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

TL;DR

This paper extends the concept of pseudo-Anosov automorphisms to higher dimensions by constructing invariant Lagrangian laminations and branched submanifolds, generalizing train tracks and geodesic laminations.

Contribution

It introduces a generalized Penner construction for symplectic automorphisms and establishes the existence of invariant higher-dimensional Lagrangian laminations under certain conditions.

Findings

Construction of invariant Lagrangian branched submanifolds.

Existence of invariant Lagrangian laminations in higher dimensions.

Generalization of geodesic laminations to higher-dimensional symplectic manifolds.

Abstract

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