Title: Topology of Attractors from Two-Piece Expanding Maps

Abstract: In this paper we study the topology of the invariant set $L_{f}$ derived from an expanding map with one discontinuity $f$. We classify the periodic orbits on the boundary of $L_{f}$ that enable $L_{f}$ to have a trapping region and that do not. We also show that when there are two eventually periodic orbits on the boundary of $L_{f}$, the components of $L_{f}$ form a Markov partition, and form its transition matrix we can calculate the precise entropy of the map. Furthermore, we show that the set of attractors derived from two piece expanding maps is open and dense in the set of all invariant sets derived by the maps. ision orbits.