# Dynamical systems and ergodic theory > 12 AI-made results in dynamical systems and ergodic theory, each explained in plain English with its trust label, Lean status and sources. Canonical: https://closedform.nishachay.tech/science/mathematics/dynamical-systems-and-ergodic-theory/ Last updated: 2026-10-11 12 results in OpenAI Math 2026. 12 AI-made results in dynamical systems and ergodic theory, from 19 papers in the OpenAI math collection, each explained in plain English. 9 have a formal Lean proof listed; 3 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1927: [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/). Results rated Landmark include [Hilbert's sixteenth problem: uniform bounds for limit cycles](https://closedform.nishachay.tech/e/openai-math-2026-143/) and [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/). ## Results (12) - [Hilbert's sixteenth problem: uniform bounds for limit cycles](https://closedform.nishachay.tech/e/openai-math-2026-143/) - Entry 143 · Claims a finite cap, set by degree alone, on limit cycles of planar polynomial systems. (Formal proof listed) - [Banach’s simple Lebesgue-spectrum problem](https://closedform.nishachay.tech/e/openai-math-2026-144/) - Entry 144 · A smooth map of a 3D torus where one measurement’s past and future build every average-zero measurement. (Formal proof listed) - [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/) - Entry 145 · If long waits make any two moments independent, they make every finite group of moments independent. (Formal proof listed) - [Positive metric entropy for the standard map](https://closedform.nishachay.tech/e/openai-math-2026-146/) - Entry 146 · The standard map is chaotic on a region of positive area for every large enough parameter. (Formal proof listed) - [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/) - Entry 147 · Two papers claim that a smooth billiard table with an unbroken band of caustics along its edge must be an ellipse. (Claimed, not checked) - [The entropy-rate dimension formula for self-similar measures](https://closedform.nishachay.tech/e/openai-math-2026-148/) - Entry 148 · A single formula gives the dimension of every self-similar measure on the line, even when its pieces collide. (Formal proof listed) - [Classwise permanence for weakly reversible mass-action systems](https://closedform.nishachay.tech/e/openai-math-2026-149/) - Entry 149 · With fixed rates, every weakly reversible chemical network herds each class of states into one common window. (Formal proof listed) - [Weak mixing of triangular billiards with an irrational angle](https://closedform.nishachay.tech/e/openai-math-2026-150/) - Entry 150 · A paper claims billiards in every triangle with an irrational angle forget their start: they are weakly mixing. (Formal proof listed) - [A C 1 counterexample to the entropy conjecture](https://closedform.nishachay.tech/e/openai-math-2026-151/) - Entry 151 · A smooth-enough map can double a shape's homology while its orbits stay orderly. (Formal proof listed) - [Zero entropy does not guarantee a smooth positive-volume model](https://closedform.nishachay.tech/e/openai-math-2026-152/) - Entry 152 · An abstract zero-entropy system with no smooth, volume-keeping copy in any dimension. (Formal proof listed) - [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://closedform.nishachay.tech/e/openai-math-2026-153/) - Entry 153 · Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter. (Claimed, not checked) - [Pointwise multiple ergodic averages for mixing transformations](https://closedform.nishachay.tech/e/openai-math-2026-154/) - Entry 154 · For mixing systems, averages along k, 2k, …, nk are claimed to settle at almost every starting point, for every n. 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