  We propose a conjecture for the exact expression of the dynamical zeta
function for a family of birational transformations of two variables, depending
on two parameters. This conjectured function is a simple rational expression
with integer coefficients. This yields an algebraic value for the topological
entropy. Furthermore the generating function for the Arnold complexity is also
conjectured to be a rational expression with integer coefficients with the same
singularities as for the dynamical zeta function. This leads, at least in this
example, to an equality between the Arnold complexity and the exponential of
the topological entropy. We also give a semi-numerical method to effectively
compute the Arnold complexity.
