5.2. Dynamical capacity estimates [034B]

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5.2. Dynamical capacity estimates

Let $f:\mathbb{C}\mathbb{P}^{n}\rightarrow\mathbb{C}\mathbb{P}^{n}$ be an holomorphic endomorphism. We let $\omega$ denote again the Fubini-Study Kähler form. Then $f^{*}\omega$ is a smooth positive closed $(1,1)$ -form of mass $\lambda=\int_{\mathbb{C}\mathbb{P}^{n}}f^{*}\omega\wedge\omega^{n-1}=:$ the first algebraic degree of $f$ . Thus $\lambda^{-1}f^{*}\omega=\omega+dd^{c}\varphi$ , where $\varphi$ is a smooth $\omega$ -psh function on $\mathbb{C}\mathbb{P}^{n}$ . Iterating this functional equation yields

\frac{1}{\lambda^{j}}(f^{j})^{*}\omega=\omega+dd^{c}g_{j},\;g_{j}=\sum_{l=0}^{j-1}\frac{1}{\lambda^{l}}\varphi\circ f^{l}.

We assume $\lambda\geq 2$ . Thus the sequence $(g_{j})$ uniformly converges on $\mathbb{C}\mathbb{P}^{n}$ towards a continuous function $g_{f}\in PSH(X,\omega)$ called the Green function of $f$ . We refer the interested reader to [33] for a detailed study of the properties of the Green current $T_{f}=\omega+dd^{c}g_{f}$ .

Dynamical volume estimates have revealed quite useful in establishing ergodic properties of the Green current $T_{f}$ (see [20], [22] and references therein). We establish herebelow very simple dynamical capacity estimates and show how to derive from them dynamical volume estimates.

Proposition 5.3.

There exists $0<\alpha<1$ such that for all Borel subsets $K$ of $X$ , for all $j\in\mathbb{N}$ ,

\left[\alpha T_{\omega}(K)\right]^{\lambda^{j}}\leq T_{\omega}(f^{j}(K)).

Proof.

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. ∎

Corollary 5.4.

Let $\varphi\in PSH(X,\omega)$ . Then the sequence $(\lambda^{-j}\varphi\circ f^{j})$ is relatively compact in $L^{1}(\mathbb{C}\mathbb{P}^{n})$ .

Proof.

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. ∎

Corollary 5.5.

Fix $\delta_{0}>0$ . There exists $\gamma_{0}\in]0,1[$ such that for all Borel subset $K$ of $\mathbb{C}\mathbb{P}^{n}$ with $Vol_{\omega}(K)\geq\delta_{0}$ , one has

Vol_{\omega}(f^{j}(K))\geq\gamma_{0}^{\lambda^{j}},\;\;\forall j\in\mathbb{N}.

In other words the volume of a given set can not decrease too fast under iteration. Such volume estimates are used in complex dynamics to prove fine convergence results towards the Green current $T_{f}$ (see [20], [22]). One may hope that dynamical capacity estimates will allow to establish convergence results in higher codimension.

Proof.

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. ∎