Positive entropy problems - Quantum Calculus

A rather general positive entropy problem is that Baire generically, a volume preserving diffeomorphism T on a 2-dimensional manifold M has positive entropy. A much bolder conjecture is that every non-integrable volume preserving diffeomorphism on a compact Riemannian 2-manifold has positive entropy. It would mean that the Standard map $T(x,y) = (2x-y+c \sin(x),x)$ had positive entropy for all non-zero c. The mathocalypse dump from October 6 has only proven positive entropy for large enough $c$ . To the definitions: entropy is the $\int_M \lambda(x,y) dV(x,y)$ , where $\lambda(x,y) = {\rm liminf}_{n \to \infty} \int_M \frac{1}{n} \log||dT^n(x,y)|| \; dV(x)$ and $dT^n$ is the Jacobian matrix of the n’th iterate $T^n$ . Pesin’s theorem from the 70ies assures that this is the same than the Kolmogorov-Sinai entropy in ergodic theory the entropy growth of partitions refined by the automorphism of the probability space. The map T is integrable if every $T$ -invariant measure has a Koopman operator with pure point spectrum. The Standard map is $T(x,y) = (2x-y+c \sin(x),x)$ is a first order reformulation of the second order nonlinear recurrence equation $x_{n+1} - 2 x_n + x_{n-1} = c \sin(x_n)$ which is a time discretization of the pendulum $x'' = c \sin(x)$ . Linearizing this nonlinear relation produces a one-dimensional Schroedinger operator $(L u)_n=u_{n+1} - 2 u_n + u_{n-1} + c \cos(x_n) u_n$ , where the orbit $x_n$ drives the potential $V_n=c \cos(x_n)$ . The operator L is an element in a type $II_1$ von Neumann algebra with finite trace. The algebra is the crossed product of the commutative von-Neumann algebra $L^{\infty}(\mathbb{T}^2)$ with the unitary Koopman automorphism $f \to f(T)$ , which makes it a non-commutative deformation. I got fond of Banach algebras, when learning it from a course of Christian Blatter.

[Non-commutative deformation also matters in single variable calculus, where $T$ is the translation by a Planck constant h. In the limit when h goes to zero, one obtains the traditional calculus. I taught such quantum calculus the first time in 2011 and the last time in 2024. The non-commutative deformation of course does not have to be advertised as such, it is just business calculus where the derivative is $f'(x)=D f(x)=f(x+1)-f(x)$ and the integral is $S f(x)=f(0)+f(1)+\cdots +f(n-1)$ and the algebra is deformed in such a way that $D x^n = n x^{n-1}$ and the fundamental theorem of calculus is essentially a tautology. It is intuitive because the calculus is deformed in such a way to the discrete that all formulas are the same than in usual calculus. (This approach turned out not to be transferable to other teachers. Transferability compatibility is the reason why most calculus texts are so damn boring, down to the numbering of the chapters! There is little imagination allowed as the most common denominator has to be able to teach it. Maybe the Mathocalypse event will have the effect that education wakes up and becomes more creative. We also in research will be forced to become much, much more creative. Boring stuff is snatched up by the machines very quickly as the October 6., 2026 mathocalypse has shown. We can no more afford to be boring. And the worst reaction would be to teach in the same way than the machines trying to replace us or to write papers as the machines would do now. We already see wastelands appear like the hexagon wastland. ]

The bounded linear functional on $C(R)$ given by $f \to {\rm tr}(f(L))$ produces by the Riesz representation theorem the density of states measure $dk$ . Entropy is now $h(T) = \int_{\mathbb{R}} \log|E| \; dk(E) = {\rm tr}({\rm log}(L)) = {\rm log} {\rm det}(L)$ . This link to determinants makes the entropy notion very natural. The following links only refer to attempts which I got involved in over the last 40 years. This ego-centric point of view is by no means a review. It is a bit of a personal eulogy to the “positive entropy problem”.

