H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles

Pedro Duarte; Tom\'e Graxinha

REVIEW 3 major objections 5 minor 1 cited by

H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that the top Lyapunov exponent of a random matrix product is H\"older continuous in the Wasserstein metric around any measure that is quasi-irreducible and has a spectral gap.

desk verdict Real theorem, serious gap: the Hölder continuity claim is probably true, but Proposition 6.9's contraction estimate is not justified by the stated hypotheses. read the letter →

arxiv 2506.04124 v1 pith:WN2O7YQP submitted 2025-06-04 math.DS

Pedro Duarte , Tomé Graxinha This is my paper

classification math.DS MSC 37H1537A3060B20

keywords LyapunovexponentsH\"oldercontinuityrandommatrixproductsnon-invertiblecocyclesWassersteindistancequasi-irreducibilityspectralgapSchr\"odinger

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 a quantitative stability statement for the top Lyapunov exponent of random matrix products. For any probability measure on all $m \times m$ matrices (not just invertible ones, and not necessarily compactly supported) with finite exponential moments, the paper shows that if the generated semigroup is quasi-irreducible and the top two Lyapunov exponents are separated, then small perturbations of the matrix distribution in the Wasserstein metric move the top exponent by at most a power of the perturbation size. The result matters because realistic models, including Schr\"odinger operators with unbounded random potentials, fall outside the invertible, compactly supported regime where such regularity was previously known.

What carries the argument

The load-bearing object is the average contraction coefficient $\kappa_\alpha(\mu)$, the $\mu$-average of the $\alpha$-H\"older norm of the projective action $\hat g$ on $\mathbb{P}(\mathbb{R}^m)$. Proposition 6.9 shows that under quasi-irreducibility and a spectral gap, some convolution power satisfies $\kappa_\alpha(\mu^n) \le \sigma_0 < 1$; this gives uniform strong mixing of the Markov operator $Q_\mu$ on H\"older observables (Theorem 6.1) and, after a Wasserstein perturbation estimate for $Q_\mu$ (Lemma 7.3), the H\"older modulus for $L_1$. Singularities of non-invertible matrices are handled by truncating the log-norm observable and inducing the transition kernel on a compact set away from the singular locus.

What would settle it

Compute $\kappa_\alpha(\mu^n)$ for a concrete quasi-irreducible measure with $L_1>L_2$, for instance a finitely supported mixture of shear and rotation matrices in $\mathrm{Mat}_2(\mathbb{R})$; Proposition 6.9 predicts it is eventually $<1$ for some $\alpha>0$. Numerically observing it stuck at $1$ for all $\alpha$ would break the spectral step of the proof.

Watch

Extended reading notes

Core claim

On the paper's own terms: given a measure $\mu$ with finite exponential moments of order $p$, quasi-irreducibility, and a spectral gap $L_1(\mu) > L_2(\mu)$, the map $\nu \mapsto L_1(\nu)$ is H\"older continuous on a small $W_p$-neighborhood of $\mu$ (Theorem 2.1). The proof supplies the same regularity for all higher Lyapunov exponents via exterior powers (Corollary 2.2), for locally constant $\mathrm{SL}_m(\mathbb{R})$ cocycles over Bernoulli shifts in the $L^1$ distance (Corollary 2.4), and yields stable exponential large-deviation estimates (Theorem 2.2). The closing section applies these results to Schr\"odinger cocycles whose potentials are sampled from Frostman measures, obtaining an asymptotic expansion of the Lyapunov exponent as the coupling grows.

Load-bearing premise

The argument rests on the existence of a power of the measure whose expected projective contraction is uniformly below one; if no such contracting power exists, the uniform mixing and the H\"older estimate collapse.

Editorial extensions

If this is right

All Lyapunov exponents, not just the top, are locally H\"older continuous as functions of the matrix-distribution measure (Corollary 2.2).
For locally constant $\mathrm{SL}_m(\mathbb{R})$ cocycles over Bernoulli shifts, the top exponent is H\"older continuous with respect to the $L^1$ distance between cocycle maps (Corollary 2.4).
Exponential large-deviation bounds for the log norm hold uniformly for all measures in a small Wasserstein neighborhood, with rate $c(\varepsilon) \sim \varepsilon^2/\log(1/\varepsilon)$ (Theorem 2.2).
For Schr\"odinger cocycles with Frostman-distributed unbounded potentials, the Lyapunov exponent is locally H\"older jointly in the potential distribution and the energy, and satisfies $L_1(\mu_{\lambda,E,q}) = q\log\lambda + L_1(\tilde\mu_{0,E,q}) + O(\lambda^{-p})$ as $\lambda \to \infty$ (Propositions 9.1 and 9.5).
The sum of the first $m$ Lyapunov exponents of symplectic Schr\"odinger cocycles admits the asymptotic $m \log \lambda + \int \log|\det(s-E)|\, d\mu(s) + O(\lambda^{-p'})$ (Proposition 9.6).

