pith. sign in

arxiv: 2605.16718 · v1 · pith:JYZ7ML5Anew · submitted 2026-05-16 · 🧮 math.DS · math.PR

Regularity of Lyapunov exponents at one-point Lyapunov spectra: the semisimple case

Yingjian Liu , Marcelo Viana This is my paper

Pith reviewed 2026-05-19 20:10 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords Lyapunov exponentsWasserstein distancelog-Hölder continuitysemisimple measuresone-point spectrumrandom matrix productsdynamical systems
0
0 comments X

The pith

Lyapunov exponents are pointwise log-Hölder continuous with respect to the Wasserstein distance at semisimple probability measures with one-point Lyapunov spectrum.

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

This paper looks at Lyapunov exponents viewed as functions of probability measures on the general linear group GL(d, R). It focuses on the special case of semisimple measures that have only a single Lyapunov exponent. The authors establish that these exponents vary in a pointwise log-Hölder continuous manner when the measures are compared using the Wasserstein distance. The argument proceeds by splitting the linear action into virtually conformal subspaces and then applying a Berry-Esseen estimate to the random walk on those subspaces. Such quantitative continuity would let one track how small changes in the driving measure affect long-term growth rates in linear random products.

Core claim

We study the regularity of Lyapunov exponents as functions on the space of compactly supported probability measures on GL(d,R). We prove that the Lyapunov exponents are pointwise log-Hölder continuous with respect to the Wasserstein distance, at semisimple probability measures with one-point Lyapunov spectrum. The proof relies on a decomposition of the action into virtually conformal subspaces and a Berry-Esseen type estimate for the random walk towards these subspaces.

What carries the argument

Decomposition of the linear action into virtually conformal subspaces combined with a Berry-Esseen estimate for the associated random walk.

If this is right

  • Small changes in the driving measure produce only controlled changes in the Lyapunov spectrum.
  • The result supplies a quantitative version of continuity for Lyapunov exponents in the semisimple one-point case.
  • Similar decomposition techniques may yield regularity statements for other classes of measures on matrix groups.
  • The log-Hölder modulus gives an explicit rate that can be used to bound approximation errors when measures are discretized.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical sampling of nearby measures could then produce Lyapunov exponent estimates whose error is bounded by a logarithmic function of the sampling distance.
  • The same splitting into conformal subspaces might be tried on non-semisimple measures to test whether the one-point assumption can be relaxed.
  • Low-dimensional examples, such as 2-by-2 matrices, could be checked by direct computation to see the sharpness of the log-Hölder exponent.

Load-bearing premise

The probability measures admit a decomposition into virtually conformal subspaces for which a Berry-Esseen type estimate holds for the random walk.

What would settle it

A single explicit semisimple measure with one-point spectrum for which the top Lyapunov exponent changes faster than any log-Hölder modulus under a sequence of arbitrarily small Wasserstein perturbations would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.16718 by Marcelo Viana, Yingjian Liu.

Figure 1
Figure 1. Figure 1: The standard simplex ∆2 partitioned into the neighbourhoods B1 , B2 , B3 around each axis, and the central region Bc r . 2. Analytical reduction (Section 4). Via Furstenberg’s formula and the Lipschitz continuity of the Markov operator, the difference |λ1(µ ′ ) − λ1(µ)| is translated into a convergence rate estimate for the µ ′ -stationary measure under the unperturbed operator Pµ. The integral is split in… view at source ↗
read the original abstract

We study the regularity of Lyapunov exponents as functions on the space of compactly supported probability measures on $\mathrm{GL}(d,\mathbb{R})$. We prove that the Lyapunov exponents are pointwise log-H\"older continuous with respect to the Wasserstein distance, at semisimple probability measures with one-point Lyapunov spectrum. The proof relies on a decomposition of the action into virtually conformal subspaces and a Berry-Esseen type estimate for the random walk towards these subspaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that Lyapunov exponents, viewed as functions on the space of compactly supported probability measures on GL(d,R), are pointwise log-Hölder continuous with respect to the Wasserstein metric at semisimple measures possessing a one-point Lyapunov spectrum. The argument proceeds by decomposing the linear action into virtually conformal subspaces and controlling the random walk on those subspaces via a Berry-Esseen-type estimate.

