{"id":"5819f46c-902b-4f20-83c9-797bbb9b1349","arxiv_id":"1908.03177","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Small perturbations of partially hyperbolic toral automorphisms with dense center foliation and a smooth center foliation are smoothly leaf-conjugate, and any bi-Hölder conjugacy is smooth.","lead":"This paper proves that for small perturbations of certain partially hyperbolic toral automorphisms, if the perturbed system has a smooth center foliation, then it is smoothly leaf-conjugate to the linear automorphism. This is a rigidity result in smooth dynamics: weak equivalence, such as a bi-Hölder conjugacy, is upgraded to a smooth one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.1's suspension leaf-conjugacy is not well-defined; this gap affects the proof of Theorem 1.1.","rationale":"The reader correctly identifies the heavy reliance on the normal form theorems from [KS16]. I agree that [KS16, Theorem 2.3(4)] is load-bearing. However, a more concrete and less external problem appears inside the proof of Theorem 1.1: the suspension map used to convert translations in the center direction into commuting maps is not well-defined for a mere leaf conjugacy. This is not a complaint about an unproved prior theorem; it is a checkable statement about the current argument. The gap does not affect Theorem 1.3, where the conjugacy is exact, but it undermines Proposition 5.1 and hence Theorem 1.1 as written. The result may still be true, and the construction may be repairable, so I would not reject outright; the paper should be conditional on fixing this step. The reader's weakest assumption should therefore be updated to include this internal well-definedness issue.","tokens_in":19875,"tokens_out":33724,"duration_ms":354833,"concrete_test":"Check well-definedness directly: for a leaf conjugacy h, \\tilde h([x,1]) and \\tilde h([f(x),0]) coincide in M_L only if h(f(x)) = L(h(x)). Use the paper's canonical choice H_c=0 from Section 5.1: for any small perturbation f = L + F, \\bar h_c(f(x)) - L_c \\bar h_c(x) = F_c(x), which is generically nonzero. Compute the two suspension images for any point with F_c(x) ≠ 0; they are not identified in M_L, so \\tilde h is not a map. The concern is settled if the authors replace \\tilde h by a well-defined suspension construction, for example by using the flow phi^t = h^{-1} L^t h on its own mapping torus, and re-derive Proposition 5.1 with that construction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Proposition 5.1, the authors pass to the mapping tori M_f and M_L and claim that a leaf conjugacy h induces a leaf conjugacy \\tilde h:M_f→M_L by \\tilde h(x,t)=(h(x),t). For an exact conjugacy this descends, because [x,1]=[f(x),0] and [h(x),1]=[Lh(x),0]=[h(f(x)),0]. For a leaf conjugacy, h(f(x)) is only required to lie on the center leaf of L(h(x)), not to equal L(h(x)). With the choice H_c=0 made in Section 5.1, the center components differ by F_c(x): (h(f(x)))_c = L_c x_c + F_c(x), while (L(h(x)))_c = L_c x_c. Hence [Lh(x),0] and [h(f(x)),0] are distinct points of M_L, so \\tilde h is not a well-defined homeomorphism. Consequently phi^t = \\tilde h^{-1} L^t \\tilde h, g = \\tilde h^{-1} \\tilde H_v \\tilde h, and the extensions F^t, G needed for Theorem 2.1 are not defined. This is a concrete gap in the central mechanism of Theorem 1.1; the reader's concern about Theorem 2.3 is related but less specific. Theorem 1.3 is not affected, since there h is an exact conjugacy.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies C^∞ diffeomorphisms f of T^d that are C^1-close to a partially hyperbolic toral automorphism L, under the assumption that the center foliation of f is smooth. Theorem 1.1 asserts that if L is diagonalizable over C and has dense center foliation, then every sufficiently small perturbation with smooth center foliation is C^∞ leaf-conjugate to L. Theorem 1.3 asserts that a bi-Hölder conjugacy between L and a volume-preserving perturbation with sufficiently regular center foliation is necessarily C^∞, and Corollary 1.5 gives a symplectic version. The proofs use nonstationary normal forms for contracting foliations, center holonomies, suspension constructions, exponential mixing, and Journé's lemma. Further results