Definition 17.17.3ECDB430D0021

Definition 17.17.3.ECDB430D0021 An envelope ray with initial potential $\varphi $ and slope $c$ is a subgeodesic $u_ t$, $t>0$, such that

1. $u_ t\to \varphi $ in $L^1(X)$ as $t\downarrow 0$, and, with $u_0=\varphi $, the restriction of the ray to every interval $(a,b)$ with $0\leq a<b<\infty $ is the Perron segment from $u_ a$ to $u_ b$;

2. $P_\theta [u_ t]=\varphi $ for every $t>0$;

3. $\sup _ Xu_ t=ct$ for every $t>0$.