pith. sign in

arxiv: 1907.09400 · v1 · pith:RNA4O23Rnew · submitted 2019-07-18 · 🧮 math.DS

Laypunov Irregular Points With Distributional Chaos

An Chen , Xueting Tian This is my paper

Pith reviewed 2026-05-24 19:53 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lyapunov irregular pointsdistributional chaosexponential specification propertymatrix cocycleergodic measuresLyapunov spectrumdynamical systems
0
0 comments X

The pith

For dynamical systems with exponential specification property, the Lyapunov-irregular set of a Holder continuous matrix cocycle displays distributional chaos of type 1 whenever ergodic measures have distinct Lyapunov spectra.

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

The paper establishes that when a dynamical system satisfies the exponential specification property and the matrix cocycle is Holder continuous, the presence of ergodic measures with differing Lyapunov spectra forces the set of points where Lyapunov averages diverge to exhibit distributional chaos of type 1. This stands in contrast to the Oseledec theorem, which guarantees that this irregular set has zero measure with respect to every invariant probability measure. A sympathetic reader cares because the result separates measure-theoretic regularity from topological complexity, showing that zero-measure sets can still carry strong forms of chaos under specification assumptions.

Core claim

If a dynamical system f has the exponential specification property and A is a Holder continuous matrix cocycle, and if ergodic measures exist with different Lyapunov spectra, then the Lyapunov-irregular set of A displays distributional chaos of type 1.

What carries the argument

The Lyapunov-irregular set (points where Oseledec averages of the cocycle diverge) and its property of displaying distributional chaos of type 1 under the given hypotheses.

If this is right

  • The irregular set is nonempty and carries a form of chaos stronger than mere density or positive entropy.
  • Specification allows construction of orbits that switch between different Lyapunov spectra, producing the irregular points.
  • The result applies uniformly to any such cocycle once multiple spectra are present.
  • It extends the contrast between zero-measure irregular sets and their topological size.

Where Pith is reading between the lines

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

  • One could test whether the same conclusion holds when the cocycle is merely continuous rather than Holder.
  • The distributional chaos might imply that the irregular set is dense in the space and has full topological entropy.
  • Similar statements could be examined for other notions of irregularity, such as points where Birkhoff averages diverge.

Load-bearing premise

The dynamical system must have the exponential specification property and the cocycle must be Holder continuous.

What would settle it

A concrete counter-example consisting of a map with exponential specification property, a Holder continuous matrix cocycle, at least two ergodic measures with distinct Lyapunov spectra, yet an irregular set that fails to satisfy the definition of distributional chaos of type 1.

read the original abstract

It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.

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

0 major / 2 minor

Summary. The paper proves that for a dynamical system f with the exponential specification property and a Hölder continuous matrix cocycle A, the existence of ergodic measures with distinct Lyapunov spectra implies that the Lyapunov-irregular set (where Oseledec averages diverge) exhibits distributional chaos of type 1. This contrasts with the zero-measure conclusion from the Oseledec Multiplicative Ergodic Theorem or Kingman's Subadditive Ergodic Theorem.

Significance. If the central claim holds, the result is significant because it shows that measure-zero irregular sets can still carry strong chaotic properties (distributional chaos of type 1) under standard specification assumptions. The derivation relies on external standard theorems without introducing free parameters or self-referential quantities, which strengthens the contribution to the literature on irregular points and chaos in dynamical systems.

minor comments (2)
  1. [Title] Title: 'Laypunov' is a typographical error and should read 'Lyapunov'.
  2. [Abstract] Abstract, lines 4-5: the statement of the main theorem is clear, but the precise definition of 'distributional chaos of type 1' is not recalled; a one-sentence reminder or reference would improve readability for a broad audience.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, including the accurate summary of the main result and its significance in showing that measure-zero Lyapunov irregular sets can exhibit distributional chaos of type 1 under the given assumptions. The recommendation for minor revision is noted, but the report contains no specific major comments or requested changes.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems

full rationale

The paper's central claim follows from the Oseledec Multiplicative Ergodic Theorem (explicitly cited as external) combined with the stated assumptions of exponential specification property on f and Hölder continuity on the cocycle A. The conclusion that the Lyapunov-irregular set exhibits distributional chaos of type 1 when ergodic measures have distinct Lyapunov spectra is presented as a theorem under these conditions, with no reduction of any prediction or result to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain is self-contained against standard external results in ergodic theory and dynamical systems, with no equations or steps that equate the output to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on two classical ergodic theorems plus two domain assumptions about the dynamical system and cocycle; no free parameters or new entities are introduced.