Reading between the lines

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

Beyond the paper, the contraction coefficient $\kappa_\alpha$ could serve as a quantitative measure of randomness for non-invertible cocycles, giving explicit H\"older exponents for specific noise models.
Beyond the paper, the truncation-and-induction method may extend to constant-rank random cocycles, where the projective action is genuinely discontinuous and current results only give continuity.
Beyond the paper, the stable large-deviation bounds suggest a Berry-Esseen-type estimate for the log norm in the non-invertible regime, since the exponential rates now hold uniformly in a Wasserstein neighborhood.
Beyond the paper, the asymptotic separation $q\log\lambda$ plus a measure-dependent correction could be used to infer the Frostman dimension of the potential distribution from the Lyapunov exponent's subleading term.

Share X Bluesky LinkedIn Reddit HN

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 right headline: this is a serious extension of Le Page's theorem, and the main result is likely true, but the proof as written has a load-bearing gap in Proposition 6.9. I checked the stress-test concern, and it lands.

What is genuinely new: the paper treats non-invertible, non-compact random cocycles in Mat_m(R) and proves local Hölder continuity in Wasserstein distance, where the projective action degenerates on singular matrices. The machinery for handling those singularities—truncation, induced kernel, isospectral reduction—is real work and the exposition in Sections 5–7 is careful. The literature placement is honest, and the reliance on earlier published results is not, by itself, a problem.

Now the soft spots, in order.

First, Proposition 6.9. The proof needs to show ∫ log(‖∧²g‖/‖gv‖²) dμⁿ(g) ≤ −1, and for the term ∫ log‖∧²g‖ dμⁿ it uses the bound n(L₁+L₂+ε). That bound does not follow from Lemma 6.3, which controls expectations of log‖gv‖ for vectors, not the maximum over two-dimensional directions. To get it you would need Lemma 6.3 applied to the exterior-square cocycle ∧²μ, and that requires ∧²μ to be quasi-irreducible and a bound on the second exterior moment Θ²_p(μ). Neither is assumed in Theorem 2.1. Quasi-irreducibility of μ does not imply quasi-irreducibility of ∧²μ; the SO(4) example is enough to show that. Without the missing estimate, the inequality ∫ e^{αX} dμⁿ < 1 has no base, and Theorem 6.1 along with the Hölder estimate lose their key contraction assumption. This is not cosmetic.

Second, the applications have smaller but real gaps. Proposition 9.6 is stated without proof. Proposition 9.5 claims O(λ^{-p}), which is stronger than what Theorem 2.1 supplies: the Hölder exponent in the proof is at most min(α/p, 1/3) with α < p/2, so the W_p distance bound ≲ β^p only gives error O(β^{p·h}) with h < 1, not O(β^p). The exponent needs correction or a separate argument.

If Proposition 6.9 can be repaired, either by adding exterior-square quasi-irreducibility to the hypotheses or by finding another contraction argument, this becomes a strong paper. As written, the central theorem is not fully supported. The authors deserve a serious referee, but the referee should focus on Section 6 first. I would not cite it until the gap is fixed; I would bring it to a reading group to discuss whether the repair is straightforward.

Referee Report

3 major / 5 minor

Summary. The paper proves local Hölder continuity of the top Lyapunov exponent for i.i.d. random linear cocycles with values in Mat_m(R), under three assumptions: finite exponential moments, quasi-irreducibility of the generated semigroup, and a positive gap L_1(μ)>L_2(μ). The topology is the Wasserstein distance W_p on the space M_p^C of measures with bounded exponential moments. Theorem 2.1 is the central result; Corollaries extend it to higher Lyapunov exponents and to locally constant Bernoulli cocycles, Theorem 2.2 gives stable exponential large-deviation estimates, and Section 9 applies the machinery to Schrödinger cocycles with unbounded potentials. The proof follows the spectral method of Nagaev/Bougerol/Le Page, with Sections 5–7 carrying the main argument. The paper is written as a self-contained derivation under explicit assumptions and does not fit free parameters, although it relies on several published results by the same authors and collaborators.

