Integrable difference analogue of the logistic equation and Bäcklund transformation of the KP hierarchy

A difference analogue of the logistic equation, which preserves integrability, is derived from Hirotaʹs bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the Bäcklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and nonintegrable systems on an equal basis. In particular we study the behaviour of Julia set at the point where the nonintegrable map passes to the integrable map.