On the creation of conjugate points for thermostats

TL;DR

This work analyzes thermostat dynamics on closed oriented surfaces, connecting the no-conjugate-points condition to dominated (projective) hyperbolicity. It proves that the $\mathcal{C}^2$-interior of the no-conjugate-points set $B(M,g)$ is contained in the projectively Anosov set $D(M,g)$ by developing a thermostat-specific index form, damped curvature, and cotangent Green bundles, then applying a delicate perturbation argument to detect conjugate points. For reversible thermostats, the authors extend Klingenberg-type results: if the thermostat is projectively Anosov, then its non-wandering set $\Omega$ contains no conjugate points, using Maslov indices, mirror dynamics, and a Mañé-type approach. Overall, the paper clarifies the relationships among no-conjugate-points, dominated splitting, and reversibility in dissipative thermostat systems, providing tools to test or obstruct conjugate points and to identify robust hyperbolic behavior in this broader setting.

Abstract

Let $(M, g)$ be a closed oriented Riemannian surface, and let $SM$ be its unit tangent bundle. We show that the interior in the $\mathcal{C}^2$ topology of the set of smooth functions $λ:SM\to \mathbb{R}$ for which the thermostat $(M, g, λ)$ has no conjugate points is a subset of those functions for which the thermostat is projectively Anosov. Moreover, we prove that if a reversible thermostat is projectively Anosov, then its non-wandering set contains no conjugate points.

Figures (4)

• Figure 1: A schematic plot of the functions $z_{-T}$, $z_T$, and $f_T$.
• Figure 2: Illustration of the function $\phi_\epsilon$.
• Figure 3: Example of a mirrored curve on $SM$.
• Figure 4: Illustration of the function $m: \Lambda(SM)\to \mathbb{S}^1$ on a fiber over $v\in SM$.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Remark 2.2
• Corollary 2.3
• Proposition 2.4
• proof : Sketch of proof
• Lemma 2.5
• proof