Positive Lyapunov exponents beyond the settled regimes: Herman's high-regularity entropy-density question, a replacement for the refuted Kotani–Last conjecture, and measure-theoretic typicality

Artur Avila's 2011 paper 'Density of positive Lyapunov exponents for SL(2,R)-cocycles' proved that cocycles with a positive Lyapunov exponent are dense in all regularity classes over any non-periodic dynamical system, and that for Schrödinger cocycles a prevalent set of potentials has positive exponent on a dense set of energies; the paper explicitly marks out three regimes its methods cannot reach. (1) Herman's question: is positive metric entropy dense in the $C^r$ topology ($r\ge2$, in particular $r=\infty$) among area-preserving diffeomorphisms of surfaces — equivalently, can the tangent cocycle of any such diffeomorphism be smoothly approximated by one with a positive exponent? (2) Kotani–Last replacement: after counterexamples refuted the conjecture that ergodic Schrödinger operators with non-almost-periodic potentials have positive Lyapunov exponent for almost every energy, no reasonable replacement principle has been advanced; formulate and establish one. (3) Typicality dividing lines: determine for which base dynamics and regularities positive exponents are measure-theoretically typical and for which zero exponents persist (the KAM-rigidity regime). Later literature settles major neighboring cases — Berger–Turaev's proof of Herman's conjecture near non-hyperbolic periodic points, and Avila's spectral dichotomy for typical one-frequency analytic Schrödinger operators — but the literature available through 2026 leaves all three cores open.

Background

An $\mathrm{SL}(2,\mathbb{R})$-cocycle is the linearized data of a dynamical system: a pair $(f,A)$, where $f$ is a measure-preserving ergodic map of a probability space $(X,\mu)$ (ergodic meaning that every $f$-invariant measurable set is null or conull) and $A:X\to\mathrm{SL}(2,\mathbb{R})$ is a measurable family of $2\times2$ real matrices with determinant one. The cocycle is iterated by $(f,A)^n(x)=A(f^{n-1}x)\cdots A(x)$, and its top Lyapunov exponent is the exponential growth rate of these products,

$$L(f,A)=\lim_{n\to\infty}\frac{1}{n}\log\|A(f^{n-1}x)\cdots A(x)\|,$$

which exists for $\mu$-a.e. $x$ by the subadditive ergodic theorem and is constant $\mu$-a.e. when $f$ is ergodic. When $f$ is a $C^1$ diffeomorphism of a compact surface $\Sigma$ preserving an area form $\omega$, the derivative defines the tangent cocycle $(f,Df)$, which takes values in $\mathrm{SL}(2,\mathbb{R})$ in area-orthonormal frames; by Pesin's entropy formula (valid for $C^{1+}$ diffeomorphisms preserving a smooth measure), the Kolmogorov–Sinai metric entropy $h_\omega(f)$ — the invariant-measure entropy quantifying chaos — satisfies $h_\omega(f)=\int\max(L(f,Df),0)\,d\omega$, so on a surface positive metric entropy and positivity of the exponent of the tangent cocycle are equivalent statements. Crucially, in $(f,Df)$ the cocycle cannot be perturbed independently of the base: this constraint is what makes the diffeomorphism questions below far harder than the corresponding cocycle statements.

The second origin is spectral theory. An ergodic Schrödinger operator on $\ell^2(\mathbb{Z})$ acts by $(Hu)_n=u_{n+1}+u_{n-1}+V(f^nx)u_n$, where the potential $V$ is sampled along orbits of the base dynamics $f$. Its spectral type decomposes into absolutely continuous, singular continuous, and pure point parts, the absolutely continuous spectrum being the set of energies at which generalized eigenfunctions propagate as bounded non-decaying waves. This spectral type is governed by the transfer matrices

$$S(E,x)=\begin{pmatrix}E-V(x)&-1\\1&0\end{pmatrix},$$

an $\mathrm{SL}(2,\mathbb{R})$-cocycle over $f$ indexed by the energy $E$. By Kotani theory — the classical identity relating the two — absolutely continuous spectrum can occur only at energies where the exponent vanishes for $\mu$-a.e. $x$, so positivity of the exponent forces singular spectral behavior. Almost-periodic potentials (uniform limits of periodic potentials; the model case is a quasiperiodic potential over a rotation of a torus) form the one general class in which zero exponents and absolutely continuous spectrum are common, which is why the boundary of the almost-periodic world is the natural dividing line for the questions below.

The source paper (Avila, J. Amer. Math. Soc. 24 (2011), 999–1014) established two abundance results, valid when the cocycle is the free object being perturbed: $\mathrm{SL}(2,\mathbb{R})$-cocycles with positive exponent are dense in all regularity classes and for all non-periodic base dynamics (ergodic bases not consisting of a finite periodic orbit); and, for Schrödinger cocycles, a prevalent set of potentials has positive exponent for a dense set of energies. Prevalence is the Hunt–Sauer–Yorke notion of 'almost every' valid in infinite-dimensional spaces: a set is prevalent if it contains the full-measure set obtained by adding to every point a random element of some fixed compactly supported finite-dimensional measure. Both statements give density in the cocycle or in the energy parameter, and the paper is explicit about the three directions they do not touch.

