Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles

El Hadji Yaya Tall

REVIEW 3 major objections 5 minor 22 references

Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that for mixed Markov quasi-periodic cocycles, the top Lyapunov exponent is a real-analytic function of the transition probabilities under primitivity, cycle non-resonance, and a simple top exponent.

desk verdict Solid irreducible case, but the reducible induction rests on an unproved measurable contraction estimate; needs a real proof before the main theorem is complete. read the letter →

arxiv 2608.01569 v1 pith:HHNCXOPT submitted 2026-08-03 math.DS

EL Hadji Yaya Tall This is my paper · ORCID

classification math.DS MSC 37H1537A3060J10

keywords LyapunovexponentsMarkovquasi-periodiccocyclesanalyticdependenceonparameterstransitionprobabilitiesforwardtransferoperatorprojectivecontractioncyclenon-resonancemeasurableinvariantsubbundles

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the top Lyapunov exponent of a mixed Markov quasi-periodic cocycle is a real-analytic function of the transition probabilities. In such a cocycle, a finite-state Markov chain chooses, at each step, a translation of a torus and a matrix-valued function evaluated at the moving torus point; the Lyapunov exponent measures the exponential growth rate of the resulting matrix products. The main theorem shows that under three hypotheses at a reference matrix — primitivity, a cycle non-resonance condition on the translations, and simplicity of the top exponent — the exponent extends to a holomorphic function on a complex neighborhood of the transition matrix, so in particular it is real-analytic near that matrix. The result needs no global irreducibility assumption, and it extends the previously known Bernoulli random quasi-periodic analyticity to Markov-dependent randomness, the natural model when the random choices carry memory. Why this matters: analyticity turns the exponent into a locally rigid object, giving uniform control of all its derivatives with respect to the probabilities and strong perturbation stability for nearby stochastic matrices.

What carries the argument

The argument is carried by three mechanisms. (1) The forward Markov transfer operator $T_Z$ on functions of the present state, the torus point, and a projective direction, defined by $(T_Z\Phi)(i,t,v)=\sum_j z_{ij}\Phi(j,t+\theta_i,A_i(t)v)$. Its $n$-th iterate is a finite path sum whose coefficients are monomials in the complexified transition entries, so holomorphy in $Z$ is explicit. (2) The conditional projective contraction coefficients $\hat K_n(\alpha,P)$, which average the $\alpha$-th power of the projective distance contraction uniformly over the initial Markov state, the initial torus point, and the initial projective directions; submultiplicativity and a uniform averaged-growth lemma yield exponential decay, which makes the logarithmic increments of the transfer iterates locally Cauchy. (3) The cycle non-resonance condition, which the paper proves is equivalent to ergodicity of the Markov-torus skew product; this uniqueness of the stationary torus marginal (Haar measure) is what lets the contraction argument close. In the reducible case, measurable invariant subbundles are handled by a separate irreducible result for essentially bounded measurable bundle cocycles, whose integrated contraction estimate plays the role of the continuous contraction theorem.

What would settle it

Build, or compute, a primitive Markov chain and continuous matrices satisfying cycle non-resonance and $\lambda_1>\lambda_2$ such that the measurable restricted or quotient bundle cocycle obtained from an invariant family has integrated contraction coefficients $K_n(\alpha,P)$ that do not decay exponentially for any $\alpha>0$; that would violate the asserted estimate (55) and break the proof of Theorem 3.3.

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Extended reading notes

Core claim

The central discovery is Theorem 3.3, stated on the paper's own terms. Let $P$ be a primitive stochastic matrix supported on a fixed pattern $U$, with translations $\theta_i$ on the torus $\mathbb T^m$ satisfying the cycle non-resonance condition (Assumption 2.4): for every nonzero integer vector $k$, some admissible directed cycle has accumulated translation $\sum_r \theta_{i_r}$ with $\langle k,\sum_r \theta_{i_r}\rangle\notin\mathbb Z$. Let $A_i\in C^0(\mathbb T^m,\mathrm{GL}_d(\mathbb R))$ and assume only that the top Lyapunov exponent $\lambda_1(P)$ is strictly larger than $\lambda_2(P)$. Then there are a relatively open neighborhood $\Omega$ of $P$ in the complex affine space $H(U)$ of matrices with the same zero pattern and row sums one, and a holomorphic function $\Lambda$ on $\Omega$, such that $\Lambda(Q)=\lambda_+(Q)$ for every real stochastic $Q\in \Omega\cap S(U)$. Thus the top exponent is real-analytic in the transition probabilities near $P$. A corollary, obtained by exterior powers, is that when the full spectrum at $P$ is simple every Lyapunov exponent is real-analytic. The path to the theorem is to prove the irreducible case by a forward Markov transfer operator with contraction uniform in the initial Markov state, then to remove irreducibility by induction on fiber dimension along measurable invariant subbundles.

