  To test a possible relation between the topological entropy and the Arnold
complexity, and to provide a non trivial example of a rational dynamical zeta
function, we introduce a two-parameter family of two-dimensional discrete
rational mappings. The generating functions of the number of fixed points, and
of the degree of the successive iterates, are both considered. We conjecture
rational expressions with integer coefficients for these two generating
functions. and a rational expression for the dynamical zeta function. We then
deduce algebraic values for the complexity growth and for the exponential of
the topological entropy. Moreover, these two numbers happen to be equal for all
the values of the parameters. These conjectures are supported by a
semi-numerical method we explain. This method also localizes the integrable
cases.