axioms (4)
  • standard math Oseledec Multiplicative Ergodic Theorem
    Used to conclude that the irregular set has zero measure with respect to invariant probabilities.
  • standard math Kingman's Subadditive Ergodic Theorem
    Alternative cited for the zero-measure statement.
  • domain assumption exponential specification property of f
    Required hypothesis for the chaos conclusion.
  • domain assumption Holder continuity of the matrix cocycle A
    Required hypothesis for the chaos conclusion.

pith-pipeline@v0.9.0 · 5604 in / 1274 out tokens · 31145 ms · 2026-05-24T19:53:29.608249+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Abdenur, C

    F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of C1-generic diffeomorphisms , Israel Journal of Mathematics, 2011, 183 (1): 1-60

    work page 2011

  2. [2]

    I. S. Baek and L. Olsen, Baire category and extremely non-normal points of invarian t sets of IFSs , Discrete Contin. Dyn. Syst., 27 (2010), 935-943

    work page 2010

  3. [3]

    Barreira and Y

    L. Barreira and Y. B. Pesin, Nonuniform hyperbolicity. Cambridge Univ. Press, Cambridge (2007)

    work page 2007

  4. [4]

    Barreira and J

    L. Barreira and J. Schmeling, Sets of non-typical points have full topological entropy an d full Haus- dorff dimension , Israel J. Math. 116 (2000), 29-70

    work page 2000

  5. [5]

    Balibrea, B

    F. Balibrea, B. Schweizer, A. Sklar, and J. Sm ´ital, Generalized specification property and distribu- tional chaos , Internat. J. Bifur. Chaos 13 (2003), 1683C1694. MR2015618

    work page 2003

  6. [6]

    Bowen, Periodic orbits for hyperbolic flows, Amer

    R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30

    work page 1972

  7. [7]

    Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans

    R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397

    work page 1971

  8. [8]

    E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points , Ergodic Theory & Dy- namical Systems, 2005, 25(4):pages. 1173-1208

    work page 2005

  9. [9]

    Chen and X

    A. Chen and X. Tian, Distributional chaos in multifractal analysis, recurrenc e and transitivity, Ergodic Theory & Dynamical Systems, to appear

    work page

  10. [10]

    Downarowicz, Positive topological entropy implies chaos DC2 , Proceedings of the American Math- ematical Society, 2012, 142(1):pages

    T. Downarowicz, Positive topological entropy implies chaos DC2 , Proceedings of the American Math- ematical Society, 2012, 142(1):pages. 137-149

    work page 2012

  11. [11]

    Denker, C

    M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space , Lecture Notes in Mathematics 527

    work page

  12. [12]

    Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Annales de l’Institut Henri Poincar´ e (B) Probability and Statistics, 1997, 33(6): 797- 815

    A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Annales de l’Institut Henri Poincar´ e (B) Probability and Statistics, 1997, 33(6): 797- 815

    work page 1997

  13. [13]

    Gogolev, Diffeomorphisms H ¨older conjugate to Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, Vol

    A. Gogolev, Diffeomorphisms H ¨older conjugate to Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, Vol. 30, no. 2 (2010), 441-456

    work page 2010

  14. [14]

    J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesaro averages, frequencies of digits and Baire category , Acta Arith., 144 (2010), 287-293

    work page 2010

  15. [15]

    Huang, X

    Y. Huang, X. Tian and X. Wang, Transitively-Saturated Property, Banach recurrence and L yapunov Regularity, Nonlinearity, 32 (7), 2019, 2721-2757

    work page 2019

  16. [16]

    Kalinin, Livˇsic Theorem for matrix cocycles , Annals of mathematics, 2011, 173 (2), 1025-1042

    B. Kalinin, Livˇsic Theorem for matrix cocycles , Annals of mathematics, 2011, 173 (2), 1025-1042

    work page 2011

  17. [17]

    Katok and B

    A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems , Encyclopedia of Mathematics and its Applications 54, Cambridge Univ. Press, Camb ridge (1995)

    work page 1995

  18. [18]

    Li and M

    J. Li and M. Wu, The sets of divergence points of self-similar measures are r esidual, J. Math. Anal. Appl., 404 (2013), 429-437

    work page 2013

  19. [19]

    Li and M

    J. Li and M. Wu, Generic property of irregular sets in systems satisfying th e specificaiton property, Discrete and Continuous Dynamical Systems 34(2014), 635-645

    work page 2014

  20. [20]

    T. Y. Li and J. A. Yorke, Period Three Implies Chaos , American Mathematical Monthly, 1975, 82(10):985-992

    work page 1975

  21. [21]

    Ma˜ n´ e,Ergodic theory and differentiable dynamics , 1987, Springer-Verlag (Berlin, London)