Load-bearing premise

The load-bearing premise is the asserted exponential contraction estimate for measurable bundle cocycles, inequality (55) in Proposition 7.2, which the paper sketches but does not fully prove; if that estimate fails the dimension-reduction induction that removes irreducibility collapses.

Editorial extensions

If this is right

Near any primitive reference matrix satisfying cycle non-resonance and $\lambda_1>\lambda_2$, the top Lyapunov exponent is real-analytic in the transition probabilities on the fixed-support face.
If the entire Lyapunov spectrum at $P$ is simple, every individual exponent is real-analytic there, since partial sums of exponents are top exponents of exterior-power cocycles.
No irreducibility of the cocycle is needed in the final statement; reducible cocycles are covered by induction on fiber dimension after a dominant invariant block is selected.
Analyticity survives passage to measurable invariant subbundles and quotients, even though such bundles need not admit continuous trivializations.
For a strictly positive transition matrix, the cycle non-resonance condition is automatically satisfied when at least one translation is rationally independent, since a self-loop supplies the needed cycle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the holomorphic extension should give explicit Cauchy bounds on derivatives of the top exponent with respect to individual transition probabilities; the paper establishes the extension but does not compute such bounds.
Editorial inference: the cycle non-resonance criterion is a purely combinatorial and arithmetic condition on the Markov graph and translations, so the same analyticity statement should hold verbatim for any support graph satisfying it, suggesting the condition is the right general replacement for 'at least one irrational translation'.
Editorial inference: a natural next question is what happens at the boundary $\lambda_1(P)=\lambda_2(P)$; the proof's contraction estimates degenerate as the spectral gap closes, so one would expect at most Hölder continuity there, consistent with the broader regularity hierarchy.
Editorial inference: because Proposition 7.2 is stated for essentially bounded measurable bundle cocycles, the induction should apply to cocycles that are only measurably dependent on the torus from the start, not merely to measurable reductions of continuous cocycles, provided the same hypotheses hold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves real-analyticity of the top Lyapunov exponent in transition probabilities for primitive Markov quasi-periodic cocycles, with a cycle non-resonance condition and simple top Lyapunov exponent. That is a genuine extension of the Bernoulli case in Bezerra–Sánchez–Tall, and the irreducible proof in Sections 4–6 is detailed and largely convincing. The cycle non-resonance criterion (Theorem 2.2) is new and clean. The paper is also honest that the measurable-bundle result is not a formal corollary of the continuous case, which I appreciate.

The soft spot is exactly where your reader put it: Proposition 7.2's contraction estimate (55). It is the load-bearing step of the reducible induction, and it is asserted, not proved. The instruction to 'repeat the three stages of Proposition 4.6' is not automatic. Proposition 4.6 uses continuity of the projective kernel, the Feller property, and weak compactness on a compact phase space. In the class B_r(U), the kernel is not continuous and the observable g is only measurable and bounded. The stress-test note is right: nothing in the paper shows that an extremal stationary measure with alpha < lambda_1 would force the state-local invariant family (23). Without that step, (55) is unsupported. Since Section 7.3 needs (55) for the terminal block after dimension reduction, the main theorem is not fully established as written. This is the same gap your reader identified, and I agree with it.

Proposition 7.3's continuity proof is also sketchier than I would like: the uniform-in-section convergence L_n(Q,xi) -> lambda_+(Q) is asserted. That is a smaller issue, but it is in the same territory.

Overall, the result is likely true and the overall strategy is sound. The irreducible case stands on its own. This is for people working on regularity of Lyapunov exponents for random and quasi-periodic cocycles; they will want the theorem if the gap is fixed. I would send this to a serious referee, with the request that the author either prove (55) for measurable bundle cocycles or find a route around it.

