This paper considers the one-dimensional advection and diffusion of a passive scalar in the context of baker's maps of the unit interval. Our main interest is the thermal transport between two points held at fixed temperatures, when a deterministic sequence of maps of various scales are involved. Molecular diffusion occurs during the periods of rest between maps. We focus on the behavior of the transport in the limit of infinite P6clet number (or small molecular diffusion). Various asymptotic results are presented and compared with numerical calculations. Convergence to turbulent transport independent of molecular diffusion is observed as the number of scales is increased.