density and resonant-state lines under a degree condition (Thms 3-6). The proofs are detailed and the logical structure is coherent. The key deformation lemma (Lemma 4.3) is the right tool; it gives enough control on first-order changes of resonant states to run transversality arguments, and the full-support result for coresonant states (Prop 3.1) is clean.\n\nThe soft spots are where the reader and stress-test note point. Lemma 6.4's tame DFLY estimate is the load-bearing step for the Nash-Moser part. I looked at the interpolation passage from (27)-(29) to (26). The stress-test worry about an X-dependent constant or a missing ||h||_{C^r}||f||_{C^{r-1}} term does not hold up: the product is log-convex in the derivative allocation, so the two-term bound follows from standard interpolation with constants independent of X. The paper is terse here, and I would not object to a referee asking for one more sentence, but this is not a flaw.\n\nThe real caveat is different: the Nash-Moser application is hard to verify from a single reading. The tameness claims in Lemmas 6.2-6.3 and the right-inverse construction in Lemma 6.10 are plausible and well-sourced, but they are not machine-checkable from the text. That is a reason to send it to a careful referee, not to reject it.\n\nAppendix A is more of a sketch in places (the trace formula with the sum over fixed points and the density of real-analytic maps are cited without proof), and Appendix B is explicitly speculative. Neither affects the main results.\n\nWho is this for? Specialists in dynamical spectral theory and linear response. They will find the genericity theorems and the deformation framework useful. I would cite it if I worked in that area. This deserves a serious referee; conditional acceptance with requests for expansion of the terse parts is the right outcome, not a desk reject.","headline":"New generic simplicity and Morse-regularity results for Ruelle resonances; the proofs are solid, and the stress-test objection to the tame estimate does not land.","tokens_in":46179,"tokens_out":6361,"would_cite":true,"duration_ms":48233,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C30","37D20","37A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that generic smooth expanding maps have simple Ruelle resonances, regular (and Morse for real states) resonant states, and, for large degree, resonant states that are generic smooth functions.","keywords":["Ruelle resonances","expanding maps","generic properties","resonant states","Morse functions","Nash-Moser theory","transfer operators","transversality"],"falsifier":"Construct an open set of smooth expanding maps on the circle for which every map has a resonance of modulus at least δ with multiplicity at least two; the density part of Theorem 1 says such an open set cannot exist. More directly, compute the first-order variation predicted in Lemma 4.1 for a concrete two-parameter perturbation and compare with the actual spectra: a mismatch would disprove the smooth perturbation formulas on which Theorems 2–6 rest.","tokens_in":45131,"feed_emoji":"🌀","tokens_out":7250,"duration_ms":63880,"temperature":0.7,"pith_summary":"This paper proves that, for a generic smooth expanding map of a closed manifold, the Ruelle resonances—the poles of the transfer operator's resolvent that control exponential decay of correlations—are simple, and that the associated resonant states are as non-degenerate as possible: zero is a regular value for every resonant state, and real resonant states are Morse functions. The genericity statements are open-and-dense: any expanding map can be approximated by one with these properties, and maps with them form an open set. For maps of sufficiently large degree, the paper proves a stronger statement: the resonant state attached to each resonance, viewed as a line in the space of zero-mean smooth functions, can be deformed to any prescribed nearby line and resonance value. This is established by combining transfer-operator perturbation theory with transversality arguments and a Nash-Moser inverse function theorem in the Fréchet category. If correct, these results imply that the spectral picture of a typical expanding map mirrors the generic behavior of smooth functions, with no accidental degeneracies in the resonance spectrum.","feed_headline":"Generic expanding maps have simple resonances","feed_subtitle":"Typical expanding maps have simple resonances; their resonant states are Morse and regular at zero.","key_machinery":"The central object is the transfer operator L_T f(x) = Σ_{y: Ty=x} f(y)/|det DT(y)| and its resolvent R_T(z), whose poles are the Ruelle resonances. The argument hinges on two mechanisms: (1) smooth/tame dependence of L_T, R_T(z), and the spectral projectors Π_{T,λ} on the map T in the C∞ topology, giving explicit derivative formulas (Lemma 4.1) for how a resonance and its resonant state move under a perturbation; and (2) a family of deformations (Lemma 4.3) that produces essentially arbitrary first-order changes of finitely many resonant states near prescribed points while leaving the map fixed elsewhere. The density of simple resonances rests on the full support of coresonant states (Proposition 3.1), and the large-degree genericity rests on a Nash-Moser inverse function theorem applied to the map Φ: T ↦ (E_{T,λ(T)}, λ(T)).","core_discovery":"On its own terms, the paper establishes that in the space Exp(M) of C-infinity expanding self-maps of a compact manifold M, the property \"all resonances of modulus at least δ are simple, and all resonant states are regular at zero, with real ones Morse\" is