In a previous paper ( Degli Esposti, Del Magno and Lenci 1998 An infinite step billiard Nonlinearity 11 991-1013) we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given sequence of non-negative numbers {p_n}_n∈, such that p_n↘0, there corresponds a table P: = ⋃_n∈[n,n + 1]×[0,p_n]. In this paper, first we generalize the main result of Degli Esposti et al to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the α- and ω-limit of every other trajectory. Then, following the recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities, and the entropy with respect to a wide class of measures is zero.