{"id":"8a978f9d-32a2-4e54-a3c2-24372b418c3b","arxiv_id":"2411.19665","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For volume-preserving partially hyperbolic Anosov diffeomorphisms on T^3, smooth rigidity is equivalent to the stable and center distributions exceeding their critical Hölder exponents; otherwise these distributions have fractal graphs.","lead":"The paper proves a sharp dichotomy for partially hyperbolic maps on the 3-torus: an invariant distribution is either smoother than a critical Hölder threshold and forces the map to be smoothly rigid, or its graph has fractal dimension. It introduces a non-fractal invariance principle that converts high regularity of invariant sections into holonomy invariance and smoothness along unstable foliations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.6's C^2 straightening of W^c is unjustified since W^c is only C^1; losing Lipschitz control along W^s threatens the proof that W^c is C^1, a key step in Theorem 1.12.","rationale":"The reader correctly identified two proof gaps: the unjustified use of W^u-minimality in Theorem 3.2 and the C^2 straightening of W^c in Proposition 6.6. I agree that both are real. However, I judge the C^2-straightening issue as more load-bearing for the paper's central claim. The minimality gap is confined to the general compact-manifold formulation of Theorem 3.2; for the headline T^3 results, W^u-minimality is known for Anosov diffeomorphisms of T^3 via Franks–Manning conjugacy and is explicitly available in Proposition 4.4. In contrast, Proposition 6.6 is used directly in Theorem 1.8 to conclude that W^c is a C^1 foliation from the C^1-along-W^s regularity of Ec, and Theorem 1.12's 'if' direction relies on this conclusion to match s-periodic data and obtain smooth rigidity. The flaw is specific: a C^2 chart rectifying a C^1 foliation would imply the foliation is C^2, which contradicts the known existence of C^∞ partially hyperbolic Anosov diffeomorphisms on T^3 with W^c not C^2. With only a C^1 chart, the Lipschitz property of the transformed vector field along the stable leaves is not preserved, so the Gronwall argument in Proposition 6.6 does not go through as written. The core non-fractal invariance principle (Theorem 3.1) appears structurally sound in itself, and the T^3 applications may be repairable, but as written the regularity bootstrap for Ec carries an unjustified assumption. This does not change the reader's CONDITIONAL verdict; it reinforces the need for repair before the general and rigidity claims can be accepted.","tokens_in":23732,"tokens_out":21074,"duration_ms":176488,"concrete_test":"Construct the rectifying chart in Proposition 6.6 explicitly: fix a C^2 transversal to W^c, project along W^c leaves to define the horizontal coordinate, and check whether the chart is C^2 or only C^1. If the chart is only C^1, verify whether the transformed center vector field e remains uniformly Lipschitz along the C^2 stable leaves; if the Lipschitz bound fails, the displayed Gronwall estimate in the proof of Proposition 6.6 is not justified. A successful test either exhibits the missing Lipschitz bound or exposes the need for an additional hypothesis on W^c.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is in Proposition 6.6 (§6.4), where the proof states: 'Since W^s is a C^2 subfoliation of W^cs [42], we can reduce the problem to R^2 via a C^2 coordinate change such that x=(0,0), W^c(x)={0}×R and the stable leaves are parallel to the x-axis.' This C^2 straightening is not available in general: W^c is only a C^1 foliation and may fail to be C^2 even for C^∞ f (Proposition 4.4, citing [48]). A C^2 chart whose level sets are the W^c leaves would force W^c to be C^2 by the implicit function theorem. With only a C^1 chart, the derivative of the chart along the C^2 stable leaves is merely continuous, not Lipschitz, so the proof's Gronwall inequality |dζ/dt| = |e(z(t))-e(y(t))| ≤ L|ζ(t)| may fail: the transformed vector field e can lose Lipschitz regularity along W^s. This matters because Proposition 6.6 is the step that upgrades 'Ec uniformly C^1 along W^s' to 'W^c is a C^1 foliation', used in Theorem 