Significance. If the uniformity of the Berry-Esseen constants can be established, the result supplies a concrete modulus of continuity for Lyapunov exponents at a natural class of measures, extending existing continuity statements in random dynamical systems and providing a quantitative tool for perturbation analysis near semisimple one-point spectra.

major comments (1)
  1. [Abstract and §3] Abstract and §3 (proof outline): the Berry-Esseen estimate for the projected random walk on the virtually conformal subspaces is invoked to obtain the log-Hölder modulus, yet no quantitative statement is given showing that the implicit constants (e.g., those depending on minimal expansion rates or spectral gaps) remain controlled under small Wasserstein perturbations of the driving measure. Without such uniformity, the claimed pointwise continuity at the semisimple point does not follow.
minor comments (2)
  1. [§2] The definition of 'virtually conformal subspaces' should be stated explicitly in §2 before the decomposition is used.
  2. [Abstract and §1] Notation for the Wasserstein distance and the one-point spectrum should be fixed consistently between the abstract and the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to establish uniformity of constants in the Berry-Esseen estimates. This is a substantive point that strengthens the argument for pointwise continuity. We address it below and will revise the manuscript to incorporate the required quantitative control.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (proof outline): the Berry-Esseen estimate for the projected random walk on the virtually conformal subspaces is invoked to obtain the log-Hölder modulus, yet no quantitative statement is given showing that the implicit constants (e.g., those depending on minimal expansion rates or spectral gaps) remain controlled under small Wasserstein perturbations of the driving measure. Without such uniformity, the claimed pointwise continuity at the semisimple point does not follow.

    Authors: We agree that a quantitative uniformity statement is necessary to pass from the fixed-measure Berry-Esseen estimate to the claimed pointwise log-Hölder continuity under Wasserstein perturbations. In the revised version we will insert a new lemma (placed after the proof outline in §3) establishing that the constants appearing in the Berry-Esseen bound—including those controlled by the minimal expansion rates on the virtually conformal subspaces and by the spectral gaps—depend continuously on the driving measure in the Wasserstein topology. The key observation is that semisimplicity together with the one-point Lyapunov spectrum at the reference measure μ₀ implies that, for all measures μ sufficiently close to μ₀, the Lyapunov exponents remain simple and the associated expansion rates stay uniformly bounded away from zero and infinity; these bounds are continuous in the Wasserstein distance because the top Lyapunov exponent is itself continuous at μ₀. Consequently the implicit constants remain bounded in a neighborhood of μ₀, yielding a uniform log-Hölder modulus and completing the proof of pointwise continuity. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via standard estimates; no reduction to inputs by construction

full rationale

The paper presents a direct proof that Lyapunov exponents are pointwise log-Hölder continuous at semisimple one-point measures, relying on a decomposition into virtually conformal subspaces followed by a Berry-Esseen estimate for the associated random walk. No quoted step equates a claimed continuity modulus to a fitted parameter, a self-citation chain, or a renamed input; the argument invokes external quantitative tools whose uniformity is asserted under the stated assumptions rather than derived tautologically from the target result. The central claim therefore retains independent content outside its own definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on background facts about Lyapunov exponents for random matrix products and on the existence of virtually conformal decompositions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Lyapunov exponents exist and are well-defined for compactly supported probability measures on GL(d,R)
    Invoked implicitly when treating the exponents as functions on the space of measures.
  • domain assumption Semisimple measures admit a decomposition into virtually conformal subspaces
    Central to the proof strategy stated in the abstract.

pith-pipeline@v0.9.0 · 5593 in / 1349 out tokens · 42641 ms · 2026-05-19T20:10:18.384081+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    Aaronson , TITLE =

    J. Aaronson , TITLE =. 1997 , PAGES =

    work page 1997

  2. [2]

    Abdenur and L

    F. Abdenur and L. J. D\' az. Shadowing in the C^1 topology

    work page

  3. [3]

    Abdenur and A

    F. Abdenur and A. Avila and J. Bochi , TITLE =. Proc. Amer. Math. Soc. , VOLUME =. 2004 , PAGES =

    work page 2004

  4. [4]

    F. Abdenur. Generic robustness of spectral decompositions. Annales Scient. E.N.S. 2003

    work page 2003

  5. [5]