Significance. If Theorem 2.1 is correct after repair, it is a substantial extension of Le Page's Hölder regularity theory to non-invertible and non-compact random cocycles, in the Wasserstein topology rather than a parametric family. The treatment of non-invertible matrices and of non-compact support is a genuine novelty, and the paper also provides stable large-deviation estimates and applications to Schrödinger cocycles with unbounded potentials. The manuscript is transparent about its hypotheses and gives many of the key estimates explicitly. The main theorem would be a valuable addition to the Lyapunov regularity literature.

major comments (3)

[Section 6, Proposition 6.9] The proof of Proposition 6.9 needs the estimate ∫ log||∧²g|| dμ^n(g) ≤ n(L_1(μ)+L_2(μ)+ε), but this estimate is not a consequence of Lemma 6.3 as the text claims. Lemma 6.3 supplies uniform convergence of ∫(1/n)log||gv|| dμ^n(g) for fixed projective vectors v; the quantity log||∧²g|| is a supremum over two-dimensional directions and cannot be bounded by a supremum of vector expectations after integration. Applying Lemma 6.3 to the exterior-square cocycle ∧²μ would require quasi-irreducibility of ∧²μ and a bound on Θ²_p(μ), neither of which is assumed in Theorem 2.1. The estimate is probably true by subadditivity of log||∧²A_n|| together with Oseledets a.s. convergence and the moment bounds in M_p^C, but that argument is absent. Since Proposition 6.9 supplies the contraction κ_α(μ^n)≤σ_0 on which Theorem 6.1 and the Hölder estimate of Section 7 depend, this is a load-bearing gap that must be fixed.
[Section 9.2, Proposition 9.5] The proof establishes W_p(µ̃_β, µ̃_0)≲β^p and then invokes Theorem 2.1 to conclude an error O(λ^{-p}). However, Theorem 2.1 only provides Hölder continuity with an unspecified exponent θ>0; in the proof of Theorem 2.1 the exponent is α/p with α<p/2, which is strictly smaller than 1/2. Consequently the displayed rate O(λ^{-p}) does not follow from the stated theorem. The asymptotic formula should either be weakened to O(λ^{-θ}) for some θ>0, or the proof must provide a uniform lower bound on the Hölder exponent along this family.
[Section 9.3, Proposition 9.6] Proposition 9.6 is a numbered proposition in the applications section, but no proof is given anywhere in the manuscript. The asymptotic formula for L_1(∧^m A_{λ,E}) is therefore an unproved assertion. It should either be proved with analogues of Lemmas 9.3 and 9.4 for exterior powers and the preceding Wasserstein estimates, or it should be explicitly stated as a conjecture/removed from the paper.

minor comments (5)

[Definition 2.2] The displayed stationarity condition contains a typo: the set written as {ˆv ∈ Prob(P(Rm)) : g v=0} should be a subset of P(Rm), not Prob(P(Rm)), and the condition should read gv=0 for v∈P(Rm).
[Section 5.2, Proposition 5.5] The statement of Proposition 5.5, '|L_1(µ)| ≤ C−1 p', is garbled; from the proof the intended bound appears to be |L_1(µ)| ≤ C/p.
[Section 7, proof of Corollary 2.4] The subsection heading 'Proof of Theorem 2.4' refers to Corollary 2.4; the numbering should be corrected.
[Corollary 2.4, footnote/reference] The attribution to Barrientos and Malicet cites '[6, Proposition X]', where 'X' appears to be a placeholder rather than an actual proposition number.
[Lemma 9.2] In the proof of Lemma 9.2 the integral is written with |t−a|^{-p}, while the statement of the lemma and the final estimate use |t−a|^{-d}; clarify which exponent is meant in each occurrence.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the Hölder-continuity theorem is derived from explicit assumptions via standard spectral theory, and the self-citations used are published external support, not circular inputs.

