Topological entropy and Arnold complexity for two-dimensional mappings

To test a possible relation between topological entropy and Arnold complexity, and to provide a nontrivial examples of rational dynamical zeta functions, we introduce a two-parameter family of discrete birational mappings of two complex variables;We conjecture rational expressions with integer coefficients for the number of fixed points and degree generating functions. We then deduce equal algebraic values for the complexity growth and for the exponential of the topological entropy. We also explain a semi-numerical method which supports these conjectures and localizes the integrable cases. We briefly discuss the adaptation of these results to the analysis of the same birational mapping seen as a mapping of two real variables. (C) 1999 Elsevier Science B.V. All rights reserved.