{"id":"33f21690-5c70-4e42-8d1e-3e6a4407aedc","arxiv_id":"2501.11656","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-uniformly expanding maps with additive noise and a fully-positive-Lyapunov stationary measure are asserted to have dense random horseshoes, a random version of Smale's horseshoe.","lead":"This paper claims that a non-uniformly expanding map driven by additive noise must contain dense random horseshoes whenever it has an ergodic stationary measure with all Lyapunov exponents positive. If the proof were completed, it would give the random analogue of Katok's classical deterministic theorem, tying positive expansion rates to symbolic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is unproven: §6.2 obtains a limit for common Young times but never verifies (2.8)–(2.11), and the common times of two renewal processes are not i.i.d. renewals, so the law-of-large-numbers step is unjustified.","rationale":"The reader's weakest_assumption (full support of μ on Y) is a legitimate missing hypothesis: Proposition 4.1 requires the orbit to be ε-dense in Y, which only follows if supp μ = Y. However, the more load-bearing obstruction is in §6.2: even after fixing full support, the construction of the horseshoe is incomplete and, as written, the renewal argument is invalid. The common times of two renewal processes are not i.i.d. renewals, and a common Young time at n_k does not make the interval [n_k,n_{k+1}] a Young time for the shifted system. Therefore conditions (2.9)–(2.11) are unproved. This is a correctness risk in the central claim, not a mere exposition gap. I agree with the reader's REJECT verdict; no verdict change is needed. The full-support issue could be fixed by adding supp μ=Y to (H2), but the §6.2 gap requires a substantially different argument or a major rewrite.","tokens_in":13427,"tokens_out":24337,"duration_ms":251165,"concrete_test":"Attempt to verify condition (2.9) for the times n_k defined in §6.2: using only the fact that n_k and n_{k+1} are Young times for I_i, derive the inclusion f^{n_{k+1}-n_k}_{θ^{n_k}(ω)}(I_i) ⊇ I_0∪I_1. If this fails, test instead the recursive definition m^{i,j}_{k+1}=min{n>m^{i,j}_k: n is a Young time for both balls under θ^{m^{i,j}_k(ω)}}; check whether (6.13) can be replaced by Birkhoff's theorem and whether (2.9)-(2.11) then hold. The paper's current argument does neither.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6.2 aims to build a horseshoe from two balls I_i, I_j having infinitely many common Young times. For Definition 2.1, one must produce times n_k satisfying (2.8) and, crucially, (2.9)–(2.11). The text stops at (6.13), a limit statement for the common times, without deriving (2.9)–(2.11). This is not merely an omitted routine check: (2.9) requires f^{n_{k+1}-n_k}_{θ^{n_k}(ω)}(I_i) ⊇ I_0∪I_1. A Young time for I_i at time n (from time 0) gives a subball of I_i whose full n-step image is the reference set J; it gives no control on the tail map from time n_k to n_{k+1} applied to the whole ball I_i. Thus the return map between consecutive common times is not shown to be a horseshoe map. Moreover, the proof of (6.13) is flawed: the sequence n^{i,j}_k of common times of the two renewal processes {m(I_i)_k} and {m(I_j)_k} is not a renewal process with i.i.d. increments; the waiting times between successive common events depend on the phases of both processes, so Proposition 6.1 (i.i.d. SLLN) does not apply. The claimed equality m^{i,j}_k = n^{i,j}_k is therefore unjustified, as is the 'law of large numbers' (6.13). Consequently Theorem 2.2 is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a notion of random horseshoe (Definition 2.1) and claims in Theorem 2.2 that, for non-uniformly expanding random dynamical systems with additive noise satisfying hypotheses (H1)-(H2), the set of κ-horseshoes is dense. The proof strategy is: (i) establish large deviation estimates for Birkhoff averages and recurrences to the critical set (Section 3); (ii) prove annealed properties of orbits of large balls (Section 4); (iii) introduce Young times and derive tail estimates for their occurrence (Section 5); and (iv) use a combinatorial argument to find two balls with infinitely many common Young times, which are then asserted to yield a random horseshoe (Section 6). The main theorem is presented as a generalization of Katok's theorem to random systems, avoiding the need for periodic orbits.","tokens_in":13776,"tokens_out":4460,"duration_ms":39499,"significance":"If the main theorem were established, it would be a substantial contribution to the random dynamical systems literature, providing a random analogue of Katok's dense horseshoes under