full rationale

The derivation of Theorem 2.1 is a forward proof from the stated assumptions μ ∈ M_p^C, quasi-irreducibility, and L1(μ) > L2(μ). No parameter is fitted to the quantity being predicted: the Lyapunov exponent is not defined through a fitted Hölder modulus, and the Wasserstein continuity is obtained from explicit inequalities (Lemma 7.3 and the proof of Theorem 2.1) rather than from a regularity assumption. The paper cites prior work by the same authors ([4], [5], [19]) for standard ingredients: Oseledets splittings, projective-action estimates, and isospectral reduction of Markov operators. These citations are not circular inputs: they are published, peer-reviewed results whose statements do not include the Hölder-continuity conclusion of this paper. In particular, the core contraction step, Proposition 6.9, is proved from Lemma 6.3, which is itself proved from Furstenberg–Kifer and the quasi-irreducibility assumption. The skeptic’s concern about the unproved bound ∫ log‖∧²g‖ dμⁿ ≤ n(L1+L2+ε) inside Proposition 6.9 is a potential correctness gap, not a circularity: the missing estimate is not equivalent to the theorem’s conclusion and no fitted parameter is involved. The paper also credits Barrientos and Malicet [6] for an independently obtained corollary, which is external support rather than a load-bearing self-citation. The paper itself flags a forthcoming justification of assumption (a) and omits the proof of Proposition 2.1; these are acknowledged limitations, not circular steps. No self-definitional, fitted-input, uniqueness-imported, ansatz-smuggled, or renaming pattern was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on explicit moment, irreducibility, and spectral gap hypotheses stated in Theorem 2.1, plus standard background theorems in probability and dynamical systems. No free parameters are fitted and no new entities are postulated.

assumptions (4)

domain assumption µ ∈ M_p^C has finite exponential moments: Θ̄_p(µ) ≤ C and Θ_p(µ) := sup_v ∫ ||gv||^{-p} dµ ≤ C.
This is the integrability condition used in truncation (Lemmas 4.2 and 4.4) and throughout; the paper says necessity will be shown in future work.
domain assumption µ is quasi-irreducible (Definition 2.1).
Used in Lemma 5.2 to force β_µ(η) = L1(µ) for every stationary measure, a key step toward Theorem 5.1.
domain assumption Spectral gap L1(µ) > L2(µ).
Used in Proposition 6.9 to obtain contraction κ_α(µ^n) < 1, the core of uniform ergodicity.
standard math Furstenberg-Kifer abstract theorem (Theorem 5.2) and Oseledets theory from [19,39].
Quoted without proof as background; used in the proof of Theorem 5.1 and Lemma 6.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles." pith.science (2026). https://pith.science/paper/WN2O7YQP

@misc{pith2026250604124,
  author       = {Pith},
  title        = {Pith review of: H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WN2O7YQP}},
  note         = {Machine review of arXiv:2506.04124}
}

abstract

We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov exponents, and quasi-irreducibility of the associated semigroup - we prove that the top Lyapunov exponent is H\"older continuous with respect to the Wasserstein distance. In the final section, we apply the main results to Schr\"odinger cocycles with unbounded potentials.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Integral representation of Lyapunov exponents
math.DS 2026-04 unverdicted novelty 7.0 of 10

A variational principle for Lyapunov exponents is derived from Markov operators, extending projective integral formulas to singular cocycles and Markov place-dependent noise.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · cited by 1 Pith paper

[1]

Artur Amorim, Marcelo Duraes, and Aline Melo, Analiticity of the lyapunov exponents of random products of matrices , 2025

work page 2025
[2]

Artur Avila, Alex Eskin, and Marcelo Viana, Continuity of the lyapunov expo- nents of random matrix products , 2023

work page 2023
[3]

Lucas Backes, Aaron Brown, and Clark Butler, Continuity of Lyapunov expo- nents for cocycles with invariant holonomies , J. Mod. Dyn. 12 (2018), 223–260

work page 2018
[4]

Alexandre Baraviera and Pedro Duarte, Approximating Lyapunov exponents and stationary measures , J. Dynam. Differential Equations 31 (2019), no. 1, 25–48. MR 3935134

work page 2019
[5]

07, 2150025

Alexandre Baraviera, Pedro Duarte, and Maria Joana Torres, Stationary mea- sures on infinite graphs , Communications in Contemporary Mathematics 24 (2022), no. 07, 2150025

work page 2022
[6]

Barrientos and Dominique Malicet, Mostly contracting random maps, 2024

Pablo G. Barrientos and Dominique Malicet, Mostly contracting random maps, 2024. NON-INVERTIBLE AND NON-COMPACT RANDOM COCYCLES 51

work page 2024
[7]

Jamerson Bezerra, Ao Cai, Pedro Duarte, Catalina Freijo, and Silvius Klein, A dynamical Thouless formula , Adv. Math. 438 (2024), 50 (English), Id/No 109446

work page 2024
[8]

2, 829–875

Jamerson Bezerra and Pedro Duarte, Upper Bound on the Regularity of the Lya- punov Exponent for Random Products of Matrices , Communications in Math- ematical Physics 403 (2023), no. 2, 829–875

work page 2023

Show all 41 references

[9]