    R. Ma˜ n´ e,Ergodic theory and differentiable dynamics , 1987, Springer-Verlag (Berlin, London)

    work page 1987

  22. [22]

    Olsen, Extremely non-normal numbers , Math

    L. Olsen, Extremely non-normal numbers , Math. Proc. Cambridge Philos. Soc., 137(2004), 43-53

    work page 2004

  23. [23]

    V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical sys- tems, Trans. Moscow Math. Soc, 19 (1968), 197-221; translated fro m Russian. LAYPUNOV IRREGULAR POINTS WITH DISTRIBUTIONAL CHAOS 13

    work page 1968

  24. [24]

    Oprocha, Specification properties and dense distributional chaos , Discrete & Continuous Dynam- ical Systems, 2007, 17 (4) : 821-833

    P. Oprocha, Specification properties and dense distributional chaos , Discrete & Continuous Dynam- ical Systems, 2007, 17 (4) : 821-833

    work page 2007

  25. [25]

    Pesin and B

    Ya. Pesin and B. Pitskel, Topological pressure and the variation al principle for noncompact sets, Functional Anal. Appl. 18 (1984), 307-318

    work page 1984

  26. [26]

    Pollicott and H

    M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued f raction and Manneville-Pomeau transformations and applications to Di ophantine approximation, Comm. Math. Phys. 207, 1999, 145-171

    work page 1999

  27. [27]

    Ruelle, Historic behaviour in smooth dynamical systems , Global Analysis of Dynamical Systems, (2001), 63-66

    D. Ruelle, Historic behaviour in smooth dynamical systems , Global Analysis of Dynamical Systems, (2001), 63-66

    work page 2001

  28. [28]

    Schweizer and J

    B. Schweizer and J. Sm ´ital, Measures of Chaos and a Spectral Decomposition of Dynamical Systems on the Interval , Transactions of the American Mathematical Society, 1994, 344( 2):737-754

    work page 1994

  29. [29]

    Sklar and J

    A. Sklar and J. Sm ´ital, Distributional Chaos on Compact Metric Spaces via Specifica tion Properties, Journal of Mathematical Analysis & Applications, 2000, 241(2):181 -188

    work page 2000

  30. [30]

    Sm ´ital and M

    J. Sm ´ital and M. ˘Stef´ ankov´ a,Distributional chaos for triangular maps , Chaos Solitons & Fractals, 2004, 21(5):1125-1128

    work page 2004

  31. [31]

    Thompson, The irregular set for maps with the specification property ha s full topological pressure

    D. Thompson, The irregular set for maps with the specification property ha s full topological pressure . Dynamics Systems, 2008, 25(1):25-51

    work page 2008

  32. [32]

    Thompson, Irregular sets, the β-transformation and the almost specification property , Transac- tions of the American Mathematical Society, 2012, 364(10):5395- 5414

    D. Thompson, Irregular sets, the β-transformation and the almost specification property , Transac- tions of the American Mathematical Society, 2012, 364(10):5395- 5414

    work page 2012

  33. [33]

    Takens, Orbits with historic behaviour, or non-existence of averag es, Nonlinearity 21, 2008, T33- T36

    F. Takens, Orbits with historic behaviour, or non-existence of averag es, Nonlinearity 21, 2008, T33- T36

    work page 2008

  34. [34]

    Takens, E

    F. Takens, E. Verbitskiy, On the variational principle for the topological entropy of certain non- compact sets, Ergodic theory and dynamical systems, 2003, 23(1): 317-348

    work page 2003

  35. [35]

    Lyapunov `Non-typical' Points of Matrix Cocycles and Topological Entropy

    X. Tian Lyapunov ‘Non-typical’ Points of Matrix Cocycles and Topol ogical Entropy , arXiv:1505.04345

    work page internal anchor Pith review Pith/arXiv arXiv

  36. [36]

    Tian, Nonexistence of Lyapunov exponents for matrix cocycles , Annales de l¨Institut Henri Poincar, Probabilits et Statistiques, 2017, 53(1):493-502

    X. Tian, Nonexistence of Lyapunov exponents for matrix cocycles , Annales de l¨Institut Henri Poincar, Probabilits et Statistiques, 2017, 53(1):493-502

    work page 2017

  37. [37]

    M. Viana. Lectures on Lyapunov Exponents , Cambridge Studies in Advanced Mathematics, 2007, 32(4):296-308. Xueting Tian, School of Mathematical Sciences, Fudan Unive rsity, Shanghai 200433, People’s Republic of China E-mail address : xuetingtian@fudan.edu.cn URL: http://homepage.fudan.edu.cn/xuetingtian An Chen, School of Mathematical Sciences, Fudan Univ...

    work page 2007