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Cone structures from a dynamical and probabilistic viewpoint

Agustin Moreno

TL;DR

The paper investigates cone structures adapted to open book decompositions on contact manifolds, motivated by the CR3BP and the question of reachability in phase space. It develops a geometric-probabilistic framework by defining $2$-dimensional cone structures on $S^3$, deriving a geometric obstruction to reachability via the inner angle $\theta$, and introducing integrability invariants $\mathcal{I}_m(C)$ and $\mathcal{I}_M(C)$ along with probabilistic bounds on transitions between regions. It then formulates Calabi-type invariants $\mathrm{CAL}_\Gamma(A)=\int_A \tau(\Gamma)\,d\mu$ tied to return times, showing that in the Reeb-flow scenario $\mathrm{CAL}_\Gamma(P)=\operatorname{vol}(M)$ and discussing asymptotic growth, with informal ties to quantum-field path integrals. Finally, it introduces a Brownian-motion-with-drift model inside the cone structure, proving a probabilistic Poincaré recurrence and interpreting the results as a bridge between classical shadow dynamics and quantum fluctuations around classical solutions.

Abstract

The goal of this note is to explore, from a geometric and probabilistic point of view, the dynamics of cone structures adapted to open book decompositions. This is inspired by the picture which arises in the study of the circular restricted three body problem (CR3BP). This yields geometric obstructions to reaching a point from another point in the phase space of the CR3BP.

Cone structures from a dynamical and probabilistic viewpoint

TL;DR

The paper investigates cone structures adapted to open book decompositions on contact manifolds, motivated by the CR3BP and the question of reachability in phase space. It develops a geometric-probabilistic framework by defining -dimensional cone structures on , deriving a geometric obstruction to reachability via the inner angle , and introducing integrability invariants and along with probabilistic bounds on transitions between regions. It then formulates Calabi-type invariants tied to return times, showing that in the Reeb-flow scenario and discussing asymptotic growth, with informal ties to quantum-field path integrals. Finally, it introduces a Brownian-motion-with-drift model inside the cone structure, proving a probabilistic Poincaré recurrence and interpreting the results as a bridge between classical shadow dynamics and quantum fluctuations around classical solutions.

Abstract

The goal of this note is to explore, from a geometric and probabilistic point of view, the dynamics of cone structures adapted to open book decompositions. This is inspired by the picture which arises in the study of the circular restricted three body problem (CR3BP). This yields geometric obstructions to reaching a point from another point in the phase space of the CR3BP.
Paper Structure (4 sections, 4 theorems, 12 equations, 3 figures)

This paper contains 4 sections, 4 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dynamics of cone structures
  3. Invariants for cone structures
  4. Brownian motion for cone structures

Key Result

Lemma 2.1

In the upper half-plane $\mathbb R^3_+$, consider a cone structure with constant inner angle $\theta$.Here, the angle is measured between an edge of the cone, and a vertical line, see Figure fig:cono. Then the future time-$t$ region is a disk of radius $t\cdot \hbox{tan}(\theta/2)$. $$$\square$

Figures (3)

  • Figure 1: A cone structure adapted to an open book.
  • Figure 2: Orbits emanating from a point can reach, in time $t$, points lying in a disk of radius $t\cdot tan(\theta/2)$, where $\theta$ is the inner angle of the cone.
  • Figure 3: The geometric measure theory of constant cone structures.

Theorems & Definitions (5)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Proposition 4.1