Proposition 17.10.86B40B93A4363D

Proof. For each $t>0$, let $(\ell ''^ t_{s})_{s\in [0,t]}$ be the geodesic from $\phi $ to $\ell _ t\lor \ell '_ t$.

Step 1. We first show that for each fixed $s\geq 0$, $\ell ''^ t_{s}$ is increasing in $t\in [s,\infty )$.

To see this, fix $s\geq 0$ and choose $t'>t\geq s$. We need to show that

Since $(\ell ''^{t'}_ a)_{a\in [0,t]}$ is a geodesic, it suffices to show that $(\ell ''^ t_ a)_{a\in [0,t]}$ is a candidate in the Perron envelope defining the former geodesic. In other words, in verifying (17.10.88), we may assume that either $s=0$ or $s=t$. The case $s=0$ is of course trivial. So it remains to prove the following:

By symmetry, it suffices to prove

But since $(\ell _ a)_{a\in [0,t']}$ is a candidate in the Perron envelope defining $\ell ''^{t'}$, this inequality follows.

Step 2. Next, observe that for a fixed $s\geq 0$, we have

for all $t>s$. The right-hand side is bounded from above by a constant independent of $t\geq s$ by proposition 17.10.39. Let

Then proposition 17.10.77 guarantees that $\ell \lor \ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$.

Step 3. We need to show that $\ell \lor \ell '$ defined in this way is indeed the minimal ray lying above $\ell $ and $\ell '$.

First, by Step 1, we have

for any $t\geq s\geq 0$. Therefore,

for all $s\geq 0$. In other words, $\ell \lor \ell '\geq \ell $. Similarly, $\ell \lor \ell '\geq \ell '$.

Next, let $L\in \mathcal{R}^{\infty }(X,\theta ;\phi )$ be a ray lying above both $\ell $ and $\ell '$. Then we have

for all $t\geq 0$. In particular,

for all $t\geq s\geq 0$. It follows that

for all $s\geq 0$.

Step 4. It remains to argue (17.10.87). Since the energy is linear along bounded rays,

Moreover, $\ell ''^ t_1$ increases to $(\ell \lor \ell ')_1$ as $t\to \infty $. By proposition 16.5.37,

Finally, by proposition 17.10.65 applied to $(\ell ''^ t_ s)_{s\in [0,t]}$,

Combining these identities gives (17.10.87). $\square$