Maximal measures for flows with nonuniform structure

In this paper, we study ergodic optimization of continuous functions for flows by concentrating on the entropy spectrum of their maximizing measures. Precisely, over a wide family of flows with non-uniformly hyperbolic structure, we obtain a picture describing coexistence of continuous functions whose maximizing measures have large and small entropy respectively in $C^0$-topology. Our proof relies on the orbit decomposition technique, originally introduced by Climenhaga and Thompson, for flows with weakened versions of expansiveness and specification property. In particular, our results extend \cite{STY} from non-Markov shift on symbolic spaces to a considerably broad class of continuous flows with nonuniform structure. To illustrate this, we apply our general results to both geodesic flows and frame flows over closed rank one manifolds of nonpositive curvature.

💡 Research Summary

This paper investigates ergodic optimization for continuous flows on compact metric spaces, focusing on the entropy spectrum of maximizing measures for continuous observables. The authors consider a broad class of flows that possess non‑uniformly hyperbolic structure, i.e. they satisfy weakened versions of expansivity and specification rather than the classical uniform versions. Two notions of weak expansivity are introduced: (E1) entropy‑expansive flows, and (E2) flows that are not entropy‑expansive but whose obstruction entropy (h_{\perp}^{\exp}) is strictly smaller than the topological entropy (h_{\mathrm{top}}(F)).

The central technical tool is the orbit‑decomposition framework ((\mathcal P,\mathcal G,\mathcal S)) originally developed by Climenhaga and Thompson. Here (\mathcal G) is a collection of “good” orbit segments that enjoy weak controlled specification at every scale (\eta>0) and whose complexity function (h_{\mathcal G,\eta}(t)) grows sub‑logarithmically. The sets (\mathcal P) and (\mathcal S) consist of pre‑ and post‑processing pieces whose entropies are strictly below the total entropy. This decomposition allows the authors to apply the thermodynamic formalism for non‑uniform systems, in particular the existence and uniqueness of equilibrium states for potentials satisfying a Bowen‑type bounded distortion property on (\mathcal G).

For any continuous observable (\varphi) the maximal functional is (\Lambda_F(\varphi)=\sup_{\nu\in\mathcal M_F(X)}\int\varphi,d\nu) and the set of maximizing measures is (\mathcal M_{\max}(\varphi)={\mu:\int\varphi,d\mu=\Lambda_F(\varphi)}). Define
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