only positivity of Lyapunov exponents and additive noise. The paper introduces a useful definition of random horseshoe and attempts to connect Young times to random symbolic dynamics. However, the central proof is incomplete: Section 6.2 stops without verifying the defining conditions of a random horseshoe, and the probabilistic argument for common Young times is flawed. Several auxiliary results are taken from self-authored preprints, making verification difficult. Thus, the claimed theorem is not established, though some of the ancillary estimates (e.g., the large deviation results) may be of independent interest.","major_comments":[{"comment":"The proof stops at the limit statement (6.13) and never verifies the conditions (2.8)-(2.11) of Definition 2.1. In particular, (2.9) requires that f^{n_{k+1}-n_k}_{θ^{n_k}(ω)}(I_i) ⊇ I_0 ∪ I_1 for both i, which is not shown: a common Young time only gives that at time n_k a sub-ball of I_i maps onto the reference set J; it gives no control on the image of the whole ball I_i under the subsequent segment of the composition. Similarly, no sub-balls J(k,ω)_{i,j} satisfying (2.10) and (2.11) are constructed. Therefore Theorem 2.2 is not proven.","section":"Section 6.2, equations (6.9)-(6.13)"},{"comment":"The sequence n^{i,j}_k of common times of two renewal processes is not itself a renewal process with i.i.d. increments. Proposition 6.1, which is invoked to obtain the law of large numbers (6.13), applies only to sums of i.i.d. nonnegative random variables. The waiting times between successive common events of two renewal processes depend on the phases of both processes, and are not i.i.d. Hence the equality (6.11) and the limit (6.13) are unjustified.","section":"Section 6.2, paragraph after (6.10)"},{"comment":"The proof of Proposition 6.1 is flawed: it uses the Central Limit Theorem to conclude an almost-sure limit of the averages of truncated variables. The correct tool is the Strong Law of Large Numbers (or the monotone convergence theorem applied to the SLLN for truncated variables). While the statement of the proposition is true, the error matters because the proposition is the only justification for (6.13), and the sequence to which it is applied is not i.i.d. in the first place.","section":"Section 6.1, Proposition 6.1"},{"comment":"The proof assumes that the orbit of the chosen point (ω,x) under the ergodic measure μ visits every element of a δ-partition of Y and reaches a reference ball Δ_c within a fixed time N. This requires that μ have full support on Y (or at least that the support intersects every partition element). Hypothesis (H2) only postulates the existence of a unique ergodic measure on Y, not that its support is all of Y. Without full support, the Birkhoff ergodic theorem does not imply visits to every partition element, and the time N may be infinite on a set of positive measure. This is a load-bearing gap in Proposition 4.1.","section":"Section 4, Lemma 4.2"},{"comment":"Several key results are taken from self-authored preprints [3] and [12] without proof or even statement. Proposition 5.4 cites [3, Lemma 4.1]; Proposition 5.5 uses [3, Proposition 3.5] and [3, Theorem 3.7]; Proposition 5.7 relies on [12, Proposition 4.3]. Since these results are central to the Young-times estimates, the paper is not self-contained, and the referee cannot verify the main theorem without access to the full proofs of these preprints. At minimum, the relevant statements should be included as lemmas with proofs or clear references to published versions.","section":"Section 5, Propositions 5.4, 5.5, 5.7"}],"minor_comments":[{"comment":"There is a typo: 'an ball' should be 'a ball'.","section":"Section 4, definition of E_J(I,N,ι)"},{"comment":"The text says 'by Proposition 4.2' but should refer to Lemma 4.2.","section":"Section 4, proof of Proposition 4.1"},{"comment":"The condition (2.8) involves E[n_0], but n_0 is not defined before taking the limit, and the meaning of its expectation is unclear. This should be clarified.","section":"Definition 2.1"},{"comment":"The notation H^n_i is used before the set is defined, and the phrase 'there exists indices i(n,ω) and j(n,ω)' should be 'there exist indices'.","section":"Section 6.2, equations (6.3)-(6.8)"},{"comment":"The paper mixes notation S^1 and X (e.g., Definition 5.1 and Proposition 5.2 mention S^1 while the setup is X ⊂ R^n). This should be made consistent.