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Thermostats without conjugate points

Javier Echevarría Cuesta, James Marshall Reber

TL;DR

The paper extends Hopf's no-conjugate-points results to thermostats by introducing the thermostat curvature $\\mathbb{K}$ and a gauge-based scalar $\\kappa_p$, establishing that $\\kappa_p \\le 0$ guarantees no conjugate points and that equality forces $\\mathbb{K}=0$; it further shows that no conjugate points imply a dominated splitting iff the Green bundles $G^*_s$ and $G^*_u$ are transversal. By lifting the dynamics to the cotangent bundle, the authors define and analyze Green bundles, Riccati equations, and Lyapunov exponents, connecting curvature, stability, and entropy. They prove that transversal Green bundles imply a continuous dominated-splitting structure and provide a counterexample on the 2-torus where $\\mathbb{K}=0$ and a dominated splitting exist without Anosov behavior, demonstrating limits of Hopf rigidity for thermostats. The results illuminate the nuanced interplay between curvature-like invariants, Green bundles, and dynamical dichotomies in non-conservative flows, and they introduce the damped thermostat curvature as a finer diagnostic for entropy and stability.

Abstract

We generalize Hopf's theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive, and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points, and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transversal everywhere if and only if it admits a dominated splitting. Finally, we provide an example showing that Hopf's rigidity theorem on the 2-torus cannot be extended to thermostats. It is also the first example of a thermostat with a dominated splitting which is not Anosov.

Thermostats without conjugate points

TL;DR

The paper extends Hopf's no-conjugate-points results to thermostats by introducing the thermostat curvature and a gauge-based scalar , establishing that guarantees no conjugate points and that equality forces ; it further shows that no conjugate points imply a dominated splitting iff the Green bundles and are transversal. By lifting the dynamics to the cotangent bundle, the authors define and analyze Green bundles, Riccati equations, and Lyapunov exponents, connecting curvature, stability, and entropy. They prove that transversal Green bundles imply a continuous dominated-splitting structure and provide a counterexample on the 2-torus where and a dominated splitting exist without Anosov behavior, demonstrating limits of Hopf rigidity for thermostats. The results illuminate the nuanced interplay between curvature-like invariants, Green bundles, and dynamical dichotomies in non-conservative flows, and they introduce the damped thermostat curvature as a finer diagnostic for entropy and stability.

Abstract

We generalize Hopf's theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive, and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points, and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transversal everywhere if and only if it admits a dominated splitting. Finally, we provide an example showing that Hopf's rigidity theorem on the 2-torus cannot be extended to thermostats. It is also the first example of a thermostat with a dominated splitting which is not Anosov.
Paper Structure (13 sections, 27 theorems, 94 equations, 4 figures)

This paper contains 13 sections, 27 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Conjugate points and thermostat curvature
  3. Green bundles in the cotangent bundle
  4. Remaining questions
  5. Organization of the paper
  6. Conjugate points and Green bundles
  7. Dynamics on the characteristic set
  8. Cocyles
  9. Conjugate points
  10. Green bundles
  11. Lyapunov exponents
  12. Transversal Green bundles
  13. Examples and counterexamples on the $2$-torus

Key Result

Theorem 1.1

Let $(M, g, \lambda)$ be a thermostat. If $\kappa_p\leq 0$ for some function $p:SM\to \mathbb{R}$ which is differentiable along the flow, then there are no conjugate points.

Figures (4)

  • Figure 1: The relevant subbundles in cotangent space.
  • Figure 2: The lifted dynamics on $\Sigma$.
  • Figure 3: The bases $\{\beta, \psi_\lambda\}$ and $\{\beta, \phi_p\}$ for $\Sigma$.
  • Figure 4: Illustration when $m=2$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 42 more