We try to find chaotic dynamical systems by extending some simple continuous functions defined on ℚundefined, to functions defined on Aℤ, where A = {0,1,2, , p — 1}. A combination of the powers of the shift map with addition of a constant in ℚundefined, gives rise to chaotic dynamical systems, conjugate to powers of the shift map. By extending the addition in ℤundefined, to the whole of Aℤ, and combining with powers of the shift map, we get an expansive map with the same topological entropy as that of powers of the shift. We also obtain two more positively expansive maps with positive entropy.