First, for area-preserving surface diffeomorphisms, positive metric entropy is dense in the $C^1$ topology — one can embed renormalized 'cooked examples' (a construction of Katok producing diffeomorphisms with a Bernoulli invariant measure and positive entropy) away from Anosov maps (diffeomorphisms that are uniformly hyperbolic on the whole manifold) — but, the paper notes, 'whether this is true for reasonably smooth dynamics seems to be far beyond our current knowledge'. This is a question from Herman's 1998 ICM problem list: is positive metric entropy dense in the $C^\infty$ (more generally $C^r$, $r\ge2$) topology among area-preserving diffeomorphisms of surfaces? Berger and Turaev isolated a Mañé-type weak-stability expectation for this smooth case — that a conservative surface diffeomorphism which cannot be $C^\infty$-approximated by positive-entropy ones must be uniformly hyperbolic — from which the density statement would follow, and that expectation is itself open. Second, the paper recalls the far-reaching Kotani–Last conjecture for Schrödinger cocycles — except for almost periodic potentials, the Lyapunov exponent should be positive for almost every energy — reports that it was recently shown to be false by the author himself, and that 'currently no reasonable replacement has been advanced'. Third, the paper asks whether positive exponents could be 'not merely dense, but measure-theoretically typical' and observes two opposite regimes: for analytic quasiperiodic cocycles with Diophantine frequency (the frequency vector $\alpha$ satisfies a small-divisor condition $|\langle k,\alpha\rangle|\ge c|k|^{-\tau}$ for all nonzero integer vectors $k$ — the regime underlying KAM theory, the Kolmogorov–Arnold–Moser persistence of invariant tori under small perturbations of integrable systems), reducibility theorems conjugate the cocycle to a constant rotation on a positive-measure set of energies, making zero exponents non-negligible; over hyperbolic base dynamics (shifts, or diffeomorphisms with a uniformly expanding direction), almost all cocycles have nonzero exponents by a theorem of Viana. Where the true dividing lines lie — which bases and which regularities admit typicality of positive exponents — is left open.

Problem Statement

The three questions below are the open core left by Avila's 2011 density and prevalence theorems; they share one scientific objective — to delimit the regimes in which positive Lyapunov exponents are abundant (dense, or measure-theoretically typical) beyond the regimes those theorems reach — and they address that objective at three levels: the constrained tangent cocycle of conservative dynamics (Question 1), the Schrödinger subclass in which exponents carry spectral meaning (Question 2), and the general landscape of bases and regularities (Question 3). Questions 1–3 are independent: none of the three is a prerequisite for the others.

Question 1 (Herman's high-regularity entropy-density question). Let $\Sigma$ be a compact surface (closed, or the disk) with area form $\omega$, and let $\mathrm{Diff}^r_\omega(\Sigma)$ be the group of $C^r$ diffeomorphisms of $\Sigma$ preserving $\omega$, equipped with the $C^r$ topology. Is the set of $f\in\mathrm{Diff}^\infty_\omega(\Sigma)$ with positive metric entropy $h_\omega(f)>0$ dense in the $C^\infty$ topology — the case posed by Herman — and, more generally, for which $r\ge2$ does the analogous $C^r$ density hold in $\mathrm{Diff}^r_\omega(\Sigma)$? Equivalently: can every area-preserving $C^r$ surface diffeomorphism be $C^r$-approximated by one whose tangent cocycle has a positive Lyapunov exponent? The $C^\infty$ case posed by Herman and the general $r\ge2$ question are independent instances of the same density question.

Question 2 (replacement for the Kotani–Last conjecture). Formulate, and establish, a correct general principle replacing the refuted Kotani–Last conjecture: a characterization, valid for ergodic Schrödinger operators on $\ell^2(\mathbb{Z})$ with non-almost-periodic potentials (or for a natural, explicitly specified subclass not already covered by a known dichotomy), of when the Lyapunov exponent is positive for almost every energy, and of when zero exponents — hence possible absolutely continuous spectrum — can occur. The target is a replacement principle for the refuted Kotani–Last conjecture together with its proof, for potentials outside the classes already covered by a known spectral dichotomy.

Question 3 (typicality dividing lines). Determine the dividing lines between classes of $\mathrm{SL}(2,\mathbb{R})$-cocycles — specified by their base dynamics and regularity — in which positive Lyapunov exponents are measure-theoretically typical (in the precise sense of prevalence, the Hunt–Sauer–Yorke notion of 'almost every' defined above) and classes in which zero exponents persist on non-negligible sets. The one-frequency analytic quasiperiodic case is settled by Avila's spectral dichotomy; the question remains open for multi-frequency quasiperiodic bases, for lower-regularity (smooth or Hölder) potentials, and for general base dynamics. A complete determination for any one of these three open regimes resolves Question 3.

