对于二维环面抛物型映射，给出部分可逆环面抛物型映射的同构分类,证明了极限圆映射有稠密的周期点集,且某些有理抛物型映射具有任意周期的周期点.对于整数抛物型映射,证明了其拓扑熵为零.通过比较极限圆映射分别在环面拓扑和平面拓扑下的符号熵、复杂度,展现了同一个映射在不同拓扑下量的差异. 

This paper discusses isomorphism between invertible torus parabolic maps and periodicity where periodic point sets in horocyclic case are dense in the torus topology, and some semi-rational cases possess periodic points of all periods. For the integral parabolic maps on the torus, the topological entropy is zero. Symbolic entropy and complexity of planar piecewise parabolic maps are investigated. Their difference under the plane topology and the torus topology respectively for the same torus parabolic maps is also discussed.