New geometric examples of Anosov actions (Q1892340)

The present paper constructs a new class of Anosov actions. The concrete construction is as follows. Let \(S = G/K\) be a Riemannian symmetric space and \(x_ 0\) be a base point of \(S\). Let \(g\) (resp. \(k\)) be the Lie algebra of \(G\) (resp. \(K\)), \(g = k + p\) be its Cartan decomposition, \(a\) be a maximal Abelian subspace of \(p\), and \(R\) be a basis of simple roots. \(R\) defines an open Weyl chamber \(a^ + = \{H \in a \mid \alpha(H) > 0, \forall \alpha \in R\}\). Each directed geodesic \(\gamma\) in \(S\) is congruent to a unique geodesic \(\gamma_ 0(s) = \text{Exp}_{x_ 0} (sH)\), \(H \in a^ +\), and the latter determines a subset \(E = \{\alpha \in R \mid \alpha(H) = 0\} \subset R\). We say that the geodesic \(\gamma\) is of type \(E\). Set \(a_ E = \{X \in a \mid \alpha(X) = 0,\;\bigvee \alpha \in E\}\), \(A_ E = \text{exp }a_ E\), \(k = \dim A_ E\). Two geodesics in \(S\) are parallel if their Hausdorff distance is finite. For a given directed geodesic \(\gamma\) of type \(E\), the parallel set \(S_ \gamma\) is the union of all directed geodesics which are parallel to \(\gamma\). Let \(L_ E\) be the subgroup of \(G\) consisting of all isometries which leave \(S_ \gamma\) invariant, then \(S_ E = L_ E \cdot x = S(E) \times A_ E \cdot x_ 0\), where \(S(E)\) is a Riemann symmetric space of non-compact type with rank \(S(E) = \text{rank }S - k\). Let \(U_ E\) be the largest normal subgroup of \(L_ E\) which acts trivially on \(S(E)\). Then \(G/U_ E\) is a homogeneous space and \(L_ E/U_ E\) is a reductive group. The main result of the present paper is: for any subset \(E \subset R\), the action \(\varphi_ E : L_ E/U_ E \times G/U_ E \to G/U_ E\), \((h,gU_ E) \mapsto ghU_ E\) is an Anosov action, i.e., there exists an element \(h \in G/U_ E\) such that the differential \((d\varphi_ E)_ h\) is normally hyperbolic to the \((L_ E/U_ E)\)-orbits. The present paper also yields the assertion: the space \(G/L_ E\) of directed parallel set of type \(E\) of a symmetric space \(G/K\) of non- compact type is a homogeneous symplectic manifold. If the rank of the symmetric space reduces to 1, then all actions coincide with the geodesic flow.