Referee Report

3 major / 5 minor

Summary. The paper proves a local real-analyticity theorem for the top Lyapunov exponent of mixed Markov quasi-periodic cocycles, viewed as a function of the transition probabilities on a fixed support face. Under the assumptions that the reference transition matrix is primitive, that the Markov torus extension is non-resonant along every directed cycle (Assumption 2.4), and that the top Lyapunov exponent is simple, the main result (Theorem 3.3) asserts that there exists a complex neighborhood of the reference matrix on which the top exponent extends to a holomorphic function of the complexified transition entries. The proof has two main parts. First, for irreducible cocycles, the paper develops a forward Markov transfer operator, a state-conditioned projective contraction estimate uniform in the initial state and torus point (Proposition 4.6), and a Cauchy estimate that yields uniform convergence of holomorphic logarithmic increments (Propositions 6.4 and 6.5). Second, for reducible cocycles, the paper argues by induction on fiber dimension: an invariant family is decomposed into restricted and quotient bundle cocycles, the dominant block is selected using simplicity and continuity, and the terminal irreducible block is handled by a measurable-bundle version of the irreducible argument (Proposition 7.2).

Significance. If the main theorem is correct, it substantially generalizes the analyticity results of Bezerra-Sánchez-Tall from Bernoulli to Markov transition probabilities, and it removes the irreducibility hypothesis from the final statement. The cycle non-resonance condition is explicit and checkable, and the proof is mostly self-contained, with no fitted parameters and with a genuine mixed random-quasi-periodic mechanism rather than a trivial reduction. The paper also correctly identifies the genuinely new difficulty: measurable invariant sections need not have continuous trivializations, so the reducible induction requires a measurable-bundle version of the irreducible argument. The two key inputs, Proposition 4.6 and Proposition 7.2, are the heart of the paper, and the strength of the claimed result depends entirely on whether they are fully justified.

major comments (3)

[§7.2, Proposition 7.2, equation (55)] The estimate K_n(α,P) ≤ C_b e^{-ζ_b n} is the load-bearing contraction estimate for the measurable-bundle irreducible case, but it is asserted rather than proved. The text says to 'repeat the three stages of Proposition 4.6,' yet Proposition 4.6 inherits Lemma 4.4, whose proof uses weak compactness of the space of probability measures on the compact projective space, continuity of the logarithmic observable g, and the Feller property of the projective Markov kernel. None of these ingredients is available for an essentially bounded measurable bundle cocycle in B_r(U) with merely measurable edge maps. In particular, the Krylov-Bogolyubov argument in Lemma 4.4 does not go through in this category. Since Section 7.3 applies Proposition 7.2 to a merely measurable restricted or quotient cocycle after dimension reduction, a failure of (55) would leave Theorem 3.3 unproved in the reducible case. The proof needs either a complete proof of (55) from the non-random filtration theorem, or a precise citation of a theorem in the literature that contains exactly this estimate for measurable bundle cocycles.
[§4, Lemma 4.4, equations (22)–(24)] The step from α(η) < λ_1(P) to the existence of a proper state-local invariant family Vi(t) satisfying Ai(t)Vi(t)=Vj(t+θ_i) for every admissible edge is not justified. The non-random filtration theorem cited from Kifer yields invariant subspaces that may depend on the full trajectory of the Markov chain, not merely on the current state and the torus point. Definition 3.1 forbids only such state-indexed families. The text asserts the stronger conclusion without proof. This matters both for the irreducible continuous case and for the measurable-bundle case in Proposition 7.2, where the same gap recurs. The authors either need to prove that the Furstenberg-Kifer filtration in this Markov setting is automatically state-local, or they need to replace the irreducibility condition by a quasi-irreducibility hypothesis, or add a separate argument showing that a lower extremal stationary measure forces a state-local invariant family.
[§7.3, Proof of Theorem 3.3; §7.2, Proposition 7.3] The induction in Section 7.3 uses Proposition 7.3 to obtain a real neighborhood W on which the dominant block remains strictly dominant. Proposition 7.3 asserts continuity for essentially bounded measurable bundle cocycles and claims uniform convergence of L_n(Q,ξ) to λ_+(Q) in ξ, locally uniformly in Q. This uniform convergence is stated without proof; the text says 'The Furstenberg-Kifer theorem gives' the convergence, but the locally uniform dependence on Q is not a direct consequence of the cited theorem, since the path weights and the stationary vector π(Q) vary with Q. Since the selection of the dominant block in the proof of Theorem 3.3 depends on this continuity, the argument needs a more explicit proof of Proposition 7.3, or a reference supplying the locally uniform convergence.

