Dynamical systems and ergodic theory
12 families · 9 verified by lean
This page gathers the 12 result families that the openai/math repository catalogues under Dynamical systems and ergodic theory. Each claim comes from a manuscript; 9 of them have a Lean formalization checked by software, and the other 3 are still to be verified. Open any card to read a plain-language explainer, the original statement and its sources.
Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
Banach’s simple Lebesgue-spectrum problem
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space.
Rokhlin’s multiple-mixing problem
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.
Positive metric entropy for the standard map
Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.
The near-boundary Birkhoff conjecture
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.
The entropy-rate dimension formula for self-similar measures
For every self-similar measure on the line generated by finitely many contracting similarities, proves , where is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.
Classwise permanence for weakly reversible mass-action systems
Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.
Weak mixing of triangular billiards with an irrational angle
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.
A C1 counterexample to the entropy conjecture
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.
Zero entropy does not guarantee a smooth positive-volume model
Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in , giving examples beyond reciprocal Pisot parameters.
Pointwise multiple ergodic averages for mixing transformations
Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.