    F. Abdenur. Attractors of generic diffeomorphisms are persistent. Nonlinearity

    work page

  6. [6]

    N. H. Abel. Ueber einige bestimmte I ntegrale. J. Reine Angew. Math. 1827

    work page

  7. [7]

    Abraham and S

    R. Abraham and S. Smale. Nongenericity of -stability. Global analysis. 1970

    work page 1970

  8. [8]

    Bashforth and J

    F. Bashforth and J. C. Adams. An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid, by F. B ashforth, with an explanation of the method of integration employed in constucting the tables which give the theoretical forms of such drops, by J. A dams. 1883

    work page

  9. [9]

    Adler and A

    R. Adler and A. Konheim and M. McAndrew , TITLE =. Trans. Amer. Math. Soc. , VOLUME =. 1965 , PAGES =

    work page 1965

  10. [10]

    Adl-Zarabi

    K. Adl-Zarabi. Absolutely continuous invariant measures for piecewise expanding C ^2 transformations in ^n on domains with cusps on their boundaries. Ergod. Th. & Dynam. Sys. 1996

    work page 1996

  11. [11]

    V. S. Afraimovich and S.-C. Chow and W. Liu , TITLE =. J. Dynam. Differential Equations , VOLUME =. 1995 , PAGES =

    work page 1995

  12. [12]

    V. S. Afraimovich and Ya. B. Pesin. The Dimension of L orenz type attractors. Sov. Math.Phys. Rev. 1987

    work page 1987

  13. [13]

    V. S. Afraimovich and V. V. Bykov and L. P. Shil'nikov. On attracting structurally unstable limit sets of L orenz type attractors. Trans. Moscow Math. Soc. 1983

    work page 1983

  14. [14]

    V. S. Afraimovich and V. V. Bykov and L. P. Shil'nikov. On attracting structurally unstable limit sets of L orenz attractor type. Trudy Moskov. Mat. Obshch. 1982

    work page 1982

  15. [15]

    V. S. Afraimovich and V. V. Bykov and L. P. Shil'nikov. On the appearence and structure of the L orenz attractor. Dokl. Acad. Sci. USSR. 1977

    work page 1977

  16. [16]

    Ahlfors , TITLE =

    L. Ahlfors , TITLE =. 1973 , PAGES =

    work page 1973

  17. [17]

    Ahlfors , TITLE =

    L. Ahlfors , TITLE =. 1966 , PAGES =

    work page 1966

  18. [18]

    Ahlfors and L

    L. Ahlfors and L. Bers , TITLE =. Ann. of Math. , VOLUME =. 1960 , PAGES =

    work page 1960

  19. [19]

    Ahlfors and L

    L. Ahlfors and L. Sario , TITLE =. 1960 , PAGES =

    work page 1960

  20. [20]

    G. B. Airy. On the intensity of light in the neighbourhood of a caustic. Trans. Cambridge Philos. Soc. 1838

    work page

  21. [21]

    V. M. Alekseev and M. V. Jakobson. Symbolic dynamics and hyperbolic dynamical systems. Phys. Rep. 1981

    work page 1981

  22. [22]

    Alexander , TITLE =

    J. Alexander , TITLE =. Trans. Amer. Math. Soc. , VOLUME =. 1915 , PAGES =

    work page 1915

  23. [23]

    C. B. Allendoerfer. The E uler number of a R iemannian manifold. Amer. J. Math. 1940

    work page 1940

  24. [24]

    C. B. Allendoerfer and A. Weil. The G auss-- B onnet theorem for R iemannian polyhedra. Trans. Amer. Math. Soc. 1943

    work page 1943

  25. [25]

    Alvarez , TITLE =

    S. Alvarez , TITLE =. Ergodic Theory Dynam. Systems , VOLUME =. 2018 , PAGES =

    work page 2018

  26. [26]

    Alvarez , TITLE =

    S. Alvarez , TITLE =. Israel J. Math. , VOLUME =. 2017 , PAGES =

    work page 2017

  27. [27]

    J. F. Alves , TITLE =. Discrete Contin. Dyn. Syst. , VOLUME =. 2006 , PAGES =

    work page 2006

  28. [28]

    J. F. Alves , TITLE =. Nonlinearity , VOLUME =. 2004 , PAGES =

    work page 2004

  29. [29]

