J.-E. Chang's conjectures on the period-energy map

For a parameter $\lambda\in\mathbb{R}$, let $V_\lambda^0$ denote the lower endpoint of the relevant energy interval, let $E$ be the energy, and let $\Theta_\lambda(E)$ be the corresponding period-energy map.

J.-E. Chang's conjectures. If $\lambda\geq 0$, then $E\mapsto\Theta_\lambda(E)$ is strictly monotonically decreasing. If $\lambda<0$, there is a value $V_\lambda^\ast>V_\lambda^0$ such that $E\mapsto\Theta_\lambda(E)$ is decreasing on $(V_\lambda^0,V_\lambda^\ast)$ and increasing on $(V_\lambda^\ast,\infty)$. Furthermore,

$$\lim_{\lambda\nearrow0}V^\ast_\lambda=\infty\qquad\text{and}\qquad\lim_{\lambda\to-\infty}(V^\ast_\lambda-V^0_\lambda)=0.$$

Monotonicity of these period-energy maps is central to classifying the associated shrinkers. The paper reports strong numerical evidence for the $\lambda\geq0$ case, but does not establish the conjectures; monotonicity is known for small energies by a classical-mechanics criterion.