Definition 17.10.382E335019C3A7

Definition 17.10.38.2E335019C3A7 Let $\ell =(\ell _ t)_{t\geq 0}$ be a family in $\mathcal{E}(X,\theta ;\phi )$. We say $\ell $ is a geodesic ray in $\mathcal{E}(X,\theta ;\phi )$ emanating from $\ell _0$ if for each $0\leq a\leq b$, the restriction $(\ell _ t)_{t\in [a,b]}$ is a geodesic.

The set of geodesic rays in $\mathcal{E}(X,\theta ;\phi )$ emanating from $\phi $ is denoted by $\mathcal{R}(X,\theta ;\phi )$.

We say a geodesic ray $\ell \in \mathcal{R}(X,\theta ;\phi )$ has finite energy if $\ell _ t\in \mathcal{E}^1(X,\theta ;\phi )$ for all $t>0$. The set of geodesic rays with finite energy is denoted by $\mathcal{R}^1(X,\theta ;\phi )$.

We say a geodesic ray $\ell \in \mathcal{R}(X,\theta ;\phi )$ is bounded if $\ell _ t\in \mathcal{E}^{\infty }(X,\theta ;\phi )$ for all $t\geq 0$. The set of bounded geodesic rays is denoted by $\mathcal{R}^{\infty }(X,\theta ;\phi )$.

Given $\ell ,\ell '\in \mathcal{R}(X,\theta ;\phi )$, we write $\ell \leq \ell '$ if $\ell _ t\leq \ell '_ t$ for each $t\geq 0$.

When $\phi =V_{\theta }$, we usually omit it from the notations and write $\mathcal{R}(X,\theta )$, $\mathcal{R}^1(X,\theta )$ and $\mathcal{R}^{\infty }(X,\theta )$ respectively.