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper's heavy reliance on two self-authored preprints ([3] and [12]) for the core Young-times results makes verification difficult. The main theorem is not established because the final construction never verifies the defining conditions of a random horseshoe, and the common-times argument rests on an invalid application of Proposition 6.1. These are load-bearing errors that cannot be fixed by minor edits. The manuscript would need a substantially revised proof to be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — here's my take on 2501.11656. The punchline: the paper's main theorem is a natural random analogue of Katok's horseshoe-density result, and I think the statement is probably true — but as written the proof stops one step short, and the last step is not a routine detail.\n\nWhat is genuinely good: the authors replace the 'predominantly expanding' assumptions of their earlier paper with control of modified Young times, and they get large deviations in the multidimensional non-uniformly expanding setting via the Nagaev method. The definition of random horseshoe is clean, and the strategy — use ergodicity to make large balls eventually cover a reference set, then look at common Young times of two balls — is plausible and clearly explained. Hypotheses (H1)-(H2) are mild, and the paper is honest about citing self-authored preprints for several key propositions.\n\nThe soft spots, in order of severity.\n\nFirst, the construction in Section 6.2 really does stop at (6.13). It never verifies conditions (2.8)-(2.11) of Definition 2.1. In particular, (2.9) asks that the image of the whole ball I_i under the tail map between consecutive common times contains I0 ∪ I1. A Young time for the ball I_i only controls a smaller subball that maps onto the reference set; it gives no statement about the whole ball I_i. The text does not show how to pass from subballs to whole balls, or how the hyperbolic times between common Young times assemble into the required horseshoe maps. That is the load-bearing gap.\n\nSecond, the law of large numbers for the common times (6.13) is not justified. The common times of two independent renewal processes are not themselves a renewal process: the waiting times between successive common events depend on the current phases of both processes, so the increments are not i.i.d. Proposition 6.1 therefore does not apply, and the equality m^{i,j}_k = n^{i,j}_k in (6.11) is asserted, not proved. This is fixable in principle with a two-dimensional renewal argument, but it isn't in the manuscript.\n\nThird, Lemma 4.2 uses full support of the ergodic stationary measure on the invariant component Y, but Hypothesis (H2) only assumes existence and uniqueness of an ergodic measure. The full-support assumption needs to be stated or derived.\n\nMinor things: the proof of Proposition 6.1 invokes the CLT where the SLLN gives the result directly; and the paper depends heavily on [3] and [12] for Propositions 5.4, 5.5, and 5.7. Self-citation is not a flaw by itself, but it compounds the incompleteness because the central gap sits downstream of those unverified results.\n\nWho gets value from this: researchers working on random dynamical systems and smooth ergodic theory, especially those interested in Katok-type results with noise. The paper deserves a serious referee — the claim is important and the approach is credible — but it needs a substantial revision, not copyediting. I would not cite the main theorem until the construction is completed.","headline":"A plausible and important claim, but the horseshoe construction stops at a limit statement and never proves the defining conditions; deserves peer review, not acceptance as is.","tokens_in":14328,"tokens_out":3398,"would_cite":false,"duration_ms":34733,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37H20","37B10","37D25","60J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Additive noise plus positive Lyapunov exponents force random horseshoes in every open set.","keywords":["random horseshoe","positive Lyapunov exponents","symbolic dynamics","random dynamical systems","additive noise","non-uniformly expanding maps","Young times","large deviations"],"falsifier":"Exhibit a map satisfying (H1)--(H2) whose ergodic stationary measure $\\mu$ has an open hole $A$ inside its component $Y$, and check whether two balls in $A$ can still form a $\\kappa$-horseshoe; if they cannot, the full-support assumption is necessary and the proof's unstated assumption is exposed.","tokens_in":13207,"feed_emoji":"🎲","tokens_out":16262,"duration_ms":161488,"temperature":0.7,"pith_summary":"This paper proves that a non-uniformly expanding map driven by additive noise must contain random horseshoes, and that these horseshoes are dense: every open set in the ergodic component contains two disjoint balls whose forward images repeatedly cover a common reference