The measure $c_{1}(\overline{L})^{n}$ on $\mathrm{X}$ defined by the metrized line bundle $\overline{L}$ is a very important invariant of the dynamical system. It satisfies the functional equations

$$f^{*}c_{1}(\overline{L})^{n}=d^{n}c_{1}(\overline{L})^{n}\quad\text{and}\quad f_{*}c_{1}(\overline{L})^{n}=c_{1}(\overline{L})^{n}.$$

The first follows by a general functorial property proved in [18] ; it implies the second. The support of the canonical measure is therefore totally invariant under $f$.