are derived for totally irreducible automorphisms with two-dimensional center, using work of Rodriguez Hertz and Avila-Viana.","tokens_in":20151,"tokens_out":18155,"duration_ms":189578,"significance":"If correct, the paper establishes strong rigidity phenomena for partially hyperbolic toral diffeomorphisms: smoothness of the center foliation forces smoothness of the leaf conjugacy, and a weak bi-Hölder conjugacy automatically improves to a smooth conjugacy. These conclusions are substantially stronger than previously known results and are obtained by a coherent combination of normal-form theory and dynamical arguments. The entropy argument in Section 3.3 and the use of normal forms along stable and unstable foliations are elegant. However, there is a load-bearing gap in the suspension construction of Proposition 5.1 that affects the proof of the main Theorem 1.1; this gap must be repaired before the paper's central claims can be accepted.","major_comments":[{"comment":"The map \\tilde h:M_f→M_L defined by \\tilde h(x,t)=(h(x),t) is not well-defined when h is only a leaf conjugacy. In M_f one has [x,1]=[f(x),0], so well-definedness would require [h(f(x)),0]=[h(x),1], equivalently h(f(x))=Lh(x). For a leaf conjugacy one only has h(f(x))∈W^c(Lh(x)); with the choice \\bar H_c=0 made in Section 5.1, the center components differ by F_c(x)=(h(f(x)))_c-(Lh(x))_c, which need not vanish. Consequently the maps \\tilde h, φ^t, g_v, F^t, and G used in the rest of Proposition 5.1 are not defined, and the proof of Theorem 1.1 loses its central mechanism. Theorem 1.3 is not affected because there h is an exact conjugacy.","section":"Section 5, Proposition 5.1, first paragraph"},{"comment":"The exponential mixing estimate used to bound ⟨D^ℓ_c L^{-k}_c(G_c∘f^k), η_ε⟩ requires that the pair (G_c∘f^k, D^ℓ_c η_ε) have zero mean, or that the distributional calculus from [FKSp13, Section 8] explicitly cancels the mean term. Even if η is a zero-average test function, η_ε is a smooth convolution approximation and D^ℓ_c η_ε need not have zero average; the text does not show that G_c has zero mean. This gap can likely be repaired by centering D^ℓ_c η_ε or by spelling out the formal adjoint of the leafwise derivative, but as written the exponential decay claimed in (3.12) is not justified and the proof of smoothness of H_c in Theorem 1.3 is incomplete at this point.","section":"Section 3.5, equations (3.10)–(3.13)"}],"minor_comments":[{"comment":"The abstract states the main result without the hypotheses that L is diagonalizable over C and has dense center foliation; these hypotheses should be included in the abstract for accuracy.","section":"Abstract"},{"comment":"The claim that a leaf conjugacy can always be chosen smooth along W^c is nontrivial, and the verification that the specific map defined via (3.3)–(3.4) with \\bar h_c(x)=x_c is indeed a homeomorphism and a leaf conjugacy is omitted. A few sentences of justification would help.","section":"Section 5.1, construction of h"},{"comment":"In the statements of Theorem 1.7 and Corollary 1.8, the symbols W^s, W^u, E^s, E^u are used both for the linear automorphism and for the perturbation; the intended system should be specified each time.","section":"Theorems 1.7 and 1.8"},{"comment":"The assertion that \\hat h_x = h^s|_{W^s(x)} under the natural identification is stated as easy to see; this identification is important for concluding that h^s is uniformly C^∞ and should be written out in more detail.","section":"Section 5.2, final paragraph"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the suspension construction in Proposition 5.1; the gap appears fixable (for example, by interpolating along center leaves to define a genuine leaf conjugacy between the mapping tori), but it is central to Theorem 1.1. The paper also relies heavily on the authors' own prior results, some of which are in preprint form; the editor may wish to verify the status and validity of these references before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. The paper is worth reading and the normal-form machinery is used in a genuinely new way, but Theorem 1.1 has a concrete gap in the proof of Proposition 5.1. The stress-test note is correct: for a leaf conjugacy h, the map \\tilde h(x,t)=(h(x),t) does not descend to the mapping tori. A leaf conjugacy only gives h(f(x)) ∈ W^c(L(h(x))), not h(f(x)) = L(h(x)). Under the identifications (x,1)∼(f(x),0) and (z,1)∼(Lz,0), well-definedness would require exactly that equality. So the suspension lifts φ^t, the extensions F^t, G, and the normal-form argument built on them are not actually defined. This is not a peripheral technicality; it is the central mechanism of Theorem 1.1. The reader's conditional verdict was fair but did not pinpoint this issue.