1.8 and in Theorem 1.12 to obtain the same s-periodic data via [28] and conclude C^∞ rigidity. The minimality gap in Theorem 3.2 flagged by the reader is real but does not threaten the main T^3 theorems, where W^u-minimality follows from Franks–Manning; the C^2-straightening gap does threaten the central rigidity statement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a non-fractal invariance principle for invariant sections of partially hyperbolic bundle maps: if the fiber expansion is weaker than the base unstable direction, then any continuous invariant section either has fractal graph (dim_B graph >= dim M + 1 - A) or is uniformly C^r along W^u and invariant under the unstable holonomy, assuming W^u is minimal. The authors apply this to partially hyperbolic Anosov diffeomorphisms on T^3, obtaining sharp thresholds: theta_s^+-Hölder continuity of E^s implies E^s is C^1 and E^s ⊕ E^u is jointly integrable; theta_c^+-Hölder continuity of E^c along W^s implies W^c is a C^1 foliation; and failure of these gives fractal graphs. These are used to prove a C^∞ rigidity criterion for f ∈ Diff^∞_vol(T^3): f is C^∞-rigid iff both E^s and E^c are theta_s^+- and theta_c^+-Hölder continuous. The paper also states stable C^1/fractal conjectures.","tokens_in":24130,"tokens_out":14581,"duration_ms":112730,"significance":"The results are novel and potentially significant: they establish a link between fractal geometry and rigidity in partially hyperbolic dynamics, give essentially sharp bootstrap thresholds for Hölder regularity of invariant distributions, and show that Hölder regularity alone can imply smooth rigidity in a setting where such conclusions are rare. The core analytic lemmas (Lemmas 5.1, 5.2, 5.3, 5.5, 5.6) are coherent, and the T^3 applications rely on known minimality and rigidity theorems rather than on circular use of the target results. The proposed conjectures are well motivated. However, two load-bearing gaps (see major comments) prevent the paper from being fully validated as written.","major_comments":[{"comment":"The proof of Theorem 3.2 states 'Since W^u is minimal, by Theorem 3.1' and then applies Theorem 3.1, but the theorem's stated hypotheses only assume topological transitivity. Minimality of W^u is not derived from the stated assumptions and does not follow in general for arbitrary compact manifolds; this is load-bearing because Theorem 3.1 explicitly requires W^u-minimality. The T^3 applications are protected because W^u-minimality is known there (Proposition 4.4), but the general theorem as stated is not proved. The authors should either add W^u-minimality as an explicit hypothesis or prove that topological transitivity together with the bunching assumptions implies it, or restrict the statement to the cases where it is available.","section":"§6.1, Theorem 3.2"},{"comment":"The proof reduces to R^2 via 'a C^2 coordinate change such that x=(0,0), W^c(x)={0}×R and the stable leaves are parallel to the x-axis.' Such a coordinate change would force the leaf W^c(x) to be a C^2 curve, hence the center foliation to be C^2 at x, but Proposition 4.4 states that W^c may fail to be C^2 even for C^∞ diffeomorphisms. With only a C^1 chart, the paper does not justify the Lipschitz estimate |e(z(t))-e(y(t))| ≤ L|ζ(t)| used in the Gronwall argument, since the transformed vector field e may lose Lipschitz regularity along W^s under a non-C^2 coordinate change. This step upgrades 'E^c uniformly C^1 along W^s' to 'W^c is a C^1 foliation' and is used in Theorems 1.8 and 1.12, so it is load-bearing. The issue is likely repairable using a C^1 coordinate chart that is smooth along the C^2 stable foliation, but the proof as written is not justified.","section":"§6.4, Proposition 6.6"}],"minor_comments":[{"comment":"The proof phrase 'Let f ∈ Diff^2(T^3) be C^1-close to a volume-preserving diffeomorphism g' is confusing because the theorem is stated for all f ∈ Diff^2_vol(T^3); while every such f is C^1-close to itself, the intended verification that volume preservation on T^3 implies the hypotheses of Theorem 3.2 should be stated directly, for example using λ_s + λ_c + λ_u = 0.","section":"§6.2, proof