PomeDefinition 17.12.10.69529F41AA18 Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential.
A test curve $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$ is said to be bounded if for $\tau $ small enough, $\Gamma _{\tau }=\phi $. The subset of bounded test curves in $\mathrm{TC}(X,\theta ;\phi )$ is denoted by $\mathrm{TC}^{\infty }(X,\theta ;\phi )$. In this case, we write
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\begin{equation} \Gamma _{\min }\coloneqq \sup \{ \tau \in \mathbb {R}:\Gamma _{\tau }=\phi \} . \end{equation}
A test curve $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$ is said to have finite energy if
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\begin{equation} \mathbf{E}^{\phi }(\Gamma )\coloneqq \Gamma _{\max } \int _ X \theta _{\phi }^ n+\int _{-\infty }^{\Gamma _{\max }} \left(\int _ X\theta _{\Gamma _{\tau }}^ n-\int _ X \theta _{\phi }^ n\right)\, \mathrm{d}\tau >-\infty . \end{equation}
When $\phi =V_{\theta }$, we write $\mathbf{E}$ instead of $\mathbf{E}^{\phi }$.
The subset of test curves with finite energy in $\mathrm{TC}(X,\theta ;\phi )$ is denoted by $\mathrm{TC}^1(X,\theta ;\phi )$.
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