\n\nWhat the paper does well: Theorem 1.3 is a solid result, and the proof through Section 3 is coherent. Smoothness along stable/unstable leaves via normal forms and holonomy limits is genuinely clever, and the sub-exponential growth plus distributional estimates in Section 3.5 are plausible, though compressed. The consequences for symplectic perturbations and the two-dimensional center case (Corollary 1.5, Theorem 1.7/1.8) are attractive and go beyond prior T^3 or compact-center results. The authors are also honest about relying on [KS16] and [K19]; that is not a flaw by itself, but the hypotheses of Theorem 2.3(4) are not checked in detail, and the paper is not self-contained at exactly the point where it matters.\n\nMinor issues: Remark 1.6 and the Section 5.1 claim that a leaf conjugacy can be chosen smooth along W^c are unsupported; they should either be proved or explicitly labeled as conjectural. Section 3.5's estimates are abbreviated enough that a referee will need to fill in details.\n\nWho this is for: researchers in smooth dynamics and rigidity, especially those working on partially hyperbolic systems and normal forms. I would not cite Theorem 1.1 in its current form, but I would not desk-reject the paper. Send it to a good referee, and ask specifically whether Proposition 5.1 can be repaired — perhaps by choosing a center-section for the suspension or by working with a genuine conjugacy on the torus. If the suspension issue is fixable, this is a strong paper; if not, Theorem 1.3 and the corollaries still stand on their own.","headline":"A strong, interesting paper whose central Theorem 1.1 currently rests on a real suspension-gap in Proposition 5.1; Theorem 1.3 and the corollaries look sound, so referee it but do not accept as-is.","tokens_in":20699,"tokens_out":3654,"would_cite":false,"duration_ms":35952,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D30","37C85","37C05","37D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Smoothness of the center foliation forces a smooth leaf conjugacy to the linear automorphism.","keywords":["partially hyperbolic diffeomorphisms","center foliation","rigidity","normal forms","leaf conjugacy","toral automorphisms","Lyapunov exponents","symplectic perturbations"],"falsifier":"Compute, for a concrete small perturbation of a diagonalizable $L$ with dense center foliation and smooth center foliation, the normal-form coordinates $\\Phi_x$ of Theorem 2.3 and a center holonomy $H_{x,y}: W^s(x) \\to W^s(y)$ for $y \\in W^c(x)$; Proposition 5.1 asserts $\\Phi_y \\circ H_{x,y} \\circ \\Phi_x^{-1}$ is always in $P_{L^s}$. Finding a single pair for which this map includes a non-resonant monomial, or a homomorphism $\\eta_x: E^s \\to \\bar{P}_x$ that is not smooth, would refute Proposition 5.1 and hence Theorems 1.1 and 1.3.","tokens_in":19675,"feed_emoji":"🔁","tokens_out":13194,"duration_ms":126702,"temperature":0.7,"pith_summary":"The paper proves a rigidity theorem for small perturbations of partially hyperbolic linear maps of the torus: once the perturbation's center foliation is smooth, the whole system is smoothly equivalent to the linear model. Specifically, a $C^\\infty$ diffeomorphism close to a diagonalizable partially hyperbolic toral automorphism with dense center foliation is $C^\\infty$ leaf-conjugate to that automorphism whenever its center foliation is $C^\\infty$. It also upgrades a merely bi-Hölder conjugacy to a smooth one under a regularity condition on the center foliation, and derives further corollaries for symplectic perturbations and for automorphisms with two-dimensional center. The interest is that 'weak equivalence implies strong equivalence' holds here for a single system, not only for higher-rank group