Entropy of meromorphic maps acting on analytic sets

2018/07/17 by De Thélin, Henry, Vigny, Gabriel
#32Bxx #37B40 #37F10 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.06483

Abstract

Let f : X→ X be a dominating meromorphic map on a compact Kähler manifold X of dimension k. We extend the notion of topological entropy h^l_top(f) for the action of f on (local) analytic sets of dimension 0≤ l ≤ k. For an ergodic probability measure ν, we extend similarly the notion of measure-theoretic entropy h_ν^l(f). Under mild hypothesis, we compute h^l_top(f) in term of the dynamical degrees of f. In the particular case of endomorphisms of ℙ² of degree d, we show that h¹_top(f)= log d for a large class of maps but we give examples where h¹_top(f)≠ log d.

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