minor comments (5)

[§4.1] The subsection heading 'Uniform averaged growth' appears to be empty; the material that belongs there seems to have been absorbed into Section 4.2 under 'Stationary unit vectors.' Please either fill in the missing subsection or renumber the headings.
[§6.1, Lemma 6.1] There is a typo in the statement: 'stationary distrobution' should read 'stationary distribution.'
[§7.1, paragraph after equation (42)] There are missing spaces in the displayed text 'ThusthefinitepathexpansionsremainpolynomialintheentriesofZ, exactlyasintheirreducible proof.' Please correct the spacing.
[References [20] and [21]] The author names in references [20] and [21] are formatted inconsistently; for example, 'E. H. Y. Tall and M. Viana' versus 'A. C de Araújo.' Please make the reference style uniform.
[§2.2, Remark 2.3] The reduction to the Bernoulli criterion is clear, but it would help to state explicitly that Assumption 2.4 depends only on the support graph and the translations, not on the numerical values of the positive transition probabilities, so that it remains valid for all nearby Q in the same support face.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the analyticity proof is self-contained modulo standard transfer-operator and Furstenberg–Kifer inputs; the main gap is an asserted measurable contraction estimate, not a circular reduction.

full rationale

The paper derives the analytic continuation of the top Lyapunov exponent from (i) finite-path polynomial dependence of transfer iterates on the complexified transition entries (eq. 42), (ii) uniform projective contraction Ktilde_n(alpha,P) <= C(alpha)e^{-zeta(alpha)n} from Proposition 4.6, and (iii) the Cauchy estimate Proposition 6.4 bounding successive logarithmic increments by a summable geometric series. The identification on real stochastic matrices in Lemma 6.7 is a telescoping identity: b_n(Q,v) equals the torus-average of log(||A^{n+1}v||/||A^n v||), and it converges to lambda_+(Q) by the independently proved uniform convergence of directional averages, not by any fitted constant or by assuming the conclusion. The reducible case in Section 7.3 is an induction on fiber dimension that selects a strictly dominant block using the spectral gap lambda_1>lambda_2 and Proposition 7.3, then applies Proposition 7.2 to the terminal measurable bundle cocycle; neither step defines the conclusion into the hypotheses. Self-citations to [5] are used as templates and as references for the Bernoulli analogue, but the Markov argument is carried out in the present text, so they are not load-bearing. The genuine weakness is Proposition 7.2's estimate (55), K_n(alpha,P) <= C_b e^{-zeta_b n}, which is asserted as the 'measurable bundle version of the Furstenberg–Kifer contraction theorem' and justified only by a sketch ('To see the mechanism, repeat the three stages of Proposition 4.6'). If that estimate fails, the reducible induction is incomplete; however, this is an omitted or sketched proof, not a circularity, because (55) is not assumed as the theorem's conclusion and the paper explicitly labels the measurable case as 'a separate argument, not a formal consequence of proof for continuous cocycles.' Verdict: no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard ergodic theory (Oseledets, Furstenberg-Kifer, Kingman), on the theorem's domain assumptions (primitivity, fixed support, cycle non-resonance, simplicity), and on one new technical assertion, the measurable-bundle contraction estimate (55), which the paper sketches but does not fully prove. No fitted parameters or new physical or mathematical entities are introduced.