set with uniform expansion, yielding a set in one-to-one correspondence with binary sequences. The result matters because random systems typically have no periodic orbits, so the classical deterministic route to horseshoes via hyperbolic periodic points is unavailable; the authors replace it with a purely probabilistic construction. The proof combines ergodicity, large-deviation estimates for Lyapunov averages and critical-set recurrences, and modified Young times that mark when a small neighbourhood is expanded diffeomorphically onto a fixed reference ball $J$. In short, the theorem transfers the classical dense-horseshoe picture from deterministic diffeomorphisms to dissipative random systems under essentially no assumptions beyond diffusive additive noise and an ergodic stationary measure with $\\lambda = \\int \\log \\|df^{-1}\\|\\,d\\mu < 0$, i.e. all Lyapunov exponents positive.","feed_headline":"Noise and expansion force random horseshoes everywhere","feed_subtitle":"Every open region of a noisy expanding map is shown to hide a two-symbol symbolic subsystem.","key_machinery":"The central objects are the random $\\kappa$-horseshoe of Definition 2.1 and the modified Young times of Section 5. A Young time for $(\\omega,x)$ is a hyperbolic time followed by an event in which some sub-ball of a large ball is mapped into a fixed reference set $J$ while staying away from the critical set; Proposition 5.4 shows that at such a time $f^n_\\omega$ expands a small neighbourhood of $x$ diffeomorphically onto $J$ with uniform derivative and distortion control. The argument also rests on the annealed large-deviation estimates of Proposition 3.1, obtained by the spectral method for Markov chains applied to the family of transfer operators $P_\\theta$, and on Proposition 4.1, which uses ergodicity to show that balls of size at least $\\varepsilon$ have probability at least $\\rho$ of eventually covering $J$. The horseshoe construction takes $M$ small balls of size about $|J|/M$, forms independent sequences of their Young return times, and uses an inclusion-exclusion bound to force two of those sequences to intersect infinitely often with positive lower density; Proposition 6.3 converts the intersection density into the required upper bound on $n_k(\\omega)/k$.","core_discovery":"The central claim is Theorem 2.2: under Hypotheses (H1)--(H2), there exists $\\kappa > 0$ such that the set of $\\kappa$-horseshoes is dense. In Definition 2.1, a pair of disjoint balls $(I_0,I_1)$ is a random $\\kappa$-horseshoe when there exist random times $n_k(\\omega)$ with $\\limsup_{k\\to\\infty} n_k(\\omega)/k = \\mathbb{E}[n_0]$, the image of each ball under $f^{n_{k+1}-n_k}_{\\theta^{n_k}\\omega}$ contains $I_0 \\cup I_1$, and sub-balls inside each $I_i$ are mapped diffeomorphically onto $I_j$ with inverse derivative norm below $\\kappa^{-1}$. The set $P(\\omega)$ of points whose orbit at every $n_k$ lies in $I_0 \\cup I_1$ is then hyperbolic and in one-to-one correspondence with the full shift on two symbols. The proof establishes this via annealed large-deviation bounds for $\\log \\|df^{-1}\\|$ and for close approaches to the critical set, an 'eventually onto' statement for large balls with uniform probability, and exponential tail control on the Young times that mark good expansion events. The underlying hypotheses are only that the critical set is a nice codimension-one manifold with power-law derivative bounds and that an ergodic stationary measure $\\mu$ exists with $\\lambda < 0$, meaning all Lyapunov exponents are positive.","pith_inferences":["Beyond the paper, the proof's reliance on the annealed transition kernel suggests that the additive form of the noise is not essential; any absolutely continuous perturbation with full support and controlled density should yield the same density of random horseshoes, with different large-deviation constants.","Beyond the paper, the full-support condition flagged in the proof may be automatic for diffusive additive noise: a noise density bounded away from zero should make the transition kernel irreducible on components, forcing $\\operatorname{supp}\\mu = Y$ and closing the gap in Hypothesis (H2).","Beyond the paper, the binary coding of $P(\\omega)$ suggests that quenched positive entropy and a random symbolic dynamics description follow from the same construction; the expected return time $\\mathbb{E}[n_0]$ in Definition 2.1 is the natural rate to extract, though the paper does not compute it."],"forward_implications":["Every open set inside the ergodic component contains two disjoint balls forming a random $\\kappa$-horseshoe, so the symbolic subsystem $P(\\omega)$ is present