Set $M_{K}=\sup_{X}V_{K,\omega}$. If $M_{K}=+\infty$ then $K$ is $PSH(X,\omega)$-polar (theorem 3.2) and there is nothing to prove: $T_{\omega}(K)=Cap_{\omega}(K)=0$. So we assume in the sequel $M_{K}<+\infty$ hence $V_{K,\omega}^{*}\in PSH(X,\omega)$. If $M_{K}\geq 1$ then $u_{K}:=M_{K}^{-1}V_{K,\omega}^{*}\in PSH(X,\omega)$ with $0\leq u_{K}\leq 1$ on $X$. Since $\omega_{V_{K,\omega}^{*}}\leq M_{K}\omega_{u_{K}}$, we get

$$\frac{1}{M_{K}^{n}}=\frac{1}{M_{K}^{n}}\int_{K}(\omega_{V_{K,\omega}^{*}})^{n}\leq\int_{K}(\omega_{u_{K}})^{n}\leq Cap_{\omega}(K)$$

whence $T_{\omega}(K)\leq\exp(-Cap_{\omega}(K)^{-1/n})$.

If $0\leq M_{K}\leq 1$ then $0\leq V_{K,\omega}^{*}\leq 1$ hence

$$1=\int_{K}(\omega_{V_{K,\omega}^{*}})^{n}\leq Cap_{\omega}(K)\leq Cap_{\omega}(X)=1.$$

∎

More precisely we are going to show the following: consider $\Omega$ an open subset of $X$, $v\in PSH^{-}(\Omega)$ and $P\subset\{v=-\infty\}$. Fix $0<\varepsilon<1/n$ and $V_{t}:=V_{G_{t},\omega}$ where $G_{t}=\{x\in\Omega\,/\,v(x)<-t\}$. Then

$$\varphi_{\varepsilon}(x):=\frac{1}{\varepsilon}\int_{1}^{+\infty}\frac{1}{t^{1+\varepsilon}}[V_{t}(x)-\sup_{X}V_{t}]dt$$

is a $\omega$-psh function such that $P\subset\{\varphi_{\varepsilon}=-\infty\}$.

Indeed since $G_{t}$ is open, we have $V_{t}\in PSH(X,\omega)$ and $V_{t}=0$ on $G_{t}$ (see proposition 3.6). Observe that $\varphi_{\varepsilon}$ is a sum of negative $\omega$-psh functions hence it is either identically $-\infty$ or a well defined $A\omega$-psh function with $A=\varepsilon^{-1}\int_{1}^{+\infty}t^{-(1+\varepsilon)}dt=1$. Recall that $-C+\sup_{X}V_{t}\leq\int_{X}V_{t}\omega^{n}\leq\sup_{X}V_{t}$ (proposition 1.7). Therefore $\int_{X}\varphi_{\varepsilon}\omega^{n}\geq-C$ hence $\varphi_{\varepsilon}\in PSH(X,\omega)$.

Fix $x\in\Omega$ such that $v(x)<-1$. Observe that $V_{t}-\sup_{X}V_{t}\leq 0$ with $V_{t}(x)=0$ if $x\in G_{t}$, i.e. when $|v(x)|>t$. Therefore

$$\varphi_{\varepsilon}(x)\leq-\frac{1}{\varepsilon}\int_{1}^{|v(x)|}\frac{\sup_{X}V_{t}}{t^{1+\varepsilon}}dt.$$

Recall now that $Cap_{\omega}$ is always dominated by $Cap_{BT}$ hence $Cap_{\omega}(G_{t})\leq C_{1}/t<1$ if $t$ is large enough. We infer from the previous proposition that

$$-\sup_{X}V_{t}\leq-[Cap_{\omega}(G_{t})]^{-1/n}\leq-C_{2}t^{1/n},$$

which yields

$$\varphi_{\varepsilon}(x)\leq\frac{-C_{2}}{\varepsilon}\int_{1}^{|v(x)|}\frac{dt}{t^{1+\varepsilon-1/n}}\leq-C_{3}|v(x)|^{1/n-\varepsilon}+C_{4}.$$