assumptions (6)

standard math Oseledets theorem for invertible cocycles over the ergodic Markov torus extension (F, mu_P times Leb).
Invoked in Section 3 to define Lyapunov exponents and the Oseledets filtration; log integrability is automatic from compactness.
standard math Furstenberg-Kifer non-random filtration and the variational formula lambda_1(P) = max alpha(eta).
Used in Lemma 4.4 and Propositions 7.1 and 7.2 to connect stationary projective unit vectors to exponents; cited from [14, Chapter III] and [11].
standard math Kingman's subadditive ergodic theorem.
Used in Lemma 4.4 to identify the uniform limsup of exterior-square averages with lambda_1 + lambda_2.
domain assumption Theorem hypotheses: P primitive, cycle non-resonance (Assumption 2.4), and simplicity lambda_1(P) > lambda_2(P).
These hypotheses define the scope of Theorems 3.2 and 3.3 and guarantee ergodicity of the base and a strictly dominant block in the reducible induction.
domain assumption The support U is fixed, so every Q in S(U) has the same admissible edges and every invariant family V remains invariant for all Q.
Used in Sections 7.1 and 7.3 to keep the dominant block fixed in a neighborhood of P; it is a structural assumption on the parameter space.
ad hoc to paper Measurable-bundle contraction estimate (55): K_n(alpha,P) <= C_b e^{-zeta_b n} for irreducible measurable bundle cocycles.
This is the key unproved input of Proposition 7.2. The paper gives only a mechanism sketch, and the main theorem's terminal irreducible step depends on it.

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Pith. "Pith review of Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles." pith.science (2026). https://pith.science/paper/HHNCXOPT

@misc{pith2026260801569,
  author       = {Pith},
  title        = {Pith review of: Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHNCXOPT}},
  note         = {Machine review of arXiv:2608.01569}
}

read the original abstract

We study the dependence of the top Lyapunov exponent of a mixed Markov quasi-periodic cocycle on the transition probabilities. The transition matrix is assumed primitive, the Markov extension on the torus is assumed non-resonant along directed cycles, and the top Lyapunov exponent is assumed simple. We first prove the result in the irreducible case by a forward Markov transfer operator and contraction estimates uniform over the initial Markov state. The reducible case is then obtained, as in the Bernoulli argument of Bezerra-S\'anchez-Tall, by decomposing along measurable invariant sections into restricted and quotient bundle cocycles and inducting on the fiber dimension. A measurable invariant section need not admit a continuous trivialization, even when the original matrices cocycle are continuous. We therefore formulate the irreducible analytic case for essentially bounded measurable bundle cocycles with essentially bounded inverses.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

22 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Amorim, M

A. Amorim, M. Durães, and A. Melo, Analyticity of the Lyapunov exponents of random products of matrices, Ergodic Theory Dynam. Systems(2026), 1–27, published online, doi:10.1017/etds.2026.10287

work page arXiv 2026

[2] [2]

Avila, A

A. Avila, A. Eskin, and M. Viana, Continuity of the Lyapunov exponents of random matrix products, arXiv:2305.06009, 2023

arXiv 2023

[3] [3]

Avila, S

A. Avila, S. Jitomirskaya, and C. Sadel, Complex one-frequency cocycles,J. Eur. Math. Soc. (JEMS)16 (2014), 1915–1935

work page 2014

[4] [4]

L.Backes, A.Brown, andC.Butler, ContinuityofLyapunovexponentsforcocycleswithinvariantholonomies, J. Mod. Dyn.12(2018), 223–260

work page 2018

[5] [5]

Bezerra, A

J. Bezerra, A. Sánchez, and E. H. Y. Tall, Analyticity of the Lyapunov exponents of random products of quasi-periodic cocycles, arXiv:2111.00683, 2021

arXiv 2021

[6] [6]

Bocker-Neto and M

C. Bocker-Neto and M. Viana, Continuity of Lyapunov exponents for random two-dimensional matrices, Ergodic Theory Dynam. Systems37(2017), 1413–1442. ANALYTICITY OF LYAPUNOV EXPONENTS FOR MIXED MARKOV–QUASI-PERIODIC COCYCLES 33

work page 2017

[7] [7]

A. Cai, P. Duarte, and S. Klein, Furstenberg theory of mixed random-quasiperiodic cocycles,Commun. Math. Phys.402(2023), 447–487

work page 2023

[8] [8]

A. Cai, P. Duarte, and S. Klein, Mixed random-quasiperiodic cocycles,Bull. Braz. Math. Soc. (N.S.)53 (2022), 1469–1497

work page 2022