[The bold conjecture could work also in higher dimensions but would be much more subtle: the product system T(x,y,z)=(2x+y,x+y,z) on $T^3$ has positive entropy even so, the 3-torus is foliated by invariant 2-tori on which the map induces the Arnold cat map. It does not have positive volume measure ergodic components but still can be considered chaotic but chaos is constraint to smaller dimensional parts. The system is foliated by Anosov systems. ]

1987 Senior Thesis:relevant to Lyapunov exponents is Wojtkowski’s theorem that the existence of an eventually invariant cone bundle is equivalent to positive Lyapunov exponent. The idea is to look at invariant measures of the cocycle map $F((x,y),u) = (T(x,y),A(x,y) u)$ on the 3-torus $T^3=\mathbb{T}^2 \times \mathbb{P}^1$ . There is a Ledrapier invariant measure $\mu$ and expansion rate random variable $X$ such that the entropy is the expectation of X on the probability space $(T^3,\mu)$ . Iterating F forward pumps mass to that measure, iterating backwards gives the unstable mirror measure. It is a bit like Markov stuff.
1990 Discontinuity and the problem of positivity. For aperiodic automorphisms, there is never continuity of Lyapunov exponents. Positivity is related to cohomology of sets. See This paper from 2023 where I reprove a theorem of Hjorth about this cohomology.
1992 “Positive Lyapunov exponents for a dense set of bounded measurable SL(2,R) cocycles”. My first real theorem. About the expectation of matrix valued random variables uses Herman subharmonicity and Kakutani skyscrapers to rotate on small sets.
1994 PhD thesis 1994: Lax deformation of Random Jacobi operators and renormalization of Random Jacobi operators were motivated by the entropy problem. Also that Lyapunov exponents can become discontinuous or are dense in SL(2,\mathbb{R})$ cocycles over an aperiodic system were motivated by the entropy problem.
1996 Topological entropy of standard type monotone twist maps. This is a variational approach based on Aubry’s anti-integrable limit idea.I forgot to mention this in the 10/10/26 youtube presentation. Variational approaches were very much on my mind, especially after the course of Moser for which I wrote the script.
1998 Singular continuous spectrum and quantitative rates of weak mixing part of the Barry Simon program on singular spectrum. I should also mention the work with Bert Hof on zero dimensional singular continuous spectrum for toral flow. And then the result on weakly mixing invariant tori in dynamical systems.
1999 Attempts to prove positive entropy. The paper still contains some valuable parts like references, history, measurements, explanations of attempts for the claim $h(T) \geq \log(|c|/2)$ and even for nonzero c, where one can still measures positive entropy (motivation is Aubry duality in the almost Matthieu situation).
1999 Twisty page, contains also C and Java programs for Standard map and an entropy lab. From a time, where I thought to be able to proof positive entropy using multi-linear algebra and subharmonicity.
1999 Open problems about billiards My job talk in December 1999 at Harvard.
2000 Fluctuation bounds on subharmonic functions. This is something which I need still to work out more. Lyapunov exponents are subharmonic functions in analytic parameters like energy E.
2000 Intriguing open problems in Hamiltonian dynamics. A Talk given at 2000 Montana State university. Related is the “good-bad-ugly” problem. Can it happen that the complement of the union of Pesin and KAM sets has positive measure?
2002 A spectral notion of integrability . Talk given at Birmingham Alabama.
2004 Mathtable talk: Applications of subharmonic functions explains much of the Herman stuff.
2008 Where chaos theory meets other disciplines (slides). Talk on February 7, 2008.
2010 Analytic continuation of Dirichlet series with almost periodic coefficients was first trying to understand KAM stuff better, which is related through Green functions. I mentioned the Monograph of Bourgain.
2024 Cauchy-Binet for Pseudo determinants. published version 2014. Is of course also motivated indirectly by the entropy problem as we essentially just need to estimate determinants.
2015 Golden rotations: talk given at BU on February 23, 2015. Ledrappier manifolds are visualized on page 8. Ledrappiers picture is to see Lyapunov exponents as expectations of random variables with respect to invariant measures. That Boston paper was very expository abd a bit gossipy.
2015 Javascript applet on the Standard map Also mentions my map $T: C^4 \to C^4$ which has invariant 2-tori $T_c$ on which the map induces the standard map.
2023 The explicit integration of the Pendulum ODE using Weierstrass elliptic functions had been motivated by using subharmonic techniques for the entropy problem of systems like coupled penduli.

Author: oliverknill