actions, and the mechanism is normal-form rigidity on the contracting foliations rather than hyperbolicity.","feed_headline":"Smooth center foliations force smooth equivalence to linear maps","feed_subtitle":"For small perturbations of toral automorphisms, smoothness of the center foliation lifts both leaf conjugacies and conjugacies to C∞.","key_machinery":"The load-bearing object is the non-stationary normal form for the contracting stable and unstable foliations (Theorem 2.3): each leaf is assigned $C^\\infty$ coordinates $\\Phi_x$ in which the restricted dynamics becomes a sub-resonance-generated polynomial—a polynomial map built only from monomials compatible with the resonance relations among the contraction eigenvalues—taking values in a finite-dimensional Lie group $P_A$. The key identity is that every center holonomy $H_{x,y}$ between stable leaves is carried by these coordinates to a polynomial in $P_A$; this is proved for true conjugacies via a suspension-flow argument (Proposition 3.2) and for leaf conjugacies by replacing $f$ with a flow adjusted along center leaves (Proposition 5.1). Once holonomies are polynomial, density of the linear center foliation yields a continuous homomorphism from $E^s$ into a Lie group of polynomial diffeomorphisms, whose automatic smoothness gives uniform smoothness of the conjugacy along stable and unstable leaves. Global smoothness is assembled from uniform smoothness along the three foliations by the standard regularity lemma for functions of several variables and, for the center component, by distributional estimates powered by exponential mixing.","core_discovery":"The central discovery is that smoothness of the center foliation acts as a rigidity trigger. For a partially hyperbolic automorphism $L$ of the torus that is diagonalizable over $\\mathbb{C}$ and has dense center foliation, any $C^\\infty$ diffeomorphism $f$ sufficiently $C^1$ close to $L$ with $C^\\infty$ center foliation is $C^\\infty$ leaf-conjugate to $L$ (Theorem 1.1). The paper further shows that when $f$ is bi-Hölder conjugate to $L$ and its center foliation is $C^r$ for $r>r(L)$, the conjugacy itself is $C^\\infty$ (Theorem 1.3). In the symplectic case, any bi-Hölder conjugacy between such $f$ and $L$ must be smooth (Corollary 1.5). For totally irreducible $L$ with two-dimensional center, several dynamical conditions—vanishing or equality of center Lyapunov exponents, non-accessibility, topological conjugacy, joint integrability of the stable and unstable foliations—are shown to be equivalent to smooth conjugacy (Theorem 1.7).","pith_inferences":["The normal-form argument does not use compact center leaves or one-dimensional stable and unstable bundles, so a natural extension is to partially hyperbolic diffeomorphisms on other homogeneous spaces, or to higher-rank abelian actions, where the same 'dense center leaves plus polynomial holonomies' mechanism should produce smooth rigidity whenever the relevant normal-form theorem holds.","The threshold $r(L)$, defined solely by ratios of stable and unstable eigenvalue moduli, suggests a quantitative finite-regularity version: for fixed $L$, the guaranteed differentiability of the leaf conjugacy should grow linearly in the regularity $r$ of the center foliation, so one could state explicit $C^q$ bounds rather than the $C^r$ or $C^{r-\\varepsilon}$ version noted in Remark 1.2.","The use of exponential mixing to regularize the center component hints that the mixing rate controls the gain: automorphisms with slower mixing would still yield $C^r$ conjugacies but the smoothness upgrade might fail, giving a testable family of examples where the distributional estimate is sharp."],"forward_implications":["If Theorem 1.1 holds as stated, every $C^\\infty$ perturbation of a diagonalizable partially hyperbolic toral automorphism with dense center foliation and smooth center foliation belongs to a single smooth leaf-conjugacy class, so classification of such perturbations reduces to classification of smooth center foliations.","Under Theorem 1.3, the Hölder modulus of the conjugacy is irrelevant: the only obstruction to smoothness is the regularity of the center foliation, with the explicit threshold $r>r(L)$.","Corollary 1.5 gives a 'weak