Bicycle tracks with hyperbolic monodromy -- results and conjectures

TL;DR

This work analyzes when the bicycle monodromy $b(\Gamma)$ of a closed planar front track is hyperbolic. It develops the hyperbolic-development framework, translating the no-skid bicycle constraint into hyperbolic-geometry data; this enables precise criteria and proofs for hyperbolicity. The authors prove a new sufficient condition for hyperbolicity that applies beyond convex curves (bounded curvature $|\kappa|\le 1$, not identically 1) and establish a necessary condition for convex curves, namely that the length $L$ exceed $2\pi$ (equivalently the average curvature $2\pi/L<1$). They also present a complementary contact-geometry perspective, deriving a bound $L \ge 2\pi|\rho(\gamma)|$ relating rear- and front-track rotation numbers and formulating conjectures based on computer experiments about the rotation number $\rho(\gamma)$ in convex cases. Taken together, these results advance understanding of hyperbolic monodromy in bicycle models and connect planar curve geometry with hyperbolic- and contact-geometric techniques, with potential implications for the hatchet planimeter and related planar dynamics.

Abstract

We find new necessary and sufficient conditions for the bicycling monodromy of a closed plane curve to be hyperbolic. Our main tool is the ``hyperbolic development" interpretation of the bicycling monodromy of plane curves. Based on computer experiments, we pose two conjectures concerning the bicycling monodromy of strictly convex closed plane curves.

Figures (25)

• Figure 1: The front ($\Gamma$) and back ($\gamma$) bicycle tracks and the circle map $b(\Gamma)$ generated by $\Gamma$.
• Figure 2: Bicycle monodromies: (a) elliptic; (b) and (c) hyperbolic.
• Figure 3: bicycling monodromy types of rectangles of size $a\times b$
• Figure 4: Monodromy types for ellipses with semi-axes $a,b$.
• Figure 5: Hyperbolic development with a corner.

Theorems & Definitions (37)

• Theorem 1: R. Foote F
• Theorem 2
• Theorem 3
• Remark 1.1
• Theorem 4
• Proposition 2.1
• proof
• Corollary 2.2
• Proposition 3.1
• Definition 3.2