of Theorem 1.5"},{"comment":"Lemma 6.5 is ambiguously stated: 'Df is Hölder continuous on a set of full Lebesgue measure' should require a uniform Hölder constant on that set and an explicit argument that the derivative extends to a Hölder function on U; as written, the claim is not generally justified. The application in Proposition 6.6 appears to have such uniform estimates, but they should be spelled out.","section":"§6.4, Lemma 6.5"}],"recommendation":"major_revision","confidential_remarks":"The two major gaps are local and appear repairable, and the main T^3 theorems are likely sound. The general Theorem 3.2 is overclaimed as stated, and the C^2-straightening step in Proposition 6.6 needs a substantive fix. I would not recommend rejection, but the revisions are substantial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Best quick read: this is the first paper I know that turns fractal geometry into a tool for partially hyperbolic rigidity. The non-fractal invariance principle (Thm 2.1/3.1) is the real contribution — if an invariant section is not fractal, it has to be uniformly C^r along W^u and holonomy-invariant. That is a clean and substantial statement, and the machinery in Section 5 (obstruction functions, box-dimension lower bounds) is coherent and mostly convincing. The T^3 applications giving sharp thresholds θs and θc and the rigidity criterion in Theorem 1.12 are the payoff, and they are new.\n\nThe weak spots are two.\n\nFirst, and this is the one that matters: Proposition 6.6 claims to prove W^c is C^1 by straightening W^s and W^c simultaneously with a C^2 coordinate change. But W^c is only C^1 in general, even for C^∞ f (Prop 4.4, citing [48]). A C^2 chart with W^c as level sets would force W^c to be C^2. With only a C^1 chart, the vector field e loses Lipschitz control along W^s, so the Gronwall estimate |dζ/dt| ≤ L|ζ| is not justified. Since Prop 6.6 is the step that gets the s-periodic data used in Theorem 1.12, this gap directly threatens the main rigidity statement. It looks repairable — maybe one can use a C^1 chart that is C^2 along W^s and argue more carefully — but as written it is a genuine hole.\n\nSecond, Theorem 3.2 assumes only topological transitivity but the proof invokes minimality of W^u. On T^3 that is known, so Theorems 1.5 and 1.8 survive. But the general statement as written is not proved.\n\nThe rest of the argument — the Livsic step in Appendix A, the use of [24,28,29,30] for periodic data — is fine, and I do not see circularity. The paper ships no code or formal proofs, but the mathematics is concrete enough.\n\nWho should read it: anyone working on regularity of invariant distributions or smooth rigidity. It deserves a serious referee, but I would send it back for a major revision to fix Prop 6.6 before I would trust Theorem 1.12.","headline":"A genuinely new fractal-vs-smoothness principle with sharp T^3 applications, but the proof that the center foliation is C^1 has a regularity gap that needs fixing.","tokens_in":24640,"tokens_out":6096,"would_cite":true,"duration_ms":43464,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D30","37D20","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Hölder thresholds decide smooth rigidity on the 3-torus","keywords":["partially hyperbolic dynamics","invariant distributions","fractal graphs","Hölder regularity","rigidity","Anosov diffeomorphisms","box dimension","non-fractal invariance principle"],"falsifier":"Take a topologically transitive but not $W^u$-minimal partially hyperbolic $f$ on a compact manifold, any bundle map $F$ satisfying the fiber-expansion bound, and any continuous $F$-invariant section $\\Phi$ that is $\\alpha^+$-Hölder along $W^u$ but not invariant under the unstable holonomy; Theorem 3.1 would be false, and the failure would show up as $\\Delta_\\delta$ vanishing along a dense set of $W^u$-leaves while remaining positive on a wandering leaf. A more numerical check: in the family of Example 6.10, estimate the lower box dimension of $\\mathrm{Graph}(E^s)$; the theorem predicts $\\dim_B > 3$ whenever $E^s$ is not $C^1$, so an open parameter interval with $E^s \\notin C^1$ and $\\dim_B = 3$ would refute the sharp dichotomy.","tokens_in":23505,"feed_emoji":"📐","tokens_out":17867,"duration_ms":125005,"temperature":0.7,"pith_summary":"This paper establishes a dichotomy for invariant distributions of partially hyperbolic systems: a continuous invariant section of a bundle map that expands fibers more slowly than the base's unstable direction either has a fractal graph or is smooth along the unstable foliation and invariant under unstable holonomy. Applied to partially hyperbolic Anosov diffeomorphisms of the 3-torus with contracting center, this yields a sharp phase transition: if the stable or center distribution exceeds its critical Hölder exponent, it jumps to $C^1$ and forces rigidity, while below that threshold its graph has lower box dimension strictly larger than the manifold dimension. The main rigidity theorem says that a $C^\\infty$ volume-preserving partially hyperbolic Anosov diffeomorphism of $\\mathbb{T}^3$ is $C^\\infty$-conjugate to a linear automorphism if and only if both the stable and center distributions beat their critical exponents. The authors propose that generically every invariant distribution is either stably $C^1$ or stably fractal, with center distributions typically fractal.","feed_headline":"Hölder thresholds decide smooth rigidity on the 3-torus","feed_subtitle":"Crossing critical Hölder thresholds yields a smooth conjugacy; missing them yields fractal graphs.","key_machinery":"The load-bearing mechanism is the obstruction function $\\Delta_\\delta(x) = \\sup_{t \\in W^u_\\delta(x)} d(\\Phi(t), \\gamma_x(t))$, where $\\gamma_x(t)$ is the point obtained by pushing $\\Phi(x)$ along the unstable holonomy of the bundle map $F$. Minimality of $W^u$ makes $\\Delta_\\delta$ either identically zero or uniformly bounded away from zero: if $\\Delta_\\delta(x) = 0$ at one point, iterating backward spreads vanishing along the dense unstable leaf. If $\\Phi$ is $\\alpha^+$-Hölder and $\\alpha > \\alpha(k,x)$, backward contraction estimates force $\\Delta_\\delta(x)=0$, hence $\\Phi$ is holonomy-invariant and $C^r$ along $W^u$. If instead $\\Delta_\\delta$ is uniformly positive, the same backward estimates yield oscillations of $\\Phi$ at every scale whose packing count gives the lower bound $\\dim_B \\mathrm{Graph}(\\Phi) \\geq \\dim M + 1 - A$. The applications identify $F$ with the Grassmannian bundle over $E^{cs}$ induced by $Df$, so that $E^s$ and $E^c$ become invariant sections; the critical exponents $\\alpha_s$, $\\alpha_c$ are shown in Appendix A to coincide with the pinching coefficients $\\theta_s$, $\\theta_c$, and a standard regularity lemma upgrades $C^1$ along $W^u$ and $W^{cs}$ to global $C^1$.","core_discovery":"The central discovery is the non-fractal invariance principle (Theorems 2.1 and 3.1). Let $f$ be a $C^r$ partially hyperbolic diffeomorphism with minimal unstable foliation $W^u$, and let $F$ be a bundle map over $f$ that expands fibers more weakly than $f$ along $W^u$. Then every continuous $F$-invariant section $\\Phi$ either has a fractal graph, with lower box dimension at least $\\dim M + 1 - A$, or is uniformly $C^r$ along $W^u$ and invariant under the unstable holonomy $h_F^u$. The quantitative version uses $\\alpha = \\inf \\alpha(k,x)$ as the critical Hölder exponent: $\\Phi$ being $\\alpha^+$-Hölder along $W^u$ forces holonomy invariance and $C^r$ leaves, while failure implies the box-dimension lower bound. For a $C^2$ volume-preserving partially hyperbolic Anosov diffeomorphism of $\\mathbb{T}^3$ with contracting center, the stable distribution $E^s$ is $\\theta_s^-$-Hölder in general; if it is $\\theta_s^+$-Hölder, then it is $C^1$ and $E^s \\oplus E^u$ is jointly integrable, and otherwise the graph of $E^s$ has lower box dimension greater than 3. The analogous statement for the center is that $E^c$ being $\\theta_c^+$-Hölder along $W^s$ forces $W^c$ to be a $C^1$ foliation, and otherwise $E^c$ has a fractal graph. Theorem 1.12 concludes that $f \\in \\mathrm{Diff}^\\infty_{\\mathrm{vol}}(\\mathbb{T}^3)$ is $C^\\infty$-rigid exactly when $E^s$ is $\\theta_s^+$-Hölder and $E^c$ is $\\theta_c^+$-Hölder.","pith_inferences":["Editorial