Integrable lattice equations and their growth properties

In this Letter we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the d egree of the iterates. We show that while for generic, nonintegrable equati ons, the degree grows exponentially fast, fur integrable lattice Equations the degree growth is polynomial. The growth criterion is used in order to o btain the integrable deautonomisations of the equations examined. In the ca se of linearisable lattice equations we show that the degree growth is slow er than in the case of equations integrable through inverse scattering tran sform techniques. (C) 2001 Elsevier Science B.V. All rights reserved.