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

2011/12/02 by Gabriel Vigny, Vigny, Gabriel
Mathematics · Physics and Astronomy · #32H04 #32Uxx #37Fxx #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.CV #math.DS #msc:32H04 #msc:32Uxx #msc:37Fxx

paper · pdf · doi:10.48550/arxiv.1112.0501

34 pages, to appear in Ann. Inst. Fourier

openalex publication_date 2011/12/02 · arxiv created 2014/04/09 · arxiv updated 2014/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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