Hyperbolic measure of maximal entropy for generic rational maps of P^k

TL;DR

This paper establishes the existence of hyperbolic measures of maximal entropy for a broad class of rational maps on projective space, using approximation techniques and super-potentials theory.

Contribution

It introduces a method to prove hyperbolic dynamics for generic rational maps via approximation of meromorphic graphs and super-potentials, extending results to polynomial maps.

Findings

Existence of hyperbolic measure of maximal entropy for generic maps

Explicit bounds on Lyapunov exponents for these measures

Application to polynomial maps and homogeneous extensions

Abstract

Let f be a dominant rational map of P^k such that there exists s <k, with lambda_s(f)>lambda_l(f) for all l. Under mild hypotheses, we show that, for A outside a pluripolar set of the group of automorphisms of P^k, the map f o A admits a hyperbolic measure of maximal entropy log(lambda_s(f)) with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of f to P^{k+1}. This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate the graph of a meromorphic function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.