Note that $\varphi_{\varepsilon}(x)=-\infty$ whenever $v(x)=-\infty$ hence $P\subset\{\varphi_{\varepsilon}=-\infty\}$. ∎

This follows straightforwardly from proposition 3.9:

$$T_{\omega}(f^{j}(K))\geq T_{(f^{j})^{*}\omega}(K)=\left[T_{\lambda^{-j}(f^{j})^{*}\omega}(K)\right]^{\lambda^{j}}\geq\left[\alpha T_{\omega}(K)\right]^{\lambda^{j}},$$

where the first two inequalities follow from 3.9.4 and 3.9.2 and last one follows from 3.9.3 and the fact that $\lambda^{-j}(f^{j})^{*}\omega=\omega+dd^{c}g_{j}$, where $g_{j}$ is uniformly bounded. ∎

Set $\varphi_{j}=\lambda^{-j}\varphi\circ f^{j}$. Observe that $\varphi_{j}$ is uniformly bounded from above and that $\varphi_{j}+g_{j}\in PSH(\mathbb{C}\mathbb{P}^{n},\omega)$. It follows from proposition 1.6 that either $\varphi_{j}$ converges uniformly towards $-\infty$ or it is relatively compact in $L^{1}(\mathbb{C}\mathbb{P}^{n})$. It is sufficient to show that for $A>0$ large enough, $\overline{\lim}_{j\rightarrow+\infty}T_{\omega}(\varphi_{j}<-A)<T_{\omega}(X)=1$. Observe that $f^{j}(\varphi_{j}<-A)=\{\varphi<-A\lambda^{j}\}$. Therefore

$$\left[\alpha T_{\omega}(\varphi_{j}<-A)\right]^{\lambda^{j}}\leq T_{\omega}(\varphi<-A\lambda^{j})\leq C\exp(-A\lambda^{j}),$$

where the last inequality follows from proposition 3.8. We infer

$$\overline{\lim}_{j\rightarrow+\infty}T_{\omega}(\varphi_{j}<-A)\leq\frac{1}{\alpha}\exp(-A)<1$$

for $A>-\log\alpha$ large enough. ∎

By the change of variables formula one gets

$$Vol_{\omega}(f^{j}K)=\int_{f^{j}K}\omega^{n}\geq\frac{1}{d_{t}^{j}}\int_{K}(f^{j})^{*}\omega^{n}=\frac{1}{d_{t}^{j}}\int_{K}|J_{FS}(f^{j})|^{2}\omega^{n},$$

where $d_{t}=\lambda^{n}$ denotes the topological degree of $f$ and $J_{FS}(f)$ stands for the jacobian of $f$ with respect to the Fubini-Study volume form. Observe that $\log|J_{FS}(f)|=u-v$ is a difference of two qpsh functions $u,v\in PSH(X,A\omega)$ for some $A=A(\lambda,n)$. Moreover by the chain rule,

$$\frac{1}{\lambda^{j}}\log|J_{FS}(f^{j})|=\sum_{l=0}^{j-1}\frac{1}{\lambda^{j}}\log|J_{FS}(f)\circ f^{l}|.$$

Since $\lambda^{-l}\log|J_{FS}(f)\circ f^{l}|$ is relatively compact in $L^{1}(\mathbb{C}\mathbb{P}^{n})$ (previous corollary), the concavity of the log yields

$$\frac{1}{Vol_{\omega}(K)}\int_{K}|J_{FS}(f^{j})|^{2}\omega^{n}\geq\exp\left(\frac{2\lambda^{j}}{Vol_{\omega}(K)}\int_{K}\frac{1}{\lambda^{j}}\log|J_{FS}(f^{j})|\omega^{n}\right)\geq\gamma^{\lambda^{j}}.$$

Decreasing slightly the value of $\gamma$ if necessary, this yields the desired inequality. ∎