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A team of researchers including Oliver Jenkinson, Professor of Mathematics in the Centre for Complex Systems, has recently published a breakthrough study in the highly renowned journal Inventiones Mathematicae. The paper, entitled Typical periodic optimization for dynamical systems: symbolic dynamics by Wen Huang, Oliver Jenkinson, Leiye Xu and Yiwei Zhang, addresses a foundational question in the field of ergodic optimization: In a complex system with great diversity of dynamical behaviour, what is the "best" or most efficient state in the sense that it maximizes some reward or utility function?

The study resolves long-standing questions that date back to the 1990s, tackling the conjecture that for a "typical" or generic reward function, the optimizing behaviour should be a simple, repeating cycle - a periodic orbit.

While this Typical Periodic Optimization Conjecture has been a central focus of the community for decades, providing a rigorous mathematical proof for broad classes of systems has remained an elusive challenge. By using techniques from symbolic dynamics, where complex motion is represented by sequences of symbols, the authors demonstrate that periodic optimization is indeed the rule rather than the exception. The work extends these results to a wide variety of systems, including those with so-called sofic structure. As well as providing a definitive answer to conjectures from thirty years ago, this insight also offers new tools for understanding how simple structures can emerge as the typical optimal state of highly complex, chaotic systems.

Link to the original research article: W. Huang, O. Jenkinson, L. Xu and Y. Zhang, Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. math. (2026)