extension: the lower bound $\\dim_B \\mathrm{Graph}(\\Phi) \\geq \\dim M + 1 - A$ suggests a pressure-based dimension formula for dynamically invariant sections, as in the classical theory of self-affine sets; deriving such a formula in the non-conformal, non-algebraic setting could be a route to proving the stable-fractal part of Conjecture 1.14.","Editorial inference: the dichotomy implies that invariant distributions that fail to be $C^1$ are not merely non-smooth but carry positive dimensional excess, so finite-resolution numerical approximations of such distributions should show resolution-dependent complexity; one could test this on the one-parameter family of Example 6.10 by estimating the box dimension of $E^s$ as the parameter varies","The paper leaves open whether a $C^1$ foliation can coexist with a fractal tangent distribution, as it notes explicitly; a natural test is to construct a partially hyperbolic $f$ where $W^c$ is $C^1$ but $E^c$ is not, and measure the graph dimension to separate foliation regularity from distribution regularity.","The proof of the general statement in Section 6.1 uses minimality of $W^u$ after the stated hypotheses assume only topological transitivity; on $\\mathbb{T}^3$ the missing step is supplied by known results, but a reader extending the theorems to other manifolds would need to close this gap or add minimality as an explicit hypothesis."],"forward_implications":["For every $C^2$ volume-preserving partially hyperbolic Anosov diffeomorphism of $\\mathbb{T}^3$ with contracting center, the stable distribution $E^s$ is either $C^1$ with $E^s \\oplus E^u$ jointly integrable, or its graph has lower box dimension strictly larger than 3; the non-$C^1$ case is generic outside a codimension-$\\infty$ subset.","For the center direction, the same alternative holds: $E^c$ is either uniformly $C^1$ along $W^s$ and $W^c$ is a $C^1$ foliation, or the graph of $E^c$ is fractal; in particular, any pathological (non-absolutely-continuous) center foliation forces $E^c$ to have a fractal graph.","For $C^\\infty$ volume-preserving $f$ on $\\mathbb{T}^3$, $C^\\infty$-rigidity—smooth conjugacy to a linear Anosov automorphism—is equivalent to the two sharp Hölder conditions $E^s$ being $\\theta_s^+$-Hölder and $E^c$ being $\\theta_c^+$-Hölder; this is a bootstrap from mild Hölder regularity to full smoothness.","The non-fractal invariance principle applies beyond $\\mathbb{T}^3$: under the bunching and minimality hypotheses of Theorems 3.2 and 3.3, the fractal-or-smooth alternative holds for invariant distributions of partially hyperbolic Anosov systems on general compact manifolds, and the authors state it for sections of general fiber bundles.","The paper's Conjecture 1.14 predicts a $C^1$-open, $C^r$-dense partition of partially hyperbolic diffeomorphisms into stably $C^1$ and stably fractal distributions, with the center distribution stably fractal whenever the stable and unstable distributions are both nontrivial."],"supporting_citations":[{"why":"Supplies the bunching results that make $E^{cs}$ and $E^u$ $C^1$, so the Grassmannian bundle construction and the $C^1$ unstable holonomy are available.","marker":"[37]"},{"why":"Provides the stable-manifold theorem for partially hyperbolic systems, unique integrability of $E^s$ and $E^u$, and the $C^r$ leaf and subfoliation structure used throughout.","marker":"[42]"},{"why":"Supplies the regularity lemma that upgrades $C^1$ regularity along two transverse foliations into global $C^1$ regularity, the key bootstrap step.","marker":"[50]"},{"why":"Characterizes joint integrability of $E^s \\oplus E^u$ by matching c-periodic data, used to derive the c-part of the rigidity conclusion.","marker":"[24]"},{"why":"Relates a Lipschitz center foliation to s-periodic data, used in the proof of Theorem 1.12 to obtain the same s-periodic data as the linear model.","marker":"[28]"},{"why":"Proves that matching full periodic data implies $C^\\infty$ conjugacy on the 3-torus, the final rigidity step.","marker":"[29, 30]"},{"why":"Supplies the cocycle rigidity theorem used in Appendix A to prove $\\theta_s = \\alpha_s$ and $\\theta_c = \\alpha_c$, making