Diffusion coefficient of piecewise linear maps

By using the cycle expansion, we obtain general expressions for the determination of the diffusion coefficient D of a piecewise linear map which is parametrized by k and h̃ (where the map contains 2k+5 branches of line segment, and h̃ is the height of the shortest line). By restricting h̃=β/m [β=1,...,(k+1)/2; m is the slope of the map], a closed form expression of D can be obtained and some of its consequences are discussed. The limiting form of D (k→∞) is then shown to be ${\mathit{k}}^2$. For the simplest case with k=1, we also show that more exact results can be found. A limiting case with h̃→0 is discussed where agreement with the result obtained from the invariant measure approach is established.