Ergodic properties of rational mappings with large topological degree. (English) Zbl 1088.37020

Guedj, Vincent

Ann. Math. (2) 161, No. 3, 1589-1607 (2005).

Let \(X\) be a projective algebraic manifold of dimension \(k\) and \(\omega\) a Hodge form on \(X\) normalized so that \(\int_X\omega^k=1\). Let \(f:X\rightarrow X\) be a rational mapping such that its Jacobian determinant does not vanish identically in any coordinate chart. Define the \(l\)th dynamical degree of \(f\) to be \[ \lambda_l(f)= \liminf_{n\to+\infty} \left(\int_X(f^n)^*\omega^l\wedge\omega^{k-l}\right)^{1/n}. \] Then \(d_t(f)=\lambda_k(f)\) is just the topological degree of \(f\).
The author proves that when \(d_t(f)>\lambda_{k-1}(f)\), there exists a mixing probability measure \(\mu_f\) such that if \(\Theta\) is any smooth probability measure on \(X\), \[ \frac{1}{d_t(f)^n}(f^n)^*\Theta\longrightarrow \mu_f, \] where the convergence holds in the weak sense of measures, and such that \(f^*\mu_f=d_t(f)\mu_f\). In particular, \(\mu_f\) is the unique measure of maximal entropy when \(k\leq 3\) or when \(X\) is complex homogeneous. Thus, the author establishes the foundation of dynamics on rational self-mappings of projective algebraic manifolds.

Reviewer: Pei-Chu Hu (Jinan)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A20	Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37B40	Topological entropy

Keywords:

rational mappings; dynamical degree; topological degree; mixing probability measure; smooth probability measure; maximal entropy; projective algebraic manifolds