Hyperbolic measure of maximal entropy for generic rational maps of P^k
  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. 
Gabriel Vigny
math.CV math.DS