Lemma 17.13.119C6538C4298E

Proof. It follows from 17.13.7, (17.13.2) and proposition 5.14.2 that the restriction of $\Gamma ^*$ to each compact subinterval of $(0,\infty )$ is a subgeodesic. By corollary 17.13.9, for any $t>0$, $\Gamma ^*_ t\in \mathcal{E}(X,\theta ;\phi )$.

First observe that as $t\to 0+$, we have

By 17.13.10 and proposition 5.17.3, it suffices to show each $L^1$-cluster point $\psi \in \mathrm{PSH}(X,\theta )$ of $\Gamma _ t^*$ as $t\to 0$ is equal to $\phi $.

To see this, first observe that by (17.13.2), for any fixed $t>0$,

Therefore, $\psi \leq \phi $. On the other hand, for any fixed $\tau <\Gamma _{\max }$, by (17.13.2), we have

for any $t>0$. So $\psi \geq \Gamma _{\tau }$ almost everywhere and hence everywhere by proposition 5.14.6. It follows that $\psi \geq \phi $. Therefore, $\psi =\phi $.

Assume that $\Gamma ^*$ is not a geodesic ray. Then we can find $0\leq a<b$ such that $(\Gamma ^*_ t)_{t\in (a,b)}$ differs from the geodesic $(\eta _ t)_{t\in (a,b)}$ from $\Gamma ^*_ a$ to $\Gamma ^*_ b$. The existence of $(\eta _ t)_ t$ is guaranteed by proposition 17.10.24. We consider the subgeodesic $(\ell _ t)_{t>0}$ given by $\ell _ t=\eta _ t$ for $t\in (a,b)$ and $\ell _ t=\Gamma ^*_ t$ otherwise. Note that $\ell $ is a subgeodesic due to the gluing lemma 17.10.16.

Consider the Legendre transform

Then $\Gamma '_{\tau }\geq \Gamma _{\tau }$ and $\Gamma '_{\tau }\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $ by proposition 5.14.15 for all $\tau \in \mathbb {R}$.

We claim that

Observe that $\Gamma '_{\tau }\equiv -\infty $ when $\tau >\Gamma _{\max }$ by 17.13.10. Thus it suffices to consider $\tau \leq \Gamma _{\max }$. Put

For any $s\in [a,b]$, the monotonicity of $t\mapsto \Gamma _ t^*-t\Gamma _{\max }$ from 17.13.10 gives

Since $\ell _ b=\Gamma _ b^*$, we get

For $s\notin [a,b]$, we have $\ell _ s=\Gamma _ s^*$, hence

Taking the infimum over all $s>0$ gives

Therefore, (17.13.13) follows. In particular, for any $\tau <\Gamma _{\max }$, we have $\Gamma '_{\tau }\sim \Gamma _{\tau }$. Moreover, $\sup _ X\ell _ t\leq t\Gamma _{\max }$ for all $t>0$: this is clear outside $(a,b)$ from 17.13.10, and on $(a,b)$ it follows from the convexity of subgeodesics applied to the geodesic $(\eta _ t)_{t\in (a,b)}$. Hence, for $\tau <\Gamma _{\max }$,

Letting $t\to 0+$ gives $\Gamma '_{\tau }\leq 0$. It follows from the fact that $\Gamma _{\tau }$ is a model potential that $\Gamma _{\tau }=\Gamma '_{\tau }$ for all $\tau <\Gamma _{\max }$. Therefore, by theorem 21.11.24, we have $\Gamma ^*_ t=\ell _ t$ for all $t>0$, which is a contradiction. $\square$