the critical exponents concrete.","marker":"[51]"},{"why":"Gives topological conjugacy of Anosov diffeomorphisms on tori to linear automorphisms and minimality of $W^u$ on $\\mathbb{T}^3$, supplying the global framework and the minimality hypothesis.","marker":"[23, 58]"},{"why":"Shows that joint integrability of $E^s \\oplus E^u$ implies minimality of $W^s$, needed to run the center-direction argument in Theorem 1.12.","marker":"[66]"}],"fun_headline_variants":["Hölder jump forces smooth leaves; miss it, get fractals","Sharp Hölder threshold splits smooth rigidity from fractals","Excess Hölder regularity implies smooth conjugacy on T^3","Fractal or smooth: invariant distributions obey Hölder rule","Non-fractal invariance principle yields rigidity or fractals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dichotomy depends on the unstable foliation $W^u$ being minimal so that vanishing of the obstruction function at one point propagates globally, a hypothesis verified on $\\mathbb{T}^3$ but not proved under the weaker topological-transitivity assumptions stated in the general theorems.","fun_headline_variants_meta":{"raw":{"variants":["Hölder jump forces smooth leaves; miss it, get fractals","Sharp Hölder threshold splits smooth rigidity from fractals","Excess Hölder regularity implies smooth conjugacy on T^3","Fractal or smooth: invariant distributions obey Hölder rule","Non-fractal invariance principle yields rigidity or fractals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00055,"raw_usage":{"total_tokens":2796,"prompt_tokens":1287,"completion_tokens":1509,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":903,"completion_tokens_details":{"reasoning_tokens":1423}},"tokens_in":903,"tokens_out":1509,"duration_ms":10161,"temperature":1.0,"reasoning_tokens":1423,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T06:00:25.544101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a topologically transitive but not $W^u$-minimal partially hyperbolic $f$ on a compact manifold, any bundle map $F$ satisfying the fiber-expansion bound, and any continuous $F$-invariant section $\\Phi$ that is $\\alpha^+$-Hölder along $W^u$ but not invariant under the unstable holonomy; Theorem 3.1 would be false, and the failure would show up as $\\Delta_\\delta$ vanishing along a dense set of $W^u$-leaves while remaining positive on a wandering leaf. A more numerical check: in the family of Example 6.10, estimate the lower box dimension of $\\mathrm{Graph}(E^s)$; the theorem predicts $\\dim_B > 3$ whenever $E^s$ is not $C^1$, so an open parameter interval with $E^s \\notin C^1$ and $\\dim_B = 3$ would refute the sharp dichotomy.","supporting_citations":[{"cited_title":"Hasselblatt","cited_arxiv_id":null,"evidence_quote":"Supplies the bunching results that make $E^{cs}$ and $E^u$ $C^1$, so the Grassmannian bundle construction and the $C^1$ unstable holonomy are available."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the stable-manifold theorem for partially hyperbolic systems, unique integrability of $E^s$ and $E^u$, and the $C^r$ leaf and subfoliation structure used throughout."},{"cited_title":"Journ´ e","cited_arxiv_id":null,"evidence_quote":"Supplies the regularity lemma that upgrades $C^1$ regularity along two transverse foliations into global $C^1$ regularity, the key bootstrap step."},{"cited_title":"Gan and Y","cited_arxiv_id":null,"evidence_quote":"Characterizes joint integrability of $E^s \\oplus E^u$ by matching c-periodic data, used to derive the c-part of the rigidity conclusion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Relates a Lipschitz center foliation to s-periodic data, used in the proof of Theorem 1.12 to obtain the same s-periodic data as the linear model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cocycle rigidity theorem used in Appendix A to prove $\\theta_s = \\alpha_s$ and $\\theta_c = \\alpha_c$, making the critical exponents concrete."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that joint integrability of $E^s \\oplus E^u$ implies minimality of $W^s$, needed to run the center-direction argument in Theorem 1.12."}],"review_version":1}