Uniqueness of \(u\)-Gibbs measures for hyperbolic skew products on \(\mathbb{T}^4\) (Q6593169)

The authors study the space \(\mathrm{Sk}^r(\mathbb T^2\times\mathbb T^2)\) of \(C^r\)-diffeomorphisms of the form\N\[\N\begin{split} f\colon &\mathbb T^2\times\mathbb T^2\to \mathbb T^2\times\mathbb T^2\\\N&(x,y)\mapsto(f_1(x),f_2(x,y)), \end{split}\N\]\Nwhere \(f_1\) is a \(C^r\)-diffeomorphism of \(\mathbb T^2\) and for every \(x\in\mathbb T^2\), \(f_2(x,\cdot)\) is a \(C^r\)-diffeomorphism of \(\mathbb T^2\). Moreover, they consider the subset \(\mathrm{Ph}^r\subset\mathrm{Sk}^r(\mathbb T^2\times\mathbb T^2)\) of partially hyperbolic \(C^r\)-diffeomorphisms \(f\) satisfying additional properties (see the article for full details):\N\begin{itemize}\N\item[(a)] \(f\) admits a dominated decomposition \(T\mathbb T^4=E^{\mathrm{ss}}\oplus E^{\mathrm{ws}}\oplus E^{\mathrm{wu}}\oplus E^{\mathrm{uu}}\) with \(Df|_{E^{\mathrm{ss}}\oplus E^{\mathrm{ws}}}\) uniformly contracting and \(Df|_{E^{\mathrm{wu}}\oplus E^{\mathrm{uu}}}\) uniformly expanding;\N\item[(b)] \(f\) is 2-center bunched, where \(E^{\mathrm{c}}:=E^{\mathrm{ws}}\oplus E^{\mathrm{wu}}\);\N\item[(c)] \(f\) verifies some technical condition which implies that there is \(\theta\in(0,1)\) such that \(E^{\mathrm{uu}}\) is \(\theta\)-Hölder.\N\end{itemize}\N\NThe authors study the subset \(\mathcal A^r\subset \mathrm{Ph}^r\) of Anosov diffeomorphisms which contract \(E^{\mathrm{ws}}\) and expand \(E^{\mathrm{wu}}\) uniformly (this set is \(C^1\)-open in \(\mathrm{Sk}^r(\mathbb T^2\times\mathbb T^2)\)). They contribute to the study of \emph{SRB-measures} and their relation with \emph{\(\mathrm{u}\)-Gibbs measures}. Here, an \(f\)-invariant probability measure \(\mu\) is \emph{\(\mathrm{u}\)-Gibbs} if for any \(\mu\)-measurable partition which is subordinated to the strong unstable foliation (that is, the foliation which is tangent to the strong unstable direction \(E^{\mathrm{uu}}\)) has conditional measures which are absolutely continuous with respect to the Riemannian volume induced in each leaf.\N\NBy results in [\textit{C. Bonatti} et al., Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Berlin: Springer (2005; Zbl 1060.37020)], every SRB-measure is \(\mathrm{u}\)-Gibbs. The main theorem, Theorem A, claims that for every \(r\ge3\), there exists a \(C^r\)-dense and \(C^2\)-open subset \(\mathcal U\) of \(\mathcal A^r\) such that for every \(f\in\mathcal U\) there is only one \(\mathrm{u}\)-Gibbs measure, and this measure is the unique SRB measure of \(f\). Furthermore, for every \(f\in\mathcal U\), the strong unstable foliation is minimal, that is, every leaf is dense.