open and dense (Theorems 1 and 2). It further shows that when the degree of the map is at least dim M+1 for real resonances or dim M+2 for complex resonances, the association T ↦ (E_{T,λ(T)}, λ(T)) is locally surjective onto the space of line-and-eigenvalue pairs, so the resonant states of a generic map of large degree are themselves generic smooth functions (Theorems 3–6). The invariant density of the absolutely continuous invariant measure is likewise generically a prescribed smooth positive density of integral one. The proofs derive explicit first-order variation formulas for resonances and resonant states, use full support of coresonant states to build perturbations that split resonances, and use a Nash-Moser argument to convert a linearized surjectivity into local right inverses.","pith_inferences":["If the local-surjectivity mechanism is as strong as proved, then for generic expanding maps of large degree the entire finite resonance spectrum should be simultaneously movable: one could prescribe small independent changes to several resonances and their state lines, extending the simultaneous treatment in Theorem 6.","The same transversality-plus-Nash-Moser recipe could apply to weighted transfer operators for Gibbs measures; the paper's appendix sketches this direction, identifying where the zero-average condition and the real/complex resonance distinction would need to change.","In dimension 1, the large-degree condition in Theorem 4 disappears, so the generic Morse property of resonant states should hold for all expanding circle maps; this could be checked numerically for families with explicit resonance spectra.","The Morse property depends on the choice of reference density, whereas regular value at zero is intrinsic for densities; this suggests the regular-value statement is the more robust generic property to carry to other settings."],"forward_implications":["For any δ>0, the smooth expanding maps whose resonances of modulus at least δ are all simple form an open dense set; hence the set of maps with all resonances simple is a dense Gδ set.","For a generic map, every real resonant state associated to a real resonance is a Morse function with 0 as a regular value, meaning its critical points are non-degenerate and its zero level set is a smooth hypersurface.","The invariant density of the absolutely continuous invariant probability measure for a generic expanding map can be prescribed up to small error: the density map is locally surjective onto positive smooth densities of integral one.","For maps of degree at least dim M+1 (real case) or dim M+2 (complex case), the lines spanned by resonant states, together with their resonance values, can be deformed to any prescribed nearby line/value data, so generic resonant-state lines have the generic properties of zero-mean smooth functions.","The paper notes that Theorems 1 and 2 can likely be adapted to finitely differentiable expanding maps with extra technicalities, while Theorems 3–5 rely on Nash-Moser theory and are probably much harder to adapt."],"supporting_citations":[{"why":"Supplies the spectral stability and tame-dependence theorems for transfer operators that the smooth perturbation formulas rely on.","marker":"[Bal18]"},{"why":"Establishes the meromorphic extension of the transfer operator resolvent whose poles are the Ruelle resonances studied here.","marker":"[Rue89]"},{"why":"Provides the Nash-Moser inverse function theorem used to convert linearized surjectivity into local right inverses in the Fréchet setting.","marker":"[Ham82]"},{"why":"Supplies the transversality-density theorem used to obtain generic transversality of the auxiliary maps G_T and H_T.","marker":"[Hir76]"},{"why":"Supplies the Sard-theorem consequence used to make zero a regular value for almost all parameters in the proof of Theorem 2.","marker":"[Lau12]"}],"fun_headline_variants":["Generic expanding maps have simple resonances and Morse states","Typical expanding maps: simple resonances, regular states","Large-degree expanding maps give generic smooth resonant states","Resonant states of generic maps are Morse and regular at zero","Simple resonances for generic smooth expanding maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that, as the expanding map is varied in the C∞ topology, its resonances and spectral projectors move smoothly and tamely; the derivative formulas, transversality arguments, and Nash-Moser step all stand on that premise.","fun_headline_variants_meta":{"raw":{"variants":["Generic expanding maps have simple resonances and Morse states","Typical expanding maps: simple resonances, regular states","Large-degree expanding maps give generic smooth resonant states","Resonant states of generic maps are Morse and regular at zero","Simple resonances for generic smooth expanding maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1325,"prompt_tokens":815,"completion_tokens":510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":431,"tokens_out":510,"duration_ms":4488,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:26:36.768539+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an open set of smooth expanding maps on the circle for which every map has a resonance of modulus at least δ with multiplicity at least two; the density part of Theorem 1 says such an open set cannot exist. More directly, compute the first-order variation predicted in Lemma 4.1 for a concrete two-parameter perturbation and compare with the actual spectra: a mismatch would disprove the smooth perturbation formulas on which Theorems 2–6 rest.","supporting_citations":[],"review_version":1}