Invariant currents and dynamical Lelong numbers

Let $f$ be a polynomial automorphism of ${\mathbb{C}}^k$ of degree $\lambda$, whose rational extension to ${\mathbb{P}}^k$ maps the hyperplane at infinity to a single point. Given any positive closed current $S$ on ${\mathbb{P}}^k$ of bidegree (1,1), we show that the sequence ${\lambda }^{-n}(f^n{)}^{\star }S$ converges in the sense of currents on ${\mathbb{P}}^k$ to a linear combination of the Green current $T_{+}$ of $f$ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for $f^{-1}$.