pith. sign in

arxiv: 2604.09878 · v1 · submitted 2026-04-10 · 🧮 math.DS

Discontinuity example for the Lyapunov exponents on the boundary of the uniformly hyperbolic set

Raquel Saraiva This is my paper

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lyapunov exponentslinear cocyclesuniform hyperbolicitydiscontinuityBernoulli shiftSL(2,R)C_δ-log topology
0
0 comments X

The pith

A locally constant SL(2,R) cocycle over a Bernoulli shift provides a discontinuity point for Lyapunov exponents in the C_δ-log topology.

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

The paper constructs an explicit example of a linear cocycle that sits exactly on the boundary of the uniformly hyperbolic set. For this cocycle the map sending a cocycle to its Lyapunov exponents fails to be continuous when the space of cocycles is given a topology finer than the usual C^0 topology. The authors prove that the example can be approached arbitrarily closely, in the C_δ-log topology, by cocycles whose Lyapunov exponents are exactly zero. This shows that vanishing of the exponents can occur under perturbations that are small in a logarithmic sense even though the target cocycle itself lies on the hyperbolicity boundary.

Core claim

We present an example of a discontinuity point for the Lyapunov exponents when viewed as a function of the cocycle in a topology finer than the C^0-topology. The linear cocycle taking values in SL(2,R) is locally constant, defined over a Bernoulli shift, and lies on the boundary of the uniformly hyperbolic set. In particular, we show that it can be approximated, in the C_δ-log-topology, by cocycles whose Lyapunov exponents vanish.

What carries the argument

The locally constant SL(2,R)-valued cocycle over the Bernoulli shift that lies precisely on the boundary of the uniformly hyperbolic set; this boundary location permits approximation by zero-exponent cocycles under C_δ-log perturbations.

If this is right

  • The Lyapunov-exponent map is discontinuous at certain boundary points when cocycles are equipped with the C_δ-log topology.
  • Uniform hyperbolicity can be destroyed by arbitrarily small logarithmic perturbations even for cocycles that are limits of hyperbolic ones.
  • Discontinuities of this kind are expected only at the boundary and not in the interior of the uniformly hyperbolic set.
  • Local constancy over a Bernoulli shift is sufficient to produce an explicit sequence of approximating cocycles with zero exponents.

Where Pith is reading between the lines

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

  • Similar boundary discontinuities may appear for cocycles over other base systems or with values in higher-dimensional matrix groups.
  • Continuity of Lyapunov exponents might be restored only under strictly stronger assumptions such as uniform hyperbolicity in the interior.
  • The construction suggests that the precise modulus of continuity of the exponent map depends on how close a cocycle sits to the hyperbolicity boundary.

Load-bearing premise

The chosen locally constant cocycle lies exactly on the boundary of the uniformly hyperbolic set and the C_δ-log topology is the correct finer topology in which the zero-exponent approximations exist.

What would settle it

A direct computation of the Lyapunov exponents for a sequence of C_δ-log perturbations of the example cocycle that shows the exponents remain bounded away from zero, or a proof that the example cocycle itself has vanishing exponents.

read the original abstract

We present an example of a discontinuity point for the Lyapunov exponents when viewed as a function of the cocycle in a topology finer than the $C^0$-topology. The linear cocycle taking values in SL(2,R) is locally constant, defined over a Bernoulli shift, and lies on the boundary of the uniformly hyperbolic set. In particular, we show that it can be approximated, in the $C_{\delta-\log}$-topology, by cocycles whose Lyapunov exponents vanish.

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 constructs an explicit locally constant SL(2,R)-cocycle over the Bernoulli shift that lies on the boundary of the uniformly hyperbolic set, with non-vanishing Lyapunov exponents, and shows that it can be approximated in the C_δ-log topology by cocycles whose Lyapunov exponents vanish, thereby exhibiting a discontinuity of the Lyapunov exponent map in a topology strictly finer than C^0.

Significance. If the boundary placement and approximation are rigorously established, the example supplies a concrete, constructive illustration of how Lyapunov exponents can fail to be continuous under perturbations stronger than uniform topology. This is useful for sharpening results on the stability of hyperbolicity and the continuity of Lyapunov exponents for linear cocycles, and the explicit matrix assignments plus approximation scheme constitute a verifiable contribution.

major comments (1)
  1. [Construction of the cocycle and boundary verification (main body, following the abstract statement)] The central claim requires that the given locally constant cocycle is precisely on the boundary of the uniformly hyperbolic set (positive top Lyapunov exponent yet no invariant splitting with uniform rates). The estimates establishing absence of uniform hyperbolicity (while preserving positive LE) are load-bearing; without explicit verification that the splitting constants cannot be chosen uniformly, the discontinuity in the C_δ-log topology does not follow.
minor comments (2)
  1. [Introduction] Notation for the C_δ-log topology and the precise definition of the Bernoulli shift should be introduced with a short self-contained paragraph early in the text to aid readers unfamiliar with the finer topology.
  2. [Approximation section] The approximation argument would benefit from a displayed inequality or diagram clarifying how the log-δ distance controls the perturbation while driving the exponent to zero.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the significance of our example. The main comment concerns the explicit verification that our cocycle lies on the boundary of uniform hyperbolicity. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim requires that the given locally constant cocycle is precisely on the boundary of the uniformly hyperbolic set (positive top Lyapunov exponent yet no invariant splitting with uniform rates). The estimates establishing absence of uniform hyperbolicity (while preserving positive LE) are load-bearing; without explicit verification that the splitting constants cannot be chosen uniformly, the discontinuity in the C_δ-log topology does not follow.

