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We generalize some properties of surface automorphisms of pseudo-Anosov type.First, we generalize the Penner construction of a pseudo-Anosov homeomorphismand show that a symplectic automorphism which is constructed by our generalizedPenner construction has an invariant Lagrangian branched submanifold and aninvariant Lagrangian lamination, which are higher-dimensional generalizationsof a train track and a geodesic lamination in the surface case. Moreover, if apair consisting of a symplectic automorphism $\psi$ and a Lagrangian branchedsurface $B_{\psi}$ satisfies some assumptions, we prove that there is aninvariant Lagrangian lamination $\mathcal{L}$ which is a higher-dimensionalgeneralization of a geodesic lamination.

Towards a higher-dimensional construction of stable/unstable Lagrangian laminations