Jamerson Bezerra and Mauricio Poletti, Random product of quasi-periodic co- cycles, Proc. Amer. Math. Soc. 149 (2021), no. 7, 2927–2942

2021
[10]

Jamerson Bezerra, Adriana S´ anchez, and El Hadji Yaya Tall,Analyticity of the lyapunov exponents of random products of quasi-periodic cocycles , Nonlinearity 36 (2023), no. 6, 3467

2023
[11]

Carlos Bocker-Neto and Marcelo Viana, Continuity of Lyapunov exponents for random two-dimensional matrices , Ergodic Theory and Dynamical Systems (2016), 1–30

2016
[12]

Theory Related Fields 78 (1988), no

Philippe Bougerol, Th´ eor` emes limite pour les syst` emes lin´ eaires ` a coefficients markoviens, Probab. Theory Related Fields 78 (1988), no. 2, 193–221

1988
[13]

Ao Cai, Pedro Duarte, and Silvius Klein, Statistical properties for mixing markov chains with applications to dynamical systems , 2024

2024
[14]

Ao Cai, Marcelo Duraes, Silvius Klein, and Aline Melo, H¨ older continuity of the lyapunov exponent for markov cocycles via furstenberg’s formula , 2022

2022
[15]

Walter Craig and Barry Simon, Log H¨ older continuity of the integrated density of states for stochastic Jacobi matrices , Comm. Math. Phys. 90 (1983), no. 2, 207–218. MR 714434

1983
[16]

Systems 38 (2018), no

Davor Dragiˇ cevi´ c and Gary Froyland,H¨ older continuity of oseledets splittings for semi-invertible operator cocycles , Ergodic Theory and Dynam. Systems 38 (2018), no. 3, 961–981

2018
[17]

Pedro Duarte, Marcelo Duraes, Tom´ e Graxinha, and Silvius Klein,Random 2d linear cocycles i: dichotomic behavior , 2025

2025
[18]

10, 105008

Pedro Duarte and Catalina Freijo, Continuity of the lyapunov exponents of non- invertible random cocycles with constant rank , Nonlinearity 37 (2024), no. 10, 105008

2024
[19]

3, Atlantis Press, 2016

Pedro Duarte and Silvius Klein, Lyapunov exponents of linear cocycles; continu- ity via large deviations , Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, 2016

2016
[20]

, Large deviations for products of random two dimensional matrices , Comm. Math. Phys. 375 (2020), no. 3, 2191–2257

2020
[21]

Pedro Duarte, Silvius Klein, and Mauricio Poletti, H¨lder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps , Mathematische Zeitschrift (2022), 2285

2022
[22]

Pedro Duarte, Silvius Klein, and Manuel Santos, A random cocycle with non H¨ older Lyapunov exponent, Discrete Contin. Dyn. Syst. 39 (2019), no. 8, 4841– 4861

2019
[23]

Systems 30 (2010), no

Gary Froyland, Simon Lloyd, and Anthony Quas, Coherent structures and iso- lated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 729–756. MR 2643709

2010
[24]

Furstenberg, Non-commuting random products , Trans

H. Furstenberg, Non-commuting random products , Trans. Amer. Math. Soc. 108 (1963), 377–428. 52 P. DUARTE AND T. GRAXINHA

1963
[25]

Furstenberg and H

H. Furstenberg and H. Kesten, Products of random matrices , Ann. Math. Statist. 31 (1960), 457–469

1960
[26]

Furstenberg and Yu

H. Furstenberg and Yu. Kifer, Random matrix products and measures in pro- jective spaces, Israel J. Math 10 (1983), 12–32

1983
[27]

B I Halperin, Properties of a particle in a one-dimensional random potential. , Advan. Chem. Phys., 13: 123-77 (1967)

1967
[28]

Malheiro and Marcelo Viana, Lyapunov exponents of linear cocycles over Markov shifts , Stoch

Ela ´ ıs C. Malheiro and Marcelo Viana, Lyapunov exponents of linear cocycles over Markov shifts , Stoch. Dyn. 15 (2015), no. 3, 1550020, 27

2015
[29]

S. V. Nagaev, Some limit theorems for stationary Markov chains , Teor. Veroy- atnost. i Primenen. 2 (1957), 389–416. MR 0094846 (20 #1355)

1957
[30]

V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic num- bers for dynamical systems , Trans. Moscow Math. Soc. 19 (1968), 197–231

1968
[31]