    Authors: We agree that the boundary property is central and that the estimates must be fully explicit. The manuscript establishes a positive top Lyapunov exponent by computing the almost-sure limit of (1/n) log ||A^n(x)|| for the locally constant cocycle A, using the Bernoulli measure and the specific matrix products to obtain λ > 0. Absence of uniform hyperbolicity is shown by exhibiting, for every candidate continuous splitting, a sequence of base points whose forward and backward products have norms that remain bounded away from exponential growth or decay, violating any uniform rate. To make this fully rigorous and address the referee's concern, we will add a dedicated lemma in the revision that quantifies the dependence: for any purported splitting E and any M > 0, there exists a point x such that the product norms over intervals of length n fail to satisfy the uniform hyperbolicity inequality with constant M, independently of n. This uses the local constancy and the explicit form of the matrices to construct the obstructing sequences explicitly. We believe these additions will render the argument self-contained while preserving the original construction and the approximation in the C_δ-log topology. revision: yes

Circularity Check

0 steps flagged

Direct construction of a discontinuity counterexample with no circular reductions

full rationale

The paper is a constructive example: it explicitly defines a locally constant SL(2,R)-cocycle over the Bernoulli shift, verifies it lies on the boundary of the uniformly hyperbolic set (positive but non-uniform Lyapunov exponents), and exhibits C_δ-log approximations by zero-exponent cocycles. No equations reduce the claimed discontinuity to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation consists of matrix assignments and direct estimates that stand independently of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a locally constant SL(2,R)-valued cocycle over a Bernoulli shift that is on the boundary of uniform hyperbolicity and admits approximation in C_δ-log by zero-LE cocycles. No free parameters are introduced; the construction uses standard properties of Bernoulli shifts and matrix products.

axioms (2)
  • standard math Bernoulli shift is ergodic and the cocycle is locally constant (constant on cylinder sets).
    Invoked to define the base dynamics and the cocycle in the abstract.
  • domain assumption The cocycle lies on the boundary of the uniformly hyperbolic set in the space of cocycles.
    This is the key positioning assumption that enables the discontinuity example.

pith-pipeline@v0.9.0 · 5365 in / 1431 out tokens · 19294 ms · 2026-05-10T15:55:31.174595+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

18 extracted references · 18 canonical work pages

  1. [1]

    (1998).Random Dynamical Systems

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

    work page arXiv 2023

  2. [2]

    Backes, A

    L. Backes, A. Brown, and C. Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies.Journal of Modern Dynamics, 12:223–260, 2018

    work page 2018

  3. [3]

    Barreira.Thermodynamic Formalism and Applications to Dimension Theory

    L. Barreira.Thermodynamic Formalism and Applications to Dimension Theory. Birkh¨ auser, 2012

    work page 2012

  4. [4]

    J. Bochi. Genericity of zero Lyapunov exponents.Ergodic Theory and Dynamical Systems, 22(6):1667–1696, 2002

    work page 2002

  5. [5]

    Bocker-Neto and M

    C. Bocker-Neto and M. Viana. Continuity of Lyapunov exponents for random two-dimensional matrices.Ergodic Theory and Dynamical Systems, 37(5):1413–1442, 2017

    work page 2017

  6. [6]

    C. Butler. Discontinuity of Lyapunov exponents near fiber bunched cocycles.Ergodic Theory and Dynamical Systems, 38(2):523–539, 2018

    work page 2018

  7. [7]

    Duarte, S

    P. Duarte, S. Klein, and M. Santos. A random cocycle with non H¨ older Lyapunov exponent.Discrete and Continuous Dynamical Systems, 39(8):4719–4739, 2019

    work page 2019

  8. [8]

    Friedman.Introduction to ergodic theory

    N. Friedman.Introduction to ergodic theory. Van Nostrand, 1969

    work page 1969

  9. [9]

    Furstenberg and H

    H. Furstenberg and H. Kesten. Products of random matrices.The Annals of Mathematical Statistics, 31(2):457–469, 1960

    work page 1960

  10. [10]

    Ledrappier and L.-S

    F. Ledrappier and L.-S. Young. Dimension of invariant measures.Journal of Differential Equations, 64:1–33, 1986

    work page 1986

  11. [11]

    A. M. Lyapunov.The General Problem of the Stability of Motion. Khar’kov Mathematical Society, 1892. In Russian; French translation published in 1907

    work page 1907

  12. [12]

    E. C. Malheiro and M. Viana. Lyapunov exponents of linear cocycles over markov shifts.Stochastics and Dynamics, 15:1–27, 2015

    work page 2015

  13. [13]

    Mamani and R

    E. Mamani and R. Saraiva. Discontinuity of Lyapunov exponents for SL(2,R)-valued cocycles.arXiv preprint arXiv:2407.20310v2, 2026

    work page arXiv 2026

  14. [14]

    V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems.Transactions of the Moscow Mathematical Society, 19:197–231, 1968

    work page 1968

  15. [15]

    Ya. B. Pesin. Families of invariant manifolds corresponding to nonzero characteristic exponents.Mathematics of the USSR- Izvestiya, 10(6):1261–1305, 1976

    work page 1976

  16. [16]

    Ya. B. Pesin. Lyapunov characteristic exponents and smooth ergodic theory.Russian Mathematical Surveys, 32(4):55–114, 1977

    work page 1977

  17. [17]

    Viana.Lectures on Lyapunov Exponents, volume 145 ofCambridge Studies in Advanced Mathematics

    M. Viana.Lectures on Lyapunov Exponents, volume 145 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2014

    work page 2014

  18. [18]

    L.-S. Young. Dimension, entropy and Lyapunov exponents.Ergodic Theory and Dynamical Systems, 2:109–124, 1982. 13

    work page 1982