Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map. Let's denote $f_{\ast} : \mathrm{NS}(X) \rightarrow \mathrm{NS}(X)$ the map induced by $f$ on the Néron-Severi group.
The dynamical degree of $f$ is defined as $$ \lambda(f) = \lim_{n \to + \infty} \vert \vert (f^n)_{\ast}\vert \vert^{\frac{1}{n}}, $$ and it is known that if $D$ is an ample class, then $\lambda(f) = \lim_{n \to + \infty} (D \cdot (f^n)_{\ast} D)^{\frac{1}{n}}$. A map $f$ as above is said to be algebraically stable if $(f^n)_{\ast} = f^n_{\ast}$ for every $n \in \mathbb{N}$; for algebraically stable maps we have $\lambda(f) = \rho(f_{\ast})$, where $\rho(f_{\ast})$ is the spectral radius of $f_{\ast} : \mathrm{NS}(X) \otimes \mathbb{R} \rightarrow \mathrm{NS}(X) \otimes \mathbb{R}$.
If $f$ is algebraically stable, the above statements imply that for an ample divisor class $D$ we have \begin{equation} \tag{1} \label{eqn} \rho(f_{\ast}) = \lim_{n \to +\infty} (D \cdot f_{\ast}^n D)^{\frac{1}{n}}. \end{equation}
My question is the following: is the relation \eqref{eqn} satisfied regardless of $f$ being algebraically stable?
It is a standard fact that $\rho(f_{\ast}) = \lim_{n \to +\infty} \vert \vert f_{\ast}^n \vert \vert^{\frac{1}{n}}$ where $\vert \vert - \vert \vert$ is the norm induced by the intersection product on $X$. However, this means we should take $$ \vert \vert f_{\ast}^n \vert \vert = \sup_{E \cdot E = 1} E \cdot f_{\ast}^n E, $$ which leaves us to prove that the supremum is a maximum and it is achieved at an ample divisor. I think one might prove this using the Hodge Index Theorem, but I wasn't able to.
Thanks for the help!
As a general reference for the definitions above you can have a look at Dynamical degree of binational transformations of projective surfaces, J. Blanc and S. Cantat