Le Page, Th´ eor` emes limites pour les produits de matrices al´ eatoires, Prob- ability measures on groups (Oberwolfach, 1981), Lecture Notes in Math., vol

´E. Le Page, Th´ eor` emes limites pour les produits de matrices al´ eatoires, Prob- ability measures on groups (Oberwolfach, 1981), Lecture Notes in Math., vol. 928, Springer-Verlag, 1982, pp. 258–303

1981
[32]

, R´ egularit´ e du plus grand exposant caract´ eristique des produits de ma- trices al´ eatoires ind´ ependantes et applications, Ann. Inst. H. Poincar´ e Probab. Statist. 25 (1989), 109–142

1989
[33]

Peres, Analytic dependence of Lyapunov exponents on transition probabil- ities, Lyapunov exponents (Oberwolfach, 1990), Lecture Notes in Math., vol

Y. Peres, Analytic dependence of Lyapunov exponents on transition probabil- ities, Lyapunov exponents (Oberwolfach, 1990), Lecture Notes in Math., vol. 1486, Springer-Verlag, 1991, pp. 64–80

1990
[34]

Ruelle, Characteristic exponents and invariant manifolds in Hilbert space , Annals of Math

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space , Annals of Math. 115 (1982), 243–290

1982
[35]

Adriana S´ anchez and Marcelo Viana,Lyapunov exponents of probability distri- butions with non-compact support , 2020

2020
[36]

Simon and M

B. Simon and M. Taylor, Harmonic analysis on SL(2, R) and smoothness of the density of states in the one-dimensional Anderson model , Comm. Math. Phys. 101 (1985), 1–19

1985
[37]

El Hadji Yaya Tall and Marcelo Viana, Moduli of continuity for the Lyapunov exponents of random GL(2)-cocycles, Trans. Amer. Math. Soc. 373 (2020), no. 2, 1343–1383

2020
[38]

Viana and K

M. Viana and K. Oliveira, Foundations of ergodic theory , Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016

2016
[39]

145, Cambridge University Press, Cambridge, 2014

Marcelo Viana, Lectures on Lyapunov exponents , Cambridge Studies in Ad- vanced Mathematics, vol. 145, Cambridge University Press, Cambridge, 2014. MR 3289050

2014
[40]

58, American Mathematical Society, Providence, RI, 2003

C´ edric Villani,Topics in optimal transportation , Graduate Studies in Math- ematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483

2003
[41]

338, Springer-Verlag, Berlin, 2009, Old and new

, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009, Old and new. MR 2459454 NON-INVERTIBLE AND NON-COMPACT RANDOM COCYCLES 53 Departamento de Matem´atica and CEMS.UL, F a...

2009

Pith tools

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

[1] [1]

Artur Amorim, Marcelo Duraes, and Aline Melo, Analiticity of the lyapunov exponents of random products of matrices , 2025

work page 2025

[2] [2]

Artur Avila, Alex Eskin, and Marcelo Viana, Continuity of the lyapunov expo- nents of random matrix products , 2023

work page 2023

[3] [3]

Lucas Backes, Aaron Brown, and Clark Butler, Continuity of Lyapunov expo- nents for cocycles with invariant holonomies , J. Mod. Dyn. 12 (2018), 223–260

work page 2018

[4] [4]

Alexandre Baraviera and Pedro Duarte, Approximating Lyapunov exponents and stationary measures , J. Dynam. Differential Equations 31 (2019), no. 1, 25–48. MR 3935134

work page 2019

[5] [5]

07, 2150025

Alexandre Baraviera, Pedro Duarte, and Maria Joana Torres, Stationary mea- sures on infinite graphs , Communications in Contemporary Mathematics 24 (2022), no. 07, 2150025

work page 2022

[6] [6]

Barrientos and Dominique Malicet, Mostly contracting random maps, 2024

Pablo G. Barrientos and Dominique Malicet, Mostly contracting random maps, 2024. NON-INVERTIBLE AND NON-COMPACT RANDOM COCYCLES 51

work page 2024

[7] [7]

Jamerson Bezerra, Ao Cai, Pedro Duarte, Catalina Freijo, and Silvius Klein, A dynamical Thouless formula , Adv. Math. 438 (2024), 50 (English), Id/No 109446

work page 2024

[8] [8]

2, 829–875

Jamerson Bezerra and Pedro Duarte, Upper Bound on the Regularity of the Lya- punov Exponent for Random Products of Matrices , Communications in Math- ematical Physics 403 (2023), no. 2, 829–875

work page 2023