Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows (Q6624543)

\textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)] in a seminal study of the dynamical properties of diffeomorphisms showed that any \(C^{1+\alpha}\) diffeomorphism \(f\) of a two-dimensional manifold has horseshoes of large entropy. One of the consequences is that such a system has the `intermediate entropy property': For any \(c\) between zero and the topological entropy of \(f\) there is an ergodic \(f\)-invariant measure \(\mu\) with \(h_{\mu}(f)=c\). Katok conjectured that the property is a general one in the following sense: Any \(C^{1+\alpha}\) diffeomorphism of a Riemannian manifold has the intermediate entropy property. Much progress has been made towards this conjecture through proofs under additional hypotheses. A different intermediation property was introduced by \textit{X. Tian} et al. [Discrete Contin. Dyn. Syst. 39, No. 2, 1019--1032 (2019; Zbl 1404.37023)] as follows. \N\NFor a continuous map \(f\colon X\to X\) of a compact metric space write \(M(f)\) for the convex set of \(f\)-invariant Borel probability measures on \(X\). The system is said to have the `intermediate Birhoff average property' if for any continuous function \(g\colon X\to\mathbb{R}\) and \(\alpha\) strictly between \(\inf_{\mu\in M(f)}\int g\mathrm{d}\mu\) and \(\sup_{\mu\in M(f)}\int g\mathrm{d}\mu\) there is an ergodic measure \(\mu_{\alpha}\in M(f)\) with \(\int g\mathrm{d}\mu_{\alpha}=\alpha\). They showed this property under the hypothesis of the `periodic gluing orbit property' and in some other cases. Motivated in part by the analogy with the variational principle and the conditional variational principle the problem studied here is to describe conditions under which both the intermediate entropy property and the intermediate Birkoff average property hold. These ideas are refined and discussed in terms of intersections of various dynamically defined subsets of \(M(f)\), with a particular focus on dense convex subsets (in the work of X. Tian et al. [loc. cit.] a key role is played by the fact that the ergodic invariant measures are dense in \(M(f)\)) and are also discussed in the setting of flows rather than continuous maps. The approach is to prove a `multi-horseshoe' entropy-dense property for flows, show that it holds for basic sets, and then use thermodynamic formalism to prove a conditional intermediate metric entropy property and several different conditional intermediate Birkhoff average properties for basic sets of flows in the general setting of asymptotically additive families of continuous functions. These general results then give the main results under some hyperbolicity hypotheses.