at arbitrarily small scales.","The set $P(\\omega)$ is hyperbolic and in one-to-one correspondence with binary sequences, giving a random analogue of the classical horseshoe that does not rely on periodic orbits.","The expected waiting time for a Young time of a ball of size $|I|$ is of order $\\log(1/|I|)$; consequently smaller balls wait only logarithmically longer to be expanded onto the reference set.","The large-deviation estimates hold uniformly over the state space outside a Lebesgue-exceptional set, so the annealed control of Lyapunov averages and critical-set recurrences is exponentially strong."],"supporting_citations":[{"why":"This supplies the previous random-horseshoe construction for predominantly expanding circle maps, which this paper generalizes to the non-uniformly expanding setting.","marker":"[12]"},{"why":"This introduced Young times and showed how to use them to build random Young towers; the modified Young times in this paper control the frequency of expansion events.","marker":"[3]"},{"why":"This provides the spectral method for Markov chains, adapted here to prove the large-deviation estimates for Lyapunov averages and critical-set recurrences.","marker":"[18]"},{"why":"This supplies the spectral perturbation result used to show that the transfer operators have an isolated leading eigenvalue, giving exponential tail estimates.","marker":"[4]"},{"why":"This provides the hyperbolic-time lemma that lets a hyperbolic time expand a small neighbourhood diffeomorphically onto a fixed ball, used in Proposition 5.2.","marker":"[2]"},{"why":"This gives BV estimates for stationary measures used to show that the logarithm of the inverse derivative norm is integrable, so the Lyapunov exponent $\\lambda$ is well defined.","marker":"[14]"}],"fun_headline_variants":["Noise plus expansion yields dense random horseshoes","Additive noise guarantees random horseshoes are dense","Dense random horseshoes from noisy expansion","Expansion with noise forces dense symbolic subsystems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the ergodic stationary measure to visit every open subset of its ergodic component within a fixed time, so a reference ball can be placed inside any prescribed open set $A$; this full-support property is used in Lemma 4.2 but not stated in Hypothesis (H2).","fun_headline_variants_meta":{"raw":{"variants":["Noise plus expansion yields dense random horseshoes","Additive noise guarantees random horseshoes are dense","Dense random horseshoes from noisy expansion","Expansion with noise forces dense symbolic subsystems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000703,"raw_usage":{"total_tokens":3127,"prompt_tokens":859,"completion_tokens":2268,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":2217}},"tokens_in":475,"tokens_out":2268,"duration_ms":16154,"temperature":1.0,"reasoning_tokens":2217,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:00:53.588718+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a map satisfying (H1)--(H2) whose ergodic stationary measure $\\mu$ has an open hole $A$ inside its component $Y$, and check whether two balls in $A$ can still form a $\\kappa$-horseshoe; if they cannot, the full-support assumption is necessary and the proof's unstated assumption is exposed.","supporting_citations":[{"cited_title":"Horseshoes for a class of nonuniformly expanding random dynamical systems on the circle","cited_arxiv_id":"2304.03685","evidence_quote":"This supplies the previous random-horseshoe construction for predominantly expanding circle maps, which this paper generalizes to the non-uniformly expanding setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This provides the spectral method for Markov chains, adapted here to prove the large-deviation estimates for Lyapunov averages and critical-set recurrences."},{"cited_title":"Limit theorems in dynamical systems using the spectral method","cited_arxiv_id":null,"evidence_quote":"This supplies the spectral perturbation result used to show that the transfer operators have an isolated leading eigenvalue, giving exponential tail estimates."},{"cited_title":"Alves and Helder Vilarinho","cited_arxiv_id":null,"evidence_quote":"This provides the hyperbolic-time lemma that lets a hyperbolic time expand a small neighbourhood diffeomorphically onto a fixed ball, used in Proposition 5.2."},{"cited_title":"Positive Lyapunov expone nt by a random perturbation","cited_arxiv_id":null,"evidence_quote":"This gives BV estimates for stationary measures used to show that the logarithm of the inverse derivative norm is integrable, so the Lyapunov exponent $\\lambda$ is well defined."}],"review_version":1}