References

1. Artur Avila, Density of positive Lyapunov exponents for SL(2,R)-cocycles, Journal of the American Mathematical Society 24 (2011), 999-1014, DOI 10.1090/s0894-0347-2011-00702-9, https://doi.org/10.1090/s0894-0347-2011-00702-9; arXiv:1004.4349
2. Michael Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Documenta Mathematica Extra Volume, 797-808, DOI 10.4171/dms/1-2/76, https://doi.org/10.4171/dms/1-2/76
3. Pierre Berger and Dmitry Turaev, On Herman's positive entropy conjecture, Advances in Mathematics 349 (2019), 1234-1288, DOI 10.1016/j.aim.2019.04.002, https://doi.org/10.1016/j.aim.2019.04.002 (also arXiv:1704.02473)
4. Marie-Claude Arnaud, Exposé Bourbaki 1166 : La démonstration de la conjecture de l'entropie positive d'Herman (d'après Berger et Turaev), Astérisque 448 (2022), exposé 1166, DOI 10.24033/ast.1155, https://doi.org/10.24033/ast.1155
5. Artur Avila, Sylvain Crovisier, and Amie Wilkinson, Diffeomorphisms with positive metric entropy, Publications Mathématiques de l'IHÉS 124 (2016), 319-347, DOI 10.1007/s10240-016-0086-4, https://doi.org/10.1007/s10240-016-0086-4
6. Artur Avila, Sylvain Crovisier, and Amie Wilkinson, Symplectomorphisms with positive metric entropy, Proceedings of the London Mathematical Society 124 (2022), 691-712, DOI 10.1112/plms.12437, https://doi.org/10.1112/plms.12437
7. Chao Liang and Yun Yang, The C¹ density of nonuniform hyperbolicity in C^r conservative diffeomorphisms, Proceedings of the American Mathematical Society 145 (2016), 1539-1552, DOI 10.1090/proc/13527, https://doi.org/10.1090/proc/13527
8. Pierre Berger, Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface, arXiv:2105.08354 (2021), https://arxiv.org/abs/2105.08354
9. Pierre Berger and Dmitry Turaev, On Kolmogorov-typical properties of symplectic dynamics, arXiv:2507.11375 (2025), https://arxiv.org/abs/2507.11375
10. Pierre Berger, Wild dynamics on manifolds, arXiv:2510.12929 (2025), https://arxiv.org/abs/2510.12929 (survey; to appear in the Proceedings of the ICM 2026)
11. Artur Avila, On the Kotani-Last and Schrodinger conjectures, arXiv:1210.6325 (2012), https://arxiv.org/abs/1210.6325
12. A. Volberg and P. Yuditskii, Kotani-Last problem and Hardy spaces on surfaces of Widom type, arXiv:1210.7069 (2012), https://arxiv.org/abs/1210.7069
13. David Damanik and Peter Yuditskii, Counterexamples to the Kotani–Last conjecture for continuum Schrödinger operators via character-automorphic Hardy spaces, Advances in Mathematics 293 (2016), 738-781, DOI 10.1016/j.aim.2016.02.023, https://doi.org/10.1016/j.aim.2016.02.023
14. Nyah Davis, Íris Emilsdóttir, Long Li, and Hangqi Liang, On the Kotani-Last Conjecture for the Dirac Operator, arXiv:2509.19072 (2025), https://arxiv.org/abs/2509.19072
15. Christian Remling, The absolutely continuous spectrum of Jacobi matrices, Annals of Mathematics 174 (2011), 125-171, DOI 10.4007/annals.2011.174.1.4, https://doi.org/10.4007/annals.2011.174.1.4
16. Marcelo Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Mathematics 167 (2008), 643-680, DOI 10.4007/annals.2008.167.643, https://doi.org/10.4007/annals.2008.167.643
17. Mauricio Poletti, Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems 38 (2018), 5163-5188, DOI 10.3934/dcds.2018228, https://doi.org/10.3934/dcds.2018228
18. Davi Obata and Mauricio Poletti, On the genericity of positive exponents of conservative skew products with two-dimensional fibers, arXiv:1809.03874 (2018), https://arxiv.org/abs/1809.03874
19. Disheng Xu, Density of positive Lyapunov exponents for symplectic cocycles, Journal of the European Mathematical Society 21 (2019), 3143-3190, DOI 10.4171/jems/899, https://doi.org/10.4171/jems/899
20. Artur Avila, Global theory of one-frequency Schrödinger operators, Acta Mathematica 215 (2015), 1-54, DOI 10.1007/s11511-015-0128-7, https://doi.org/10.1007/s11511-015-0128-7
21. Artur Avila, KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators, arXiv:2307.11071 (2023), https://arxiv.org/abs/2307.11071
22. A. Katok, Bernoulli Diffeomorphisms on Surfaces, Annals of Mathematics 110 (1979), 529-547, DOI 10.2307/1971237, https://doi.org/10.2307/1971237