[Paper Review] Codimension zero laminations are inverse limits

Álvaro Lozano Rojo|arXiv (Cornell University)|Apr 28, 2012

Point processes and geometric inequalities6 references3 citations

TL;DR

This paper establishes that codimension zero laminations are precisely the inverse limits of branched manifolds under a flattening condition, and proves that equicontinuous codimension zero laminations preserve a transverse metric via their inverse limit structure. The work generalizes prior results on solenoids and tiling spaces, providing a unifying framework for laminations in dynamical systems and geometric topology.

ABSTRACT

The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse limit structure allows us to show that equicontinuous codimension zero laminations preserves a distance function on transversals.

Motivation & Objective

• To characterize when an inverse limit of branched manifolds yields a codimension zero lamination.
• To extend previous results on solenoids and tiling spaces to general codimension zero laminations.
• To show that equicontinuous codimension zero laminations admit a transverse metric via their inverse limit representation.
• To unify the study of laminations, branched manifolds, and inverse limits in dynamical systems and foliation theory.

Proposed method

• Define branched manifolds as locally modeled on quotient spaces of disks and trees, with $C^r$ transition maps.
• Introduce the concept of a 'flattening' map in a projective system of branched manifolds to ensure the inverse limit is a lamination.
• Construct inverse limits $\varprojlim(S_k, f_k)$ where $f_k$ are cellular, $C^r$ maps between branched manifolds.
• Use regular covering maps and deck transformation groups $\Delta_{k,1}$ to model transversals as profinite groups.
• Represent the transversal as $T_1 \cong \varprojlim(\Delta_{k,1}, f_{k,k-1})$, giving a profinite group structure.
• Construct a left-invariant distance on the transversal using the product metric $d((x_k),(y_k)) = \sum_{k=2}^\infty \frac{1}{2^k} \delta_{\Delta_{k,1}}(x_k,y_k)$, invariant under $\Delta_{\infty,1}$-action.

Experimental results

Research questions

• RQ1 Under what conditions is the inverse limit of a system of branched manifolds a codimension zero lamination?
• RQ2 Can every codimension zero lamination be realized as an inverse limit of branched manifolds?
• RQ3 Does the inverse limit structure of a codimension zero lamination imply the existence of a transverse metric?
• RQ4 How does the dynamics of equicontinuous codimension zero laminations relate to their inverse limit representation?
• RQ5 What is the role of the deck transformation group $\Delta_{\infty,1}$ in the transverse geometry of such laminations?

Key findings

• An inverse limit of branched manifolds is a codimension zero lamination if and only if the system is flattening.
• Every codimension zero lamination is homeomorphic to an inverse limit of branched manifolds and cellular maps.
• Equicontinuous codimension zero laminations preserve a transverse metric, constructed via the inverse limit structure.
• The transversal of such a lamination is isomorphic to a profinite group $\varprojlim(\Delta_{k,1}, f_{k,k-1})$, with a canonical left-invariant distance.
• The lamination arises as the suspension of a free action of $\Delta_{\infty,1}$ on the transversal, with the action preserving the metric.
• The inverse limit construction yields a faithful representation of $\Delta_{\infty,1}$ as a subgroup of the transversal, ensuring the metric is preserved.