    J. F. Alves. Statistical analysis of non-uniformly expanding dynamical systems. 2003

    work page 2003

  30. [30]

    J. F. Alves , TITLE =. Discrete Contin. Dynam. Systems , VOLUME =. 2001 , PAGES =

    work page 2001

  31. [31]

    J. F. Alves , TITLE =. Ann. Sci. \'Ecole Norm. Sup. , VOLUME =. 2000 , PAGES =

    work page 2000

  32. [32]

    J. F. Alves. SRB measures for nonhyperbolic systems with multidimensional expansion. 1997

    work page 1997

  33. [33]

    J. F. Alves and V. Ara. Hyperbolic times: frequency versus integrability , JOURNAL =. 2004 , PAGES =

    work page 2004

  34. [34]

    J. F. Alves and V. Ara. Random perturbations of nonuniformly expanding maps , NOTE =. Ast\'erisque , VOLUME =. 2003 , PAGES =

    work page 2003

  35. [35]

    J. F. Alves and V. Ara. On the uniform hyperbolicity of some nonuniformly hyperbolic systems , JOURNAL =. 2003 , PAGES =

    work page 2003

  36. [36]

    J. F. Alves and V. Ara. Stochastic stability of non-uniformly hyperbolic diffeomorphisms , JOURNAL =. 2007 , PAGES =

    work page 2007

  37. [37]

    J. F. Alves and S. Luzzatto and V. Pinheiro , TITLE =. Ann. Inst. H. Poincar. 2005 , PAGES =

    work page 2005

  38. [38]

    J. F. Alves and S. Luzzatto and V. Pinheiro , TITLE =. Ergodic Theory Dynam. Systems , VOLUME =. 2004 , PAGES =

    work page 2004

  39. [39]

    J. F. Alves and S. Luzzatto and V. Pinheiro , TITLE =. Electron. Res. Announc. Amer. Math. Soc. , VOLUME =. 2003 , PAGES =

    work page 2003

  40. [40]

    J. F. Alves and V. Pinheiro , TITLE =. J. Stat. Phys. , VOLUME =. 2008 , PAGES =

    work page 2008

  41. [41]

    A. Amorim. Analytic regularity of L yapunov exponents for derivative cocycles and density of hyperbolicity. 2026

    work page 2026

  42. [42]

    Analyticity of the L yapunov exponents of random products of matrices

    Artur Amorim and Marcelo Dur \ a es and Aline Melo. Analyticity of the L yapunov exponents of random products of matrices. 2025

    work page 2025

  43. [43]

    Anderson

    P. Anderson. Absence of diffusion in certain random lattices. Phys. Rev. 1958

    work page 1958

  44. [44]

    Andersson

    M. Andersson. Robust ergodic properties in partially hyperbolic dynamics. Trans. Amer. Math. Soc. 2010

    work page 2010

  45. [45]

    Andersson and C

    M. Andersson and C. V\'. On mostly expanding diffeomorphisms , JOURNAL =. 2018 , PAGES =

    work page 2018

  46. [46]

    Andersson and C

    M. Andersson and C. H. V\' a squez. Statistical stability of mostly expanding diffeomorphisms

    work page

  47. [47]

    Andronov and L

    A. Andronov and L. Pontryagin. Syst \`e mes grossiers. Dokl. Akad. Nauk. USSR. 1937

    work page 1937

  48. [48]

    D. V. Anosov. Geodesic flows on closed R iemannian manifolds of negative curvature. Proc. Steklov Math. Inst. 1967

    work page 1967

  49. [49]

    D. V. Anosov and Ya. G. Sinai. Certain smooth ergodic systems. Russian Math. Surveys. 1967

    work page 1967

  50. [50]

    N. Aoki. The set of A xiom A diffeomorphisms with no cycles. Bol. Soc. Bras. Mat., Nova S\'er. 1992

    work page 1992

  51. [51]

    Aoun and Y

    R. Aoun and Y. Guivarc'h. Random matrix products when the top L yapunov exponent is simple. J. Eur. Math. Soc. 2020

    work page 2020

  52. [52]

    V. Ara. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures , JOURNAL =. 2007 , PAGES =

    work page 2007

  53. [53]

    V. Ara. Large deviations bound for semiflows over a non-uniformly expanding base , JOURNAL =. 2007 , PAGES =

    work page 2007

  54. [54]

