Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems (Q2884074)

If \(f \in \) Diff\({}^1(M)\) has a unique probability measure \(\mu_f\) of maximal entropy, then \(f\) is called intrinsically ergodic. In this paper, the authors study the intrinsic ergodicity of a class of non-Anosov partially hyperbolic diffeomorphisms, which is mainly based on \textit{R. Mañé}'s example [Topology 17, 383--396 (1978; Zbl 0405.58035)].NEWLINENEWLINEThey also study the related notion of intrinsic stability: A diffeomorphism \(f\) is intrinsically stable if there exists a \(C^1\) neighborhood \({\mathcal U} \) of \(f\) such that any \(g \in {\mathcal U}\) is intrinsically ergodic and the measure space \((g, \mu_g)\) is isomorphic to \((f, \mu_f)\).NEWLINENEWLINEIn Theorem 1.3 the authors construct a non-empty open set \({\mathcal U} \subset \) Diff\({}^1({\mathbb T}^d)\), for any \(d \geq 3\), such that any \(f \in {\mathcal U}\) is partially hyperbolic (but non Anosov), robustly transitive and intrinsically stable (but non structurally stable). The method of construction of such a set \({\mathcal U}\) has two main parts:NEWLINENEWLINEThe first part of the proof of Theorem 1.3 is posed in Theorem 1.5. It is abstract and based on topological dynamical arguments. In Theorem 1.5 the authors generalize the sufficient topological conditions of \textit{R. Bowen} [Math. Syst. Theory 8 (1974), 193--202 (1975; Zbl 0299.54031)] to obtain instrinsic ergodicity. They prove that an adequate family of homeomorphisms that are extensions of expansive ones (which satisfy the specification property) are intrinsically ergodic and intrinsically stable. Thus, in particular, the method applies to some non Anosov diffeomorphisms that are \(C^0\)-close to Anosov toral automorphisms. The instrinsic stability implies that the topological entropy is locally constant. On purpose, the authors note that \textit{Y. Hua} et al. [Ergodic Theory Dyn. Syst. 28, No. 3, 843--862 (2008; Zbl 1143.37023)] also proved (using a different method) that the topological entropy is locally constant for an example of partially hyperbolic diffeomorphism that is \(C^1\)-close to a toral automorphism.NEWLINENEWLINEAs said above, in Theorem 1.5 the authors extend Bowen's sufficient conditions for the intrinsic ergodicity to some adequate family of (non necessarily expansive) homeomorphisms. To prove Theorem 1.5 the authors consider a \(C^0\)-close extension \(g\) of an expansive homeomorphism \(f\) which satisfies the specification property. An extension of \(f\) is defined as a homeomorphism \(g\) such that there exists a continuous and surjective map \(\pi\) (a semiconjugation) satisfying \(f \circ \pi = \pi \circ g\). If the so-called saturation sets \(\pi^{-1}(\{\pi(x)\})\) behave adequately with respect to the measure \(\mu\) of maximal entropy of \(f\), then the authors prove that \(g\) is also intrinsically ergodic. Moreover, they prove that the measure \(\nu\) of maximal entropy of \(g\) is obtained as the weak\(^*\) limit (which they prove to exist) of a sequence of probability measures \(\nu_n\) that are supported and equidistributed on finite sets \(P_n\) of periodic points. These sets \(P_n\) are \(\epsilon-n\) separated (for any small enough \(\epsilon >0\)) and maximal satisfying the condition \(g^n(p) = p \;\forall \;p \in P_n\). Thus, from the proof of Theorem 1.5, the authors also deduce that the periodic orbits are equidistributed with respect to the unique measure of maximal entropy.NEWLINENEWLINEThe second part of the proof of Theorem 1.3 explicitly shows how to generalize Mañé's example in \(\mathbb{T}^3\) to dimension \(d \geq 3\). The authors first take an Anosov automorphism \(f_A : \mathbb{T}^d \mapsto \mathbb{T}^d\) with only one eigenvalue inside the unit circle. By taking an open set \({\mathcal U} \subset \) Diff\({}^1(\mathbb{T}^d)\) of robustly transitive (but non-Anosov) diffeomorphisms \(f\), they prove a shadowing proposition: each orbit \(\{f^n(x)\}_{n}\) is an \(\epsilon\)-chain of \(f_A\) and thus, it is \(\delta\)-shadowed by an orbit \(\{f^n_{A}\}_n\). This shadowing property implies the existence of the semiconjugation \(\pi\) between \(f\) and \(f_A\). Therefore, applying Theorem 1.5, the authors deduce that any \(f \in {\mathcal U}\) is intrinsically ergodic.NEWLINENEWLINEThe two ending sections of the paper (Sections 5 and 6) provide two further examples of intrinsically ergodic, partially hyperbolic (but non hyperbolic) diffeomorphisms:NEWLINENEWLINEIn Section 5, the authors construct \(f \in \) Diff\({} ^1(\mathbb{T}^m)\) for \(m \geq 4\), by mixing the generalization of Mañé's example (constructed as in the second part of the proof of Theorem 1.3) with a derived from Anosov. The intrinsically ergodic diffeomorphism \(f\) is, in this example, produced by an isotopy from a linear Anosov \(f_A\) that is \(C^0\)-close to \(f\), and besides, robustly transitive.NEWLINENEWLINEFinally, in Section 6, the authors exhibit a new example of an intrinsically ergodic diffeomorphism \(f\), which is non-robustly transitive. The construction of such an \(f\) follows the idea of Anosov, but instead of performing a deformation on a fixed point through a pitch fork bifurcation, the authors deform through a (non-generic) Hopf bifurcation. They make such a choice to obtain a non-Axiom A diffeomorphism \(f\): in fact, since the attractor includes an invariant circle (due to the Hopf bifurcation) it is non-hyperbolic.