Linearisable mappings, revisited. Issue 4 (1st October 2016) - Grammaticos, B.; Ramani, A.; Satsuma, J.
Abstract : We examine the growth properties of second-order mappings which are integrable by linearisation and which generically exhibit a linear growth of the homogeneous degree of initial conditions. We show that for Gambier-type mappings for which the growth proceeds generically with a step of 1 there exist cases where the degree increase by unity every two steps. We examine also mappings belonging to the family known as "of third kind" in relation to the approach of Diller and Favre concerning the regularisable or not character of mappings and show that the anticonfined singularities of these mappings exhibit a linear growth with step 1. (The term anticonfined is used for singularities where the singular values extend all the way to infinity on both sides with just a few regular values in the middle). Moreover we construct specific examples of Gambier-type mappings which have anticonfined singularities and where the degree of the singularity increases linearly but where the average slope can be adjusted so as to be arbitrarily small.