    Ara \'u jo

    V. Ara \'u jo. Infinitely many stochastically stable attractors. Nonlinearity. 2001

    work page 2001

  55. [55]

    Ara \'u jo

    V. Ara \'u jo. Attractors and time averages for random maps. Annales de l'Inst. Henri Poincar\'e - Analyse Non-lin\'eaire. 2000

    work page 2000

  56. [56]

    Ara \'u jo

    V. Ara \'u jo. Attractors and time averages for random maps. 1998

    work page 1998

  57. [57]

    V. Ara. Large deviations for non-uniformly expanding maps , JOURNAL =. 2006 , PAGES =

    work page 2006

  58. [58]

    V. Ara. Three-dimensional flows , SERIES =. 2010 , PAGES =

    work page 2010

  59. [59]

    Ara \'u jo and A

    V. Ara \'u jo and A. Tahzibi. Stochastic stability at the boundary of expanding maps

    work page

  60. [60]

    Arbieto and J

    A. Arbieto and J. Bochi , TITLE =. Stoch. Dyn. , VOLUME =. 2003 , PAGES =

    work page 2003

  61. [61]

    Arbieto and C

    A. Arbieto and C. Matheus. Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials. 2006

    work page 2006

  62. [62]

    Arbieto and C

    A. Arbieto and C. Matheus. A P asting L emma I : T he case of vector fields. Ergod. Th. & Dynam. Sys. 2007

    work page 2007

  63. [63]

    Arbieto and C

    A. Arbieto and C. Matheus and K. Oliveira. Equilibrium states for random non-uniformly expanding maps. Nonlinearity. 2004

    work page 2004

  64. [64]

    M.-C. Arnaud. Approximation des ensembles -limites des diff\'eomorphismes par des orbites p\'eriodiques. Annales Sci. E. N. S. 2003

    work page 2003

  65. [65]

    M.-C. Arnaud. The generic C 1 symplectic diffeomorphisms of symplectic 4 -dimensional manifolds are hyperbolic, partially hyperbolic, or have a completely elliptic point. Ergod. Th. & Dynam. Sys. 2002

    work page 2002

  66. [66]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. Ergod. Th. 2001 , PAGES =

    work page 2001

  67. [67]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , VOLUME =. 2000 , PAGES =

    work page 2000

  68. [68]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. Bol. Soc. Brasil. Mat. , VOLUME =. 1999 , PAGES =

    work page 1999

  69. [69]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , VOLUME =. 1999 , PAGES =

    work page 1999

  70. [70]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. M\'em. Soc. Math. Fr. , VOLUME =. 1998 , PAGES =

    work page 1998

  71. [71]

    Arnaud , TITLE =

    M.-C. Arnaud , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , VOLUME =. 1996 , PAGES =

    work page 1996

  72. [72]

    Arnaud and C

    M.-C. Arnaud and C. Bonatti and S. Crovisier , TITLE =. Ergodic Theory Dynam. Systems , VOLUME =. 2005 , PAGES =

    work page 2005

  73. [73]

    Arn \'e odo and P

    A. Arn \'e odo and P. Coullet and C. Tresser. A possible new mechanism for the onset of turbulence. Physical Letters. 1981

    work page 1981

  74. [74]

    Arn \'e odo and P

    A. Arn \'e odo and P. Coullet and C. Tresser. Possible new strange attractors with spiral structure. Comm. Math. Phys. 1981

    work page 1981

  75. [75]

    Arnold and J

    D. Arnold and J. Rogness , TITLE =. Notices Amer. Math. Soc. , VOLUME =. 2008 , PAGES =

    work page 2008

  76. [76]

    V. I. Arnold , TITLE =. 1992 , PAGES =

    work page 1992

  77. [77]

    I Arnold , TITLE =

    V. I Arnold , TITLE =

    work page

  78. [78]

    V. I. Arnold. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Mathematical Surveys. 1963

    work page 1963

  79. [79]

    V. I. Arnold. Small denominators I : O n the mapping of a circle to itself. Izv. Akad. Nauk. Math. Series. 1961

    work page 1961

  80. [80]

    V. I. Arnold. Mathematical methods of classical mechanics. 1978

    work page 1978

Showing first 80 references.