implies strong' rigidity result for individual symplectic maps, not just higher-rank actions: bi-Hölder conjugacy to the linear automorphism automatically upgrades to $C^\\infty$ conjugacy.","Theorem 1.7 and Corollary 1.8 imply that for totally irreducible two-dimensional-center automorphisms, several ostensibly soft dynamical properties (non-accessibility, equal center exponents, topological conjugacy, joint integrability) are all equivalent to smooth conjugacy; in the symplectic setting this is a dichotomy: either the system is non-uniformly hyperbolic or it is smoothly conjugate to "],"supporting_citations":[{"why":"Establishes Theorem 2.3, the normal-form coordinates on contracting foliations with smooth leaf dependence and the Lie-group homogeneous structure that makes all center holonomies polynomial.","marker":"[KS16]"},{"why":"Provides structural stability of partially hyperbolic systems: dynamical coherence, the initial leaf conjugacy, and the existence of the invariant foliations the proof works with.","marker":"[HPS77]"},{"why":"Prove the basic non-stationary normal form theorem for contracting extensions, yielding the polynomial normal forms and the group $P_A$ used throughout.","marker":"[GuKt98, Gu02]"},{"why":"Provides the regularity lemma for functions of several variables used to globalize smoothness from the three foliations.","marker":"[J88]"},{"why":"Supplies exponential mixing of ergodic toral automorphisms on Hölder functions, which powers the distributional estimates for the center component.","marker":"[L82]"},{"why":"Provides the distributional regularity criterion used to conclude smoothness of the center component from boundedness of its distributional derivatives against Hölder test functions.","marker":"[FKSp13]"},{"why":"Gives stable ergodicity for totally irreducible automorphisms with two-dimensional center and the bi-Hölder conjugacy produced when a perturbation is not accessible, used in Theorem 1.7.","marker":"[RH05]"},{"why":"Provides the theorem that accessible perturbations of such automorphisms have distinct center Lyapunov exponents, giving the implication (2) implies (3) in Theorem 1.7.","marker":"[AV10]"},{"why":"Gives uniqueness of the measure of maximal entropy for ergodic toral automorphisms, used to show the bi-Hölder conjugacy is volume-preserving.","marker":"[B67]"}],"fun_headline_variants":["Smooth center foliation forces smooth equivalence","Center foliation smoothness implies rigidity","Smooth center leaves yield smooth conjugacy","Rigidity from smooth center foliations","Smooth center foliation: rigidity trigger"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that every center holonomy, being only $C^{d(A)+\\varepsilon}$ along leaves, is sent by normal-form coordinates to a sub-resonance polynomial; if some center holonomy lacks this leaf regularity, or the normal-form coordinates do not have the asserted homogeneous structure, the smoothness argument along stable and unstable leaves collapses.","fun_headline_variants_meta":{"raw":{"variants":["Smooth center foliation forces smooth equivalence","Center foliation smoothness implies rigidity","Smooth center leaves yield smooth conjugacy","Rigidity from smooth center foliations","Smooth center foliation: rigidity trigger"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00037,"raw_usage":{"total_tokens":1962,"prompt_tokens":906,"completion_tokens":1056,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1005}},"tokens_in":522,"tokens_out":1056,"duration_ms":10298,"temperature":1.0,"reasoning_tokens":1005,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:21:40.182214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete small perturbation of a diagonalizable $L$ with dense center foliation and smooth center foliation, the normal-form coordinates $\\Phi_x$ of Theorem 2.3 and a center holonomy $H_{x,y}: W^s(x) \\to W^s(y)$ for $y \\in W^c(x)$; Proposition 5.1 asserts $\\Phi_y \\circ H_{x,y} \\circ \\Phi_x^{-1}$ is always in $P_{L^s}$. Finding a single pair for which this map includes a non-resonant monomial, or a homomorphism $\\eta_x: E^s \\to \\bar{P}_x$ that is not smooth, would refute Proposition 5.1 and hence Theorems 1.1 and 1.3.","supporting_citations":[],"review_version":1}