Algebraic entropy criterion for integrability of birational maps

A birational map is assumed to admit an algebraic entropy, measuring the asymptotic growth of the degrees of its iterates. Algebraic entropy conjecture. The condition that algebraic entropy is zero is equivalent to the definition of integrability for birational maps. This is asserted here for the integrable Volterra-like five-point differential-difference equations considered in the paper; the paper reports that the algebraic-entropy and generalized-symmetry tests agree whenever the algebraic entropy method applies.