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arXiv · 2512.21394

Ergodic optimization for Gauss's continued fraction map

Yinying Huang·Oliver Jenkinson·Zhiqiang Li
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Abstract

The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for H\"older continuous functions. Although a Ma\~n\'e cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $\alpha$-H\"older essentially compact functions.

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Yinying Huang, Oliver Jenkinson, Zhiqiang Li. 2025-12-24. Ergodic optimization for Gauss's continued fraction map. https://arxiv.org/abs/2512.21394

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