Entropy of Rational Selfmaps of Projective Varieties

Let X ⊂ CP be an irreducible projective variety. Assume that F : X → X is a rational continuous map. Denote by h(F ) the entropy of F . In [Fri] we showed that h(F ) = logρ(F ) if X is smooth. Here ρ(F ) is the spectral radius of the induced linear map on the homology groups of X over the rationals. In the first part of this paper (§1) we show that this result is valid for any irreducible normal projective variety X . More general, h(F ) = logρ(F ) for a regular selfmap F of an irreducible projective variety X . We conjecture that the regularity assumption of F can be replaced by the continuity assumption. The second part of this paper (§2-3) deals with the case where F : X → X is a rational but not a continuous map. One can extend naturally F to the restriction of the standard shift map to the space Ω̂(F ) which is the closure of the orbit space of F [Fri]. Using this extension we define the entropy h(F ) as in [Fri]. On the other hand one can define H(F ) the volume growth of algebraic subvarieties on X . As in the case of smooth X discussed in [Fri] our arguments show that