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

Sangjin Lee

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 $ψ$ and a Lagrangian branched surface $B_ψ$ satisfies some assumptions, we prove that there is an invariant Lagrangian lamination $\mathcal{L}$ which is a higher-dimensional generalization of a geodesic lamination.

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

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 and a Lagrangian branched surface satisfies some assumptions, we prove that there is an invariant Lagrangian lamination which is a higher-dimensional generalization of a geodesic lamination.

Paper Structure

This paper contains 20 sections, 13 theorems, 102 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Plumbing spaces
  4. Generalized Dehn twist
  5. Lagrangian branched submanifolds
  6. Lagrangian branched submanifolds.
  7. Construction of fibered neighborhoods.
  8. Associated branched manifolds and the notion of "carried by".
  9. The generalized Penner construction
  10. Proof of Theorem \ref{['branched surface thm']}
  11. Construction of Lagrangian laminations
  12. Singular and regular disks
  13. Braids
  14. Lagrangian lamination on a singular disk
  15. Lagrangian lamination on a regular disk
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

Let $M$ be a symplectic manifold and let $\psi : M \stackrel{\sim}{\to} M$ be a symplectic automorphism of generalized Penner type. Then, there exists a Lagrangian branched submanifold $\mathcal{B}_{\psi}$ such that if $L$ is a Lagrangian submanifold which is carried (resp. weakly carried) by $\math

Figures (10)

  • Figure 1: $P(\alpha \simeq S^1, \beta \simeq S^1)$ with plumbing data $(2,0)$ (left) and $(1,1)$ (right).
  • Figure 2: Black curves are part of a Lagrangian branched submanifold and the black marked points denote a connected component $\ell$ of $Locus(\mathcal{B})$. in (a), $L(\ell)$ is in red, and the fibers $F_p$, for $p \in \mathcal{B} \cap U(\ell)$, are in blue; (b) and (c) are not allowed by Equation \ref{['eqn local properties']}; and in (d), the red and green boxes are examples of $N(S)$.
  • Figure 3: Black curves are part of a Lagrangian branched submanifold and marked points denote $\ell$; in (a), $U(\ell)$ is shaded blue, the vertical line segments are fibers; (b) fiber $F_p$ for $p \notin S(\ell) \times (0,1]$ is in green; and in (c), fiber $F_p$ for $p \in S(\ell) \times (0,1]$ is in red
  • Figure 4: (a) represents $\pi:N(\mathcal{B}) \to \mathcal{B}^*$. In $N(\mathcal{B})$, the blue, red, and green represent $\pi^{-1}(S_0)$, $\pi^{-1}(S_1)$, and $\pi^{-1}(S_2)$, where $S_i$ is the corresponding sector of $\mathcal{B}^*$; (b) represents $F_x$ where $x$ is in the branch locus of $\mathcal{B}^*$ in (a).
  • Figure 5: The blue curves represent $D_p^+$ in the left hand picture and $D_p^-$ in the right hand picture, the red curves represent $N_p$ in both.
  • ...and 5 more figures

Theorems & Definitions (59)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • proof : Proof of Theorem \ref{